L-statistics for Repeated Measurements Data With Application to Trimmed Means, Quantiles and Tolerance Intervals

Abstract

The L-statistics form an important class of estimators in nonparametric statistics. Its members include trimmed means and sample quantiles and functions thereof. This article is devoted to theory and applications of L-statistics for repeated measurements data, wherein the measurements on the same subject are dependent and the measurements from different subjects are independent. This article has three main goals: (a) Show that the L-statistics are asymptotically normal for repeated measurements data. (b) Present three statistical applications of this result, namely, location estimation using trimmed means, quantile estimation and construction of tolerance intervals. (c) Obtain a Bahadur representation for sample quantiles. These results are generalizations of similar results for independently and identically distributed data. The practical usefulness of these results is illustrated by analyzing a real data set involving measurement of systolic blood pressure. The properties of the proposed point and interval estimators are examined via simulation.

1 Preliminaries

The class of linear functions of order statistics, the so-called L-statistics, plays a significant role in non-parametric statistics. Two prominent members of this class are sample quantiles and trimmed means. The sample quantiles are used for nonparametric estimation of population quantiles and their functions such as the inter-quartile range (David and Nagaraja 2003). They are also used for construction of nonparametric tolerance intervals for a population that are often sought in engineering, manufacturing and medicine (Krishnamoorthy and Mathew 2009). The trimmed means provide a robust alternative to sample mean for estimating a location parameter (Wilcox 2012). There is extensive literature on L-statistics for independently and identically distributed (i.i.d.) data — see, e.g., Serfling (1980), Huber (1981) and Fernholz (1983) for reviews. In particular, it is well-known that L-statistics are asymptotically normal for i.i.d. data. This article is concerned with generalizing this result to repeated measurements data and applying it to nonparametric estimation of trimmed means and quantiles, and construction of nonparametric tolerance intervals.

Let X_ij, j = 1, …, k_i, denote the k_i repeated measurements on the ith subject in the study, i = 1, …, n. The subjects are assumed to be independent. The design of the study need not be balanced, i.e., the k_i may not be equal. Let $N={\sum }_{i=1}^n{k}_i$ denote the total number of observations. We also assume that:

• A.1The X_ij are identically distributed as a continuous random variable X with cumulative distribution function (c.d.f.) F, probability density f and finite variance.
• A.2The vectors (X_i1, …, X_iki), i = 1, …, n, are independent, and ∃ an exchangeable sequence X̃₁, X̃₂, … such that $(X_{i1},\dots ,X_{i{k}_i})\stackrel{d}{=}({\tilde{X}}_1,\dots ,{\tilde{X}}_{k_i})$ for each i. Thus, in particular, X̃₁ and X̃₂ represent two repeated measurements on a randomly selected subject from the population. Let G be the bivariate c.d.f. of (X̃₁, X̃₂). Due to the exchangeability assumption, both the marginals of G are equal to F.

The distinctive feature of the data X_ij is that the repeated measurements on a subject are replications of the same underlying quantity. In other words, the true underlying measurement for a subject does not change during the replication process. Therefore, the measurements on the same subject are dependent. On the other hand, the measurements from different subjects are independent. This kind of repeated measurements data are common in a variety of applications, including clinical studies concerned with estimation of reliability (Fleiss 1986; Dunn 1989), gauge repeatability and reliability studies (Burdick et al. 2005) and method comparison studies (Bland and Altman 1999). These data are typically analyzed by modeling them using a one-way random-effects model (or more generally a mixed-effects model) that treats the effect of subject as a random effect and assumes normal distributions for random effects and errors. The parameters of the model are estimated by a likelihood-based method and the asymptotic theory of maximum likelihood estimators (MLEs) is used for inference (Pinheiro and Bates 2000). Specifically, for inference on quantiles and construction of tolerance intervals, the methodology described in Krishnamoorthy and Mathew (2009, chap. 4) can be used (see also Sharma and Mathew (2012)). However, the MLEs are well-known to be non-robust against the violation of the normality assumption. This violation occurs frequently in practice — see Section 7 for a real example involving measurement of systolic blood pressure that motivated this work.

When the normality assumption is not reasonable, an alternative is to use a nonparametric method to analyze the data. Olsson and Rootzen (1996) consider nonparametric estimation of quantiles from repeated measurements. Their method can deal with unbalanced as well as balanced designs. Hutson (2003) considers nonparametric estimation of normal range — a quantile interval — using repeated measurements from a balanced design. These authors show that it is not a good idea to apply the estimation methods designed for i.i.d. data to univariate summaries of within-subject repeated measurements (e.g., averages) because it may lead to substantial loss of efficiency. The authors such as Wilcox (1994), Wilcox et al. (2000) and Keselman et al. (2000) use trimmed means in repeated measures designs in place of the usual means to get robust tests of hypotheses on treatment effects in an analysis-of-variance setting. Although the estimators studied in each of these articles are special cases of L-statistics, their authors are not concerned with studying the general class of L-statistics, as we do in this article. A study of general L-statistics allows us to present a unified treatment of the separate estimators. This unified approach additionally provides a method for constructing nonparametric tolerance intervals with repeated measurements data (see Section 5).

To study general L-statistics for repeated measurements data, let X₍₁₎ ≤ X₍₂₎ ≤ … ≤ X_(N) be the order statistics associated with the N observations X_ij, j = 1, …, k_i, i = 1, …, n. We estimate the population c.d.f. F(x) by a weighted empirical c.d.f.,

$$F_n(x)=\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}I(X_{ij}\le x),$$ (1)

where w_i = w(k_i, n), 0 < w_i < 1, is the known weight of the observation X_ij and I(A) is the indicator of event A. The weights depend on subject i only through k_i — the number of repeated measurements on the subject. All observations on a given subject receive the same weight because they are exchangeable from assumption A.2. The weights are assumed to satisfy ${\sum }_{i=1}^n{k}_i{w}_i=1$, so that F_n(x) is an unbiased estimator of F(x).

The weights in F_n may be arbitrary provided they satisfy the additional assumptions A.5 and A.6 in Section 2. In particular, these assumptions hold for the two weight functions,

$$w_{i,1}=\frac{1}{n{k}_i}\text{ and }{w}_{i,2}=\frac{1}{N},i=1,\dots ,n,$$ (2)

which are of special interest due to their simplicity. The first one assigns a total of 1/n weight to each subject and distributes it equally among the repeated measurements on this subject, whereas the second one assigns equal weight to each observation in the data.

It can be seen that F_n(x) is the minimum variance unbiased estimator of F(x) if the weights are

$$\frac{{\{1+(k_i-1)\rho (x,x)\}}^{-1}}{{\sum }_{l=1}^n{k}_l{\{1+(k_l-1)\rho (x,x)\}}^{-1}},$$ (3)

where

$$\rho (x,y)=\text{corr}[I({\tilde{X}}_1\le x),I({\tilde{X}}_2\le y)]=\frac{G(x,y)-F(x)F(y)}{{[F(x)\{1-F(x)\}F(y)\{1-F(y)\}]}^{1/2}}.$$ (4)

Olsson and Rootzen (1996) refer to (3) as the “optimal weight function.” The two weight functions in (2) are its special cases obtained by taking ρ(x, x) = 1 and ρ(x, x) = 0, respectively. All these weight functions are identical for balanced designs. We do not use the optimal weight function in this article as the resulting F_n(x) is not a non-decreasing function of x and the unknown ρ(x, x) needs to be replaced with an estimate. These issues cause additional complications for the theory, but the optimal weights generally do not lead to significant gains in efficiency over the simpler weights in (2), especially w_i,1 (see, e.g., Olsson and Rootzen (1996)).

Next, for a given 0 < p < 1, let $F_n^{-1}(p)=\text{inf}\{x:F_n(x)\ge p\}$ denote the plug-in estimator of F⁻¹(p) = inf{x : F(x) ≥ p}, the pth quantile (or 100pth percentile) of the population. If we let q_s,N be the total empirical probability weight of the s smallest observations, then the order statistics are seen to be sample quantiles, i.e.,

$$X_{(s)}=F_n^{-1}(p),\text{ if }{q}_{s-1,N}<p\le q_{s,N},s=1,\dots ,N.$$ (5)

Here q_0,N = 0 and q_N,N = 1.

A general L-statistic has the form:

$$\sum _{s=1}^N{c}_{s,N}{X}_{(s)},$$ (6)

for some choice of constants c_{1, N}, ‥‥, c_N,N. Consider a fixed signed measure dM(x) = m(x)dx on [0, 1]. The function m(x) is sometimes called a weight-generating function. An important subclass of (6) wide enough for all typical applications is given by Serfling (1980, chap. 8):

$$T(F_n)={\int }_0^1{F}_n^{-1}(x)m(x)dx+\sum _{l=1}^r{a}_l{F}_n^{-1}(p_l)≕T_1(F_n)+T_2(F_n),$$ (7)

for a pre-specified positive integer r. It is also assumed that 0 < p₁ < p₂ < … < p_r < 1 are specified, and that a₁, …, a_r are known constants, not all of which are equal to zero. The statistic T(F_n) can be written in the more familiar L-statistic form (6) by using (5) and taking the coefficients as $c_s,N={\int }_{q_{s-1,N}}^{q_{s,N}}m(x)dx+a_l$, where l is such that q_s−1,N < p_l ≤ q_s,N. The form (7) shows that T(F_n) is actually a sum of two L-statistics: T₁(F_n) — the continuous part of T(F_n), obtained by weighting all observations in a continuous manner; and T₂(F_n) — the discrete part of T(F_n), which is a weighted sum of r observations. Often, the statistic of interest is T₁(F_n) alone (e.g., 100α% trimmed mean, 0 < α < 1/2) or T₂(F_n) alone (e.g., sample quantile). Upon replacing F_n in (7) with F, we get the L-functional,

$$T(F)={\int }_0^1{F}^{-1}(x)m(x)dx+\sum _{l=1}^r{a}_l{F}^{-1}(p_l)≕T_1(F)+T_2(F),$$ (8)

representing the population parameter that T(F_n) actually estimates.

In the standard i.i.d. case, which is a special case of our set up when k_i = 1 ∀i, the statistic T is known to be asymptotically normal (Serfling 1980, p. 282). We generalize this result to the case of repeated measurements data in Section 2 using a statistical functional approach (van der Vaart 1998, chap. 20). In particular, we extend the technique of Fernholz (1983, prop. 4.3.3) for i.i.d. data to show that the remainder term in the von Mises expansion of a Hadamard differentiable functional goes to zero in probability. Further, we extend the technique of Ghosh (1971, thm. 1) for the i.i.d. data to get a Bahadur representation for sample quantiles. Together these results provide the desired asymptotic normality of T. This result is applied in Sections 3, 4 and 5 respectively for location estimation using trimmed means, quantile estimation and construction of tolerance intervals. We perform a simulation study in Section 6 to examine the finite sample accuracy of the proposed confidence and tolerance intervals and also to compare the two weight functions in (2). A real data application is presented in Section 7. Section 8 is devoted to technical details.

2 Asymptotic normality of T(F_n)

First, we make the following assumptions in addition to A.1 and A.2.

• A.3max_i=1,…,n k_i ≤ k*, where k* is a known constant. Thus, in the asymptotic analysis, we let the number of subjects increase but keep the number of repeated measurements bounded.
• A.4Let μ_n(k) denote the proportion of subjects with exactly k repeated measurements, k = 1, …, k*. There exist constants μ(k) such that lim_n→∞ μ_n(k) = μ(k), k = 1, …, k*. If for some k, there is no subject in the study with k measurements, then μ_n(k) and μ(k) are simply ignored. So, without loss of generality, μ_n(k) and μ(k), k = 1, …, k*, are all taken to be positive.
• A.5Let w(k) = w(k, n), k = 1, …, k*, denote the common weight of observations from subjects with k repeated measurements. There exist constants θ(k) such that lim_n→∞ nw(k) = θ(k), k = 1, …, k*. As in A.4, the θ(k) are assumed to be positive without loss of generality.
• A.6The ratio (max_1≤i≤n w_i) / (min_1≤i≤n w_i) = o(n^δ/{2(2+δ)}) for some δ > 0.
• A.7The function m(x) has support in [γ, 1 − γ] for some 0 < γ < 1/2, and ∃C > 0 such that |m(x)| ≤ C.

Let IC(x, F, T) denote the influence curve of the functional T in (8). It is defined as: IC(x, F, T) = (d/dt)T(F + t(δ_x − F))|_t=0, where δ_x is the c.d.f. of the point mass distribution at x (van der Vaart 1998, chap. 20). In other words, δ_x(y) = I(x ≤ y), y ∈ ℝ. Since T(F) = T₁(F) + T₂(F), we can write

$$IC(x,F,T)=IC(x,F,T_1)+IC(x,F,T_2).$$ (9)

The influence curves for T₁ and T₂ have been derived, for instance, in Huber (1981, pp. 56–57). They are:

$$IC(x,F,T_1)={\int }_{-\infty }^{x}m(F(y))dy-{\int }_{-\infty }^{+\infty }(1-F(y))m(F(y))dy,$$ (10)

$$IC(x,F,T_2)=\sum _{l=1}^r{a}_l\frac{p_l-{\delta }_x(F^{-1}(p_l))}{f(F^{-1}(p_l))}.$$ (11)

Next, let ${\psi }^2(k_i)=\text{var}[{\sum }_{j=1}^{k_i}IC(X_{ij},F,T)]$. Due to the exchangeability assumption A.2, we can write

$${\psi }^2(k_i)=k_i\text{var}[IC({\tilde{X}}_1,F,T)]+2\left(\begin{array}{c}\hfill k_i\hfill \\ \hfill 2\hfill \end{array}\right)\text{cov}[IC({\tilde{X}}_1,F,T),IC({\tilde{X}}_2,F,T)]=k_{i}E[I{C}^2({\tilde{X}}_1,F,T)]+2\left(\begin{array}{c}\hfill k_i\hfill \\ \hfill 2\hfill \end{array}\right)E[IC({\tilde{X}}_1,F,T)IC({\tilde{X}}_2,F,T)],$$ (12)

where the second equality follows from the fact that an influence curve has mean zero (van der Vaart 1998, chap. 20). Also, let

$${\sigma }_n^2=n\sum _{i=1}^n{w}_i^2{\psi }^2(k_i)=\sum _{k=1}^{k^{*}}{\mu }_n(k){\{nw(k)\}}^2{\psi }^2(k),{\sigma }^2=\underset{n\to \infty }{\text{lim}}{\sigma }_n^2=\sum _{k=1}^{k^{*}}\mu (k){\theta }^2(k){\psi }^2(k),$$ (13)

where k*, w(k), μ_n, μ and θ are from A.4 and A.5. The following result gives asymptotic normality of T(F_n).

Theorem 1

Consider the L-functional T defined in (8) and σ² defined in (13). Then, under the assumptions A.1 to A.7, and the additional assumptions listed in Proposition 3 in Section 8, we have:

$$n^{1/2}[T(F_n)-T(F)]\stackrel{d}{\to }N(0,{\sigma }^2).$$

This result generalizes a similar result in Serfling (1980, thm. A, p. 282) for i.i.d. data. It can be used in the usual manner to perform large-sample inference on T(F). For example, when n is large, an approximate 100(1 − β)% confidence interval for T(F) is:

$$T(F_n)\pm z_{1-\beta /2}{\hat{\sigma }}_n/n^{1/2},$$ (14)

where ${\hat{\sigma }}_n^2$ is an estimator of ${\sigma }_n^2$ whose limit is σ², and z_1−β/2 is the (1 − β/2)th quantile of the N(0, 1) distribution. A general approach to get ${\hat{\sigma }}_n^2$ is to simply replace the population quantities in ${\sigma }_n^2$ with their sample counterparts. In particular, let ${\widehat{IC}}_{ij}=IC(X_{ij},F_n,T)$ denote the empirical influence curve, obtained by replacing F in (9) with F_n. Then the expectations needed in (12) can be estimated as:

$$Ê[I{C}^2({\tilde{X}}_1,F,T)]=\frac{1}{n}\sum _{i=1}^n\frac{1}{k_i}\sum _{j=1}^{k_i}{\widehat{IC}}_{ij}^2,$$

$$Ê[IC({\tilde{X}}_1,F,T)IC({\tilde{X}}_2,F,T)]=\frac{1}{\#\{i:k_i>1\}}\sum _{i:k_i>1}\frac{1}{k_i(k_i-1)}\sum _{1\le j\ne l\le k_i}{\widehat{IC}}_{ij}{\widehat{IC}}_{il}.$$

Plugging-in these estimates in (12) gives ψ̂²(k_i). Hence from (13), ${\hat{\sigma }}_n^2$ can be taken as ${\hat{\sigma }}_n^2=n{\sum }_{i=1}^n{w}_i^2{\hat{\psi }}^2(k_i)$. Often, however, the expression for σ² can be simplified (see, e.g., (17) in Section 4). In this case, the unknowns in the simplified expression may be replaced with their estimates to get ${\hat{\sigma }}_n^2$.

3 Estimation of population trimmed means

For a given 0 < α < 1/2, the 100α% trimmed mean can be obtained from the functional T in (8) by taking m(x) = I(α < x < 1 − α)/(1 − 2α) in its continuous part T₁, and setting its discrete part T₂ equal to zero. This gives the population and sample versions of the trimmed mean as:

$$T(F)=\frac{1}{1-2\alpha }{\int }_{\alpha }^{1-\alpha }{F}^{-1}(x)dx,T(F_n)=\frac{1}{1-2\alpha }{\int }_{\alpha }^{1-\alpha }{F}_n^{-1}(x)dx.$$

Here α is the trimming proportion on each side. To write the sample version in the familiar L-statistic form, let l* and u* be integers such that q_l*−1,N < α ≤ q_l*,N and q_u*−1,N < 1−α ≤ q_u*,N. Also, let $w_j^{*}$, an element of {w_i, i = 1, …, N} in (1), be the weight associated with the jth order statistic X_(j). Then, from (5), the sample α-trimmed mean is:

$$T(F_n)=\frac{1}{1-2\alpha }\{(q_{l*,N}-\alpha )X_{(l*)}+\sum _{j=l*+1}^{u^{*}-1}{w}_j^{*}{X}_{(j)}+(1-\alpha -q_{u^{*}-1,N})X_{(u^{*})}\}.$$ (15)

It may be noted that T(F) coincides with the median if the distribution F is symmetric. Huber (1981) gives the influence function of T(F) as:

$$IC(X,F,T)=\{\begin{array}{cc}\frac{1}{1-2\alpha }\{F^{-1}(\alpha )-W(F)\},\hfill & \text{if }X<F^{-1}(\alpha ),\hfill \\ \frac{1}{1-2\alpha }\{X-W(F)\},\hfill & \text{if }{F}^{-1}(\alpha )\le X\le F^{-1}(1-\alpha ),\hfill \\ \frac{1}{1-2\alpha }\{F^{-1}(1-\alpha )-W(F)\},\hfill & \text{if }X>F^{-1}(1-\alpha ),\hfill \end{array}$$

where W(F) = (1 − 2α)T(F) + α{F⁻¹(α) + F⁻¹(1 − α)}. This influence curve can be used as described in Section 2 to estimate the standard error of the sample trimmed mean and to get an approximate confidence interval for the population trimmed mean.

4 Estimation of population quantiles

For a given 0 < p < 1, the pth quantile is a special case of the functional T in (8) obtained by setting its continuous part T₁ equal to zero, and taking r = 1, p₁ = p and a₁ = 1 in its discrete part T₂. This gives the population and the sample pth quantile as T(F) = F⁻¹(p) and $T(F_n)=F_n^{-1}(p)$. These quantities will henceforth be denoted as Q_p and Q̂_p for notational convenience. From (11), the influence curve for Q_p is:

$$IC(X,F,Q_p)=\{p-{\delta }_X(Q_p)\}/f(Q_p).$$ (16)

Upon substituting this expression in (12) and simplifying (13), we get:

$${\sigma }^2=\frac{p(1-p)}{f^2(Q_p)}\sum _{k=1}^{k^{*}}k\{1+(k-1)\rho (Q_p,Q_p)\}\mu (k){\theta }^2(k),$$ (17)

where ρ(Q_p, Q_p) is given by (4). The asymptotic normality of Q̂_p holds from Theorem 1.

It may be noted that Olsson and Rootzen (1996) also establish asymptotic normality of Q̂_p by using in F_n an estimate of the optimal weight function (3), obtained by replacing ρ(x, x) with an estimator ρ̂(x, x) which is to be defined in (19). Although the weights in our result do not depend on x, they can be arbitrary provided they satisfy the assumptions A.5 and A.6. In this sense, our result differs from Olsson and Rootzen’s. Besides, unlike theirs, our result follows from a more general result derived for L-statistics.

Using (13), σ² given in (17) can be estimated by

$${\hat{\sigma }}_n^2=\frac{np(1-p)}{{\hat{f}}^2({\hat{Q}}_p)}\sum _{i=1}^n{k}_i\{1+(k_i-1)\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)\}{w}_i^2,$$ (18)

where f̂ is an estimator of the density f and ρ̂ is an estimator of ρ. The density may be estimated as

$$\hat{f}(x)=\frac{F_n(x+h)-F_n(x-h)}{2h},$$

with the bandwidth h chosen, e.g., according to the recommendations of Silverman (1986, chap. 4). Next, the correlation ρ(x, y) may be estimated by a simple estimator,

$$\hat{\rho }(x,y)=\frac{\widehat{\mathit{\text{cov}}}\{I({\tilde{X}}_1\le x),I({\tilde{X}}_2\le y)\}}{{\{\widehat{\mathit{\text{var}}}[I({\tilde{X}}_1\le x)]\widehat{\mathit{\text{var}}}[I({\tilde{X}}_1\le y)]\}}^{1/2}},$$ (19)

where

$$\widehat{\mathit{\text{var}}}[I({\tilde{X}}_1\le x)]=\frac{1}{\#\{i:k_i>1\}}\sum _{i:k_i>1}\frac{1}{k_i}\sum _{j=1}^{k_i}{\{I(X_{ij}\le x)-{\overline{F}}_n(x)\}}^2,$$

$$\widehat{\mathit{\text{cov}}}[I({\tilde{X}}_1\le x),I({\tilde{X}}_2\le y)]=\frac{1}{\#\{i:k_i>1\}}\sum _{i:k_i>1}\frac{1}{k_i(k_i-1)}\times \sum _{i\le j\ne l\le k_i}\{I(X_{ij}\le x)-{\overline{F}}_n(x)\}\{I(X_{il}\le y)-{\overline{F}}_n(y)\},$$

$${\overline{F}}_n(x)=\frac{1}{n}\sum _{i=1}^n\frac{1}{k_i}\sum _{j=1}^{k_i}I(X_{ij}\le x).$$

This ρ̂ is related to the estimator of an intraclass correlation given in Karlin et al. (1981) and has also been used in Olsson and Rootzen (1996). Potentially other estimators of ρ may also be considered, but ρ̂ works well in simulations.

Although ${\hat{\sigma }}_n^2$ defined in (18) can be used in (14) to get a confidence interval for Q_p, it has the unattractive feature of having to estimate the density f(Q_p). This problem can be avoided by constructing the confidence interval directly using the following result. It is proved in Section 8.4.

Theorem 2

Suppose the assumptions A.1 to A.6 hold. Assume also that the bivariate c.d.f. G of (X̃₁, X̃₂) is continuous at (Q_p, Q_p) and f (Q_p) > 0, for 0 < p < 1. Let ${\hat{r}}_n^2=np(1-p){\sum }_{i=1}^n{k}_i\{1+(k_i-1)\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)\}{w}_i^2$. Define l̂_n = p − z_1−β/2r̂_n/n^1/2 and û_n = p + z_1−β/2r̂_n/n^1/2. Then, lim_n→∞ P(Q_p ∈ [Q̂_l̂n, Q̂_ûn]) = 1 − β.

We next obtain a weak version of Bahadur representation of sample quantiles which generalizes Ghosh (1971, thm. 1) for i.i.d. data. It is proved in Section 8.5.

Theorem 3

Suppose the assumptions A.1 to A.6 hold. Assume also that the bivariate c.d.f. G of (X̃₁, X̃₂) is continuous at (Q_p, Q_p) and f(Q_p) > 0, for 0 < p < 1. Let p⁽ⁿ⁾ be a sequence of probabilities such that n^1/2 (p⁽ⁿ⁾ − p) = O(1), and ${\hat{Q}}_{p^{(n)}}=F_n^{-1}(p^{(n)})$. Then,

$${\hat{Q}}_{p^{(n)}}=Q_p+\{p^{(n)}-F_n(Q_p)\}/f(Q_p)+o_p(1/n^{1/2}).$$

5 Construction of nonparametric tolerance intervals

For given 0 < p, β < 1, an interval [L̂₁, L̂₂] computed from the sample data is called a (p, 1 − β) tolerance interval for a random variable X if

$$P(F({\hat{L}}_2)-F({\hat{L}}_1)\ge p)=1-\beta .$$ (20)

The random quantity $F({\hat{L}}_2)-F({\hat{L}}_1)={\int }_{{\hat{L}}_1}^{{\hat{L}}_2}f(x)dx$ represents the probability content of the interval [L̂₁, L̂₂] under the distribution of X. Thus, a (p, 1 − β) tolerance interval captures at least p proportion of the X population with 1 − β confidence. The interval is one-sided if either L̂₁ = −∞ or L̂₂ = ∞, otherwise it is two-sided. Tolerance intervals are common in engineering and manufacturing applications; see Guttman (1988), Vardeman (1992) and Krishnamoorthy and Mathew (2009) for an introduction to this topic.

In general, a nonparametric tolerance interval has the form [L̂₁, L̂₂] = [X_(r), X_(s)], where r and s (r < s) are chosen to satisfy (20). This notation allows the possibility of one-sided intervals by letting r be zero with X₍₀₎ = −∞ and s be N +1 with X_(N+1) = ∞, provided both r = 0 and s = N +1 are not taken at the same time. In the i.i.d. case, it is well-known that F{X_(s)}−F{X_(r)} follows a Beta (s−r, N−s+r+1) distribution (Guttman, 1988). Hence, for example, a two-sided equal-tailed tolerance interval can be obtained by taking s = N − r + 1, r < (N + 1)/2, and numerically solving (20) for r. This is equivalent to finding r such that the c.d.f. of the Beta (N − 2r + 1, 2r) distribution at p is β.

In the case of repeated measurements data, however, the distribution of F{X_(s)} − F{X_(r)} (or equivalently F(Q̂_p2) − F(Q̂_p1) for p₂ > p₁) does not have a simple form. This motivates us to search for p₁ and p₂ so that (20) holds in the limit, i.e.,

$$\underset{n\to \infty }{\text{lim}}P(F({\hat{Q}}_{p_2})-F({\hat{Q}}_{p_1})\ge p)=1-\beta .$$ (21)

We refer to the resulting (Q̂_p1, Q̂_p2) as an asymptotic (p, 1 − β) tolerance interval. This interval has approximately 1 − β confidence when n is large. As before, here we allow the possibility of one-sided intervals by letting p₁ be zero with Q̂₀ = −∞ and p₂ be one with Q̂₁ = ∞, provided both p₁ = 0 and p₂ = 1 are not taken simultaneously. To develop a procedure for constructing this interval, let

$${\nu }_l^2(p_l)=p_l(1-p_l)\sum _{k=1}^{k^{*}}k\{1+(k-1)\rho (Q_{p_l},Q_{p_l})\}\mu (k){\theta }^2(k),l=1,2,$$

$${\nu }_{12}(p_1,p_2)=p_1(1-p_2)\sum _{k=1}^{k^{*}}k[1+(k-1)\rho (Q_{p_1},Q_{p_2}){\left\{\frac{(1-p_1)p_2}{p_1(1-p_2)}\right\}}^{1/2}]\mu (k){\theta }^2(k),$$

$${\nu }^2(p_1,p_2)={\nu }_1^2(p_1)-2{\nu }_{12}(p_1,p_2)+{\nu }_2^2(p_2),$$ (22)

where ρ(x, y) is given by (4). Here ${\nu }_l^2(p_l)$ is defined for 0 < p_l < 1, and ν₁₂(p₁, p₂) and ν²(p₁, p₂) are defined for 0 < p₁ < p₂ < 1. Next, let ${\hat{\nu }}_1^2,{\hat{\nu }}_2^2$ and ν² be consistent estimators of ${\nu }_1^2,{\nu }_2^2$ and ν², respectively. They are obtained by replacing Q_p and ρ in (22) with Q̂_p and ρ̂, defined by (19). The next result shows that the probability content of (Q̂_p1, Q̂_p2) is asymptotically normal regardless of whether this interval is one- or two-sided. It is a consequence of Theorem 1 and is proved in Section 8.6.

Theorem 4

Suppose that the assumptions A.1 to A.6 hold.

1. Suppose that the assumptions listed in Proposition 3 in Section 8 also hold for r = 2 and for all 0 < p₁ < p₂ < 1. Then, $n^{1/2}[F({\hat{Q}}_{p_2})-F({\hat{Q}}_{p_1})-(p_2-p_1)]\stackrel{d}{\to }N(0,{\nu }^2(p_1,p_2))$.

2. (b) Suppose that the assumptions listed in Proposition 3 also hold for r = 1 and for all 0 < p_l < 1, separately for each l = 1, 2. Then, $n^{1/2}(F({\hat{Q}}_{p_l})-p_l)\stackrel{d}{\to }N(0,{\nu }_l^2(p_l))$, l = 1, 2.

This result implies that, when n is large, F(Q̂_p2) − F(Q̂_p1) and F(Q̂_pl) approximately follow N(p₂ − p₁, ν²(p₁, p₂)/n) and $N(p_l,{\hat{\nu }}_l^2(p_l)/n)$ distributions, respectively. Therefore, p₁ and p₂ required for the two-sided interval (Q̂_p1, Q̂_p2) to satisfy (21) can be found by solving:

$$n^{1/2}\{p-(p_2-p_1)\}/\hat{\nu }(p_1,p_2)=z_{\beta }.$$ (23)

It follows from (23) that p₁ and p₂ satisfy p₂ − p₁ ≥ p whenever 0 < β ≤ 1/2. For an equal-tailed interval one can take p₂ = 1 − p₁ in (23). Analogously, for the one-sided case, p₁ needed for the interval (Q̂_p1, ∞) and p₂ needed for the interval (−∞, Q̂_p2) can be computed by respectively solving the equations

$$n^{1/2}\{p-(1-p_1)\}/{\hat{\nu }}_1(p_1)=z_{\beta },n^{1/2}(p-p_2)/{\hat{\nu }}_2(p_2)=z_{\beta }.$$ (24)

The finite sample accuracy of these tolerance intervals can be improved by computing (p₁, p₂) after applying a logit (or log-odds) transformation to the probability content. For this, we can deduce from Theorem 4 and delta method that:

$$n^{1/2}[\text{logit}\{F({\hat{Q}}_{p_2})-F({\hat{Q}}_{p_1})\}-\text{logit}(p_2-p_1)]\stackrel{d}{\to }N(0,{\nu }^2(p_1,p_2)/{\{(p_2-p_1)(1-p_2+p_1)\}}^2),$$

$$n^{1/2}[\text{logit}\{F({\hat{Q}}_{p_l})\}-\text{logit}(p_l)]\stackrel{d}{\to }N(0,{\nu }_l^2(p_l)/{\{p_l(1-p_l)\}}^2),l=1,2.$$

Thus, the more accurate (p₁, p₂) can be computed by solving the following counterparts of (23) and (24):

$$n^{1/2}\{\text{logit}(p)-\text{logit}(p_2-p_1)\}(p_2-p_1)(1-p_2+p_1)/\hat{\nu }(p_1,p_2)=z_{\beta },$$

$$n^{1/2}\{\text{logit}(p)-\text{logit}(1-p_1)\}{p}_1(1-p_1)/{\hat{\nu }}_1(p_1)=z_{\beta },n^{1/2}\{\text{logit}(p)-\text{logit}(p_2)\}{p}_2(1-p_2)/{\hat{\nu }}_2(p_2)=z_{\beta }.$$ (25)

This is the method we recommend for use in practice.

6 A simulation study

In this section, we use Monte Carlo simulations to evaluate certain properties of sample trimmed means, sample quantiles and tolerance intervals. We also compare the two weight functions given in (2) for estimating F in the case of unbalanced designs (recall that they are equal in the case of balanced designs). Our focus is on models that have the structure of a one-way random-effects model:

$$X_{ij}=\xi +3b_i+{\epsilon }_{ij},j=1,\dots ,k_i,i=1,\dots ,n,$$ (26)

where ξ is the fixed intercept taken to equal 0 without loss of generality, b_i is the random effect of the ith subject and ε_ij is the random error term. Here b_i and ε_ij are mutually independent and they are also independent for different subjects. The coefficient of b_i in (26) is taken as 3 to have high intraclass correlation between the repeated measurements, which is a typical scenario in applications.

6.1 Trimmed Means

We first examine the asymptotic efficiency of the trimmed mean relative to the normality-based MLE of the underlying location parameter. We specifically consider a total of ten models obtained using combinations of t₃, t₅, t₃₀ and N(0, 1) as distributions for the two random terms in (26). These models are summarized in Table 1. Only symmetric distributions are considered so that the parameter T(F) that the trimmed mean estimates equals the location parameter ξ, whose true value is zero.

Table 1.

Estimated ARE of the trimmed mean estimate ξ̂_l, MSE(ξ̂_mle)/MSE(ξ̂_l), with respect to ξ̂_mle, l = 1, 2. The “N” under models refers to the N(0, 1) distribution.

	α
Models for bi, εij	0.05		0.10		0.125
	ξ̂1	ξ̂2	ξ̂1	ξ̂2	ξ̂1	ξ̂2
N, N	0.98	0.84	0.96	0.82	0.95	0.81
t5, N	1.20	1.00	1.24	1.03	1.25	1.04
t3, N	1.65	1.41	1.78	1.54	1.82	1.57
N, t3	1.01	0.87	0.99	0.86	0.97	0.85
t5, t3	1.18	1.04	1.22	1.08	1.23	1.09
t3, t3	1.62	1.36	1.77	1.49	1.80	1.52
N, t5	0.98	0.83	0.95	0.81	0.95	0.80
t5, t5	1.16	1.00	1.18	1.03	1.18	1.03
t3, t5	1.65	1.39	1.80	1.52	1.85	1.56
t30, t30	1.00	0.84	0.98	0.83	0.96	0.82

We simulate data from each model on n = 400 subjects in a way that k_i equals 1, 2, 3 and 4 for 100 subjects each. These data are used to compute three estimates of ξ — the α-trimmed mean with weights w_i,1 = 1/(nk_i) and w_i,2 = 1/N, and the MLE of ξ assuming normality for both random-effects and errors in the model (26). These estimators are denoted as ξ̂₁, ξ̂₂ and ξ̂_mle, respectively. Three values for α are used: 0.05, 0.10 and 0.125. The process of simulating data and estimating ξ is repeated 2,000 times, and the approximate mean-squared errors (MSEs) of the three estimators are computed. The ratio MSE(ξ̂_mle)/MSE(ξ̂_l) gives the estimated asymptotic relative efficiency (ARE) of ξ̂_l relative to ξ̂_mle, l = 1, 2. The computations are performed using the statistical software R (R Development Core Team 2011) and its nlme package (Pinheiro et al. 2011) is used to get ξ̂_mle.

Table 1 presents the ARE estimates. It shows that ξ̂₁ is more efficient than ξ̂₂ at all settings considered. In fact, ξ̂₁ is only slightly less efficient than the MLE. In the worst case, ξ̂₁ loses 5% efficiency over the MLE, which occurs when α = 0.125 and the model is either 3N(0, 1)+N(0, 1) or 3N(0, 1)+t₅. On the other hand, the gain in efficiency of ξ̂₁ over the MLE can be substantial in case of heavy-tailed distributions. The largest gain in the table is 85% for the model 3t₃ + t₅ and α = 0.125. It is also interesting to note that the heavy-tailedness of random-effect distribution causes more loss in efficiency of ξ̂_mle than the heavy-tailedness of error distribution. Moreover, when the model for error distribution is fixed and the random-effect distribution spans N(0, 1), t₅ and t₃ distributions, we observe the pattern that the ARE of ξ̂₁ for t₅ falls between those for N(0, 1) and t₃. But this pattern does not hold when the random-effect distribution is fixed and the error distribution varies. Additional simulations in Assaad (2012, chap. 4) for balanced designs with between 2 to 4 repeated measurements per subject show that the above conclusion regarding the relative merits of ξ̂₁ and ξ̂_mle remains unchanged. (It may be recalled that ξ̂₁ = ξ̂₂ in case of balanced designs.) Overall, these findings suggest that ξ̂₁ with α = 0.10 or 0.125 provides a strong alternative to ξ̂_mle in all models considered.

Next, we examine the coverage accuracy of two nonparametric confidence intervals for ξ — one using ξ̂₁ and the other using ξ̂₂. Simulations in Assaad (2012, chap. 4) show that n around 50 is large enough for these confidence intervals to be accurate. Moreover, just like the ARE case, the design of the study (balanced or unbalanced) and the number of repeated measurements per subject do not have any noteworthy impact on this conclusion.

6.2 Quantiles

Here we only evaluate the finite sample accuracy of the confidence interval for Q_p obtained using Theorem 2. For a comparison of asymptotic efficiencies of Q̂_p with weights w_i,1 = 1/(nk_i) and w_i,2 = 1/N, we refer the reader to Figure 1 of Olsson and Rootzen (1996). It shows that unless the correlation ρ(Q_p, Q_p), given by (4), is small, w_i,1 leads to a more efficient estimator than w_i,2.

Figure 1.

Histogram of the blood pressure data. Also marked on this graph are likelihood-based (bottom line segment) and nonparametric (top line segment) 95% confidence intervals for quantiles and (0.90, 0.95) tolerance intervals. Here “T.I.” means “tolerance interval” and “C.I.” means “confidence interval.” The measurements range between 77 to 228 mmHg.

To study the coverage accuracy, we consider three distributions — N(0, 1), t₃ and a skew-normal distribution (Azzalini 1985) with location zero, scale one and skewness parameter 5, denoted as SN(0, 1, 5) — for each of the two random terms b_i and ε_ij in (26). This results in a total of nine models. From each model, we simulate data on n = 52 subjects in a way that k_i equals 1, 2, 3, 4 for 13 subjects each in case of an unbalanced design and k_i equals 2, 3, 4 for all subjects in case of balanced designs. These data are used to compute 95% confidence intervals for median Q_0.5 and 90th percentile Q_0.9 via Theorem 2. Simulations in Assaad (2012, chap. 4) reveal that n around 50 may be large enough for these confidence intervals to be accurate. Besides, this accuracy does not seem to be affected by either the design of the study (balanced or unbalanced) or the data distribution (normal, heavy-tailed or skewed) or the number of repeated measurements. Further simulations for Q_0.99 (not presented here) show that n around 250 is needed to achieve satisfactory coverage probabilities in all the above models.

6.3 Tolerance intervals

Here we examine the finite sample accuracy of the proposed tolerance intervals. As in Section 6.2, we focus on nine models of the form (26). They are summarized in the first column of Table 2. From each model, we simulate data on n = 60 subjects in a way that k_i equals 1, 2, 3, 4 for 15 subjects each. These data are used to compute two-sided equal-tailed tolerance intervals by solving (25), using each of the two weight functions w_i,1 = 1/(nk_i) and w_i,2 = 1/N. We then compute the true probability content of each interval numerically. This process of simulating data, constructing tolerance intervals and computing their probability content is repeated 2,000 times and the proportion of times the true content exceeds p is obtained.

Table 2.

Proportion of times (in %) the probability content of an asymptotic (p, 0.95) tolerance interval exceeds p in case of an unbalanced design with n = 60. The weight functions w_i,1 and w_i,2 are given by (2). The “N” and “SN” under models refer to N(0, 1) and SN(0, 1, 5) distributions, respectively.

Models for bi, εij	p = 0.8		p = 0.9
	wi,1	wi,2	wi,1	wi,2
N, N	94.3	93.4	94.2	92.2
SN, N	93.3	93.3	93.1	91.4
t3, N	94.1	94.0	92.8	92.0
N, t3	94.0	93.0	94.1	92.3
SN, t3	93.8	92.2	93.6	92.4
t3, t3	93.6	93.4	92.9	92.1
N, SN	93.9	93.5	91.7	91.8
SN, SN	92.7	92.9	93.0	91.3
t3, SN	93.8	93.7	92.6	93.1

Table 2 presents these proportions for p = 0.80, 0.90 and 1 − β = 0.95. We see that, in general, the values are closer to 0.95 in case of p = 0.80 than p = 0.90, and with weights w_i,1 than w_i,2. Specifically with weights w_i,1, most values are around 0.94 in case of p = 0.80 and around 0.93 in case of p = 0.90, regardless of whether the distribution is normal, heavy-tailed or skewed. On the whole, these values that the tolerance intervals with weights w_i,1 have reasonable accuracy with n = 60 in case of p = 0.80, whereas a larger n (around 80, based on additional simulations in Assaad (2012, chap. 4)) is needed to achieve a similar level of accuracy in case of p = 0.90. Further simulations for balanced designs with between 2 to 4 repeated measurements per subject suggest that the accuracy of the tolerance interval does not depend on the number of repeated measurements.

7 Application to blood pressure data

In this section, we use a portion of the blood pressure data of Bland and Altman (1999) to illustrate the application of our results. These data were originally collected to evaluate agreement between three methods of measuring systolic blood pressure. However, since a comparison of two or more measurement methods is not of concern in this article, we focus only on the data from one of the methods, namely, the semi-automatic blood pressure monitor. There are 3 repeated measurements (in mmHg) of systolic blood pressure taken using the monitor in quick succession on each of 85 subjects in the study. These measurements are our X_ij, j = 1, 2, 3, i = 1, …, 85, and X represents the population from which these data are drawn. We are interested in estimating the center, the 90th and 99th percentiles and the 10% trimmed mean of the distribution of X, and also constructing a (p = 0.90, 1 − β = 0.95) tolerance interval for it. A histogram of the data presented in Figure 1 shows marked right-skewness in the distribution.

We first fit a one-way random-effects model,

$$X_{ij}=\xi +b_i+{\epsilon }_{ij},j=1,2,3,i=1,\dots ,85,$$ (27)

assuming that $b_i~N(0,{\sigma }_b^2)$ and ${\epsilon }_{ij}~N(0,{\sigma }_{\epsilon }^2)$. This model implies that $X~N(\xi ,{\sigma }_b^2+{\sigma }_{\epsilon }^2)$. The model is fit using the nlme package (Pinheiro et al. 2011) in R. The MLE of the parameter vector $(\xi ,{\sigma }_b^2,{\sigma }_{\epsilon }^2)$ and its approximate estimated variance matrix are:

$$\left(\begin{array}{c}\hfill 143.03\hfill \\ \hfill 971.30\hfill \\ \hfill 83.14\hfill \end{array}\right),\left(\begin{array}{ccc}\hfill 11.75\hfill & \hfill 0.00\hfill & \hfill 0.00\hfill \\ \hfill 0.00\hfill & \hfill 23492.14\hfill & \hfill -27.11\hfill \\ \hfill 0.00\hfill & \hfill -27.11\hfill & \hfill 81.32\hfill \end{array}\right).$$

Figure 2 presents the normal quantile-quantile plot of the estimated random-effects and the residuals. There is clear evidence of skewness in random-effects and heavier-than-normal tails in residuals, invalidating the normality assumption and justifying the need for a nonparametric analysis.

Figure 2.

Normal quantile-quantile plots for estimated random effects and residuals resulting from fitting the model (27) to the blood pressure data. A line passing through the first and third quartiles is added in each plot.

Table 3 summarizes the ML and nonparametric estimates of median Q_0.5, 90th percentile Q_0.9 and 99th percentile Q_0.99, along with their standard errors and 95% confidence intervals. It may be noted that the two weight functions in (2) used for nonparametric estimation are identical due to the balanced design of the data. In the parametric case, $Q_p=\xi +z_p{({\sigma }_b^2+{\sigma }_{\epsilon }^2)}^{1/2}$ and ${\hat{Q}}_p=\hat{\xi }+z_p{({\hat{\sigma }}_b^2+{\hat{\sigma }}_{\epsilon }^2)}^{1/2}$ is its MLE. Further, the delta method (Lehmann 1999, p. 295) is used to estimate the standard error of Q̂_p and to construct the confidence interval for Q_p. In the nonparametric case, the standard error of Q̂_p is estimated using (18), with h = 0.79(Q̂_0.75 − Q̂_0.25)n^−1/5 as the bandwidth in the density estimate f̂ (Silverman 1986, p. 47), and the confidence interval for Q_p is computed using Theorem 2. Also presented in Table 3 are the nonparametric estimate of 10% trimmed mean, its standard error and 95% confidence interval; and parametric and nonparametric (0.90, 0.95) tolerance intervals. The computations involving trimmed mean and tolerance interval are described in Sections 3 and 5, respectively. The parametric tolerance interval is computed using Mee’s approach in Krishnamoorthy and Mathew (2009, sec. 4.5). All these confidence and tolerance intervals are also marked on the histogram in Figure 1.

Table 3.

Comparison of likelihood-based and nonparametric inferences for the blood pressure data. Here “S.E.” means “standard error” and “C.I.” means “confidence interval.”

	likelihood-based			nonparametric
	estimate	S.E.	95% C.I.	estimate	S.E.	95% C.I.
Q0.5	143	3.4	(136, 150)	135	3.5	(128, 142)
Q0.9	185	4.6	(176, 194)	192	7.0	(181, 217)
Q0.99	219	6.5	(206, 231)	228	3.7	(226, 228)
10% trimmed mean	-	-	-	140	3.5	(133, 147)
	likelihood-based			nonparametric
(0.90, 0.95) tolerance interval	(81, 205)			(94, 224)

We note that there are substantial differences between the parametric and the nonparametric estimates reported in Table 3. In particular, due to the long right tail of the distribution, it is reasonable that the MLE of Q_0.5, which is the overall sample mean of the data, is greater than the nonparametric median estimate. Moreover, the nonparametric estimates of Q_0.9 and Q_0.99 and the nonparametric tolerance interval are to the right of their parametric counterparts for the same reason. Overall, these findings confirm that the nonparametric estimates are more consistent with the observed data distribution than the normality-based estimates even though the latter lead to smaller standard errors for Q̂_p and a shorter tolerance interval.

Using the nonparametric estimates, we conclude that median of the distribution of systolic blood pressure measurements made by the semi-automatic monitor is 135 (95% confidence interval: [128, 142]), its 90th percentile is 192 (95% confidence interval: [181, 217]) and its 99th percentile is 228 (95% confidence interval: [226, 228]). Further, 90% of the distribution of measurements is contained in [94, 224] with 95% confidence. The 10% trimmed mean of 140 (95% confidence interval: [133, 147]) provides another estimate of the center of the distribution — it is shifted to the right of the median due to right-skewness in the distribution.

Finally a remark is in order about the nonparametric confidence interval for Q_0.99. This interval is not expected to be accurate as the number of subjects in these data (n = 85) is considerably smaller than n = 250 needed to achieve satisfactory coverage probability (see Section 6.2). Note also that the upper endpoint of this interval coincides with Q_0.99 and the two equal 228, the largest observation in the data. This is due to the relatively small n and that the interval endpoints need to be observations in the sample (see Theorem 2).

8 Technical details and proofs

This section is devoted to proving Theorems 1–4. For the functionals T, T₁ and T₂ given by (8), we can write

$$\begin{gathered} n^{1/2}[T(F_n)-T(F)]=n^{1/2}[T_1(F_n)-T_1(F)]+n^{1/2}[T_2(F_n)-T_2(F)] \\ =n^{1/2}\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}IC(X_{ij},F,T)+n^{1/2}{\mathrm{\Delta }}_{1n}+n^{1/2}{\mathrm{\Delta }}_{2n}, \end{gathered}$$ (28)

where Δ_ln represents the remainder term

$${\mathrm{\Delta }}_{ln}=T_l(F_n)-T_l(F)-\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}IC(X_{ij},F,T_l),l=1,2,$$ (29)

and the influence curves are given by (10) and (11). The following results hold for the terms on the RHS of (28).

Proposition 1

Let σ² be as defined in (13). Then, under the assumptions A.1 to A.7,

$$n^{1/2}\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}IC(X_{ij},F,T)\stackrel{d}{\to }N(0,{\sigma }^2).$$

Proposition 2

Under the assumptions A.1 to A.7, n^1/2Δ_1n = o_p(1).

Proposition 3

Let G be the bivariate c.d.f. of (X̃₁, X̃₂). Assume that G is continuous at (Q_pl, Q_pl) and F′(Q_pl) > 0, for each l = 1, …, r. Then, under the assumptions A.1 to A.6, n^1/2Δ_2n = o_p(1).

We prove these results in the next three sections. But first let us use them to quickly establish Theorem 1.

Proof of Theorem 1

The result follows immediately from (28) by applying Propositions 1, 2 and 3, and Slutsky’s theorem.

8.1 Proof of Proposition 1

Let ${\eta }_i={\sum }_{j=1}^{k_i}IC(X_{ij},F,T)$ and T_ni = n^1/2w_iη_i, i = 1, …, n. These η_i are independent with mean zero and variance ψ²(k_i), defined in (12). The finiteness of this variance is ensured by the second part of assumption A.7 (Shao 2003, exer. 5.34). Note also that ${\sum }_{i=1}^n{T}_{ni}=n^{1/2}{\sum }_{i=1}^n{w}_i{\sum }_{j=1}^{k_i}IC(X_{ij},F,T)$, and ${\sigma }_n^2$, given by (13), is the variance of this sum. Next, for the δ > 0 assumed in A.6, we can write:

$$\frac{{\sum }_{i=1}^{n}E{|T_{ni}|}^{2+\delta }}{{\sigma }_n^{2+\delta }}=\frac{n^{\frac{2+\delta }{2}}{\sum }_{i=1}^n{w}_i^{2+\delta }E{|{\eta }_i|}^{2+\delta }}{n^{\frac{2+\delta }{2}}{({\sum }_{i=1}^n{w}_i^2{\psi }^2(k_i))}^{\frac{2+\delta }{2}}}\le \frac{{\text{max}}_{1\le i\le n}{w}_i^{2+\delta }{\sum }_{i=1}^{n}E{|{\eta }_i|}^{2+\delta }}{{\text{max}}_{1\le i\le n}{w}_i^{2+\delta }{({\sum }_{i=1}^n{\psi }^2(k_i))}^{\frac{2+\delta }{2}}}.$$ (30)

Further, from assumptions A.3 and A.4, we have:

$$\frac{{\sum }_{i=1}^{n}E{|{\eta }_i|}^{2+\delta }}{{({\sum }_{i=1}^n{\psi }^2(k_i))}^{\frac{2+\delta }{2}}}=\frac{n{\sum }_{i=1}^n\frac{E{|{\eta }_i|}^{2+\delta }}{n}}{n^{\frac{2+\delta }{2}}{\left({\sum }_{i=1}^n\frac{{\psi }^2(k_i)}{n}\right)}^{\frac{2+\delta }{2}}}~n^{-\frac{\delta }{2}}\frac{{\sum }_{k=1}^{k^{*}}\mu (k)E{|{\eta }_k|}^{2+\delta }}{{({\sum }_{k=1}^{k^{*}}\mu (k){\psi }^2(k))}^{\frac{2+\delta }{2}}}.$$

The rightmost ratio is free of n. From (30) and assumption A.6, this means ${\sum }_{i=1}^{n}E{|T_{ni}|}^{2+\delta }=o({\sigma }_n^{2+\delta })$. Therefore, from the Liapounov theorem (Shao 2003, p. 69), ${\sum }_{i=1}^n{T}_{ni}/{\sigma }_n\stackrel{d}{\to }N(0,1)$. The result now holds from Slutsky’s theorem since σ² is the limit of ${\sigma }_n^2$.

8.2 Proof of Proposition 2

In this section, we deduce the desired n^1/2Δ_1n = o_p(1) from a more general result, which extends the results of Fernholz (1983, chap. 4) about the remainder term in the von Mises expansion of a Hadamard differentiable functional from i.i.d. data to repeated measurements data.

Let Y_ij = F(X_ij), so that the Y_ij are distributed uniformly on [0, 1]. Also, let U be the c.d.f. of this uniform distribution. Define the counterpart of F_n for the Y_ij as:

$$U_n(x)=\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}I(Y_{ij}\le x).$$ (31)

Next, let 𝔻[0, 1] be the space of cadlag functions (i.e., right continuous functions with left-hand limits) on [0, 1]. We assume that 𝔻 is equipped with the sup norm ‖·‖_∞. Suppose we have a functional τ : 𝔻[0, 1] ⟼ ℝ that is Hadamard differentiable at U ∈ 𝔻[0, 1] with derivative ${\tau }_U^{\prime }$. From the von Mises expansion, we have:

$$n^{1/2}[\tau (U_n)-\tau (U)]=n^{1/2}{\tau }_U^{\prime }(U_n-U)+n^{1/2}\text{ Rem}(U_n-U).$$ (32)

The remainder term converges in probability to zero from the following result.

Proposition 4

Suppose the assumptions A.1 to A.5 hold. Then, for the remainder term in the von Mises expansion (32) of a Hadamard differentiable functional τ, we have: n^1/2 Rem(U_n − U) = n^1/2Δ_1n = o_p(1).

Before proving this result, let us first use it to establish Proposition 2.

Proof of Proposition 2

Since F_n = U_n ◦ F and F = U ◦ F, the statistical functional T₁, defined in (8), induces a functional 𝔻[0, 1] ⟼ ℝ. Take τ to be this functional, i.e.,

$$T_1(F)=T_1(U○F)≕\tau (U),T_1(F_n)=T_1(U_n○F)≕\tau (U_n).$$

This τ is known to be Hadamard differentiable at U ∈ 𝔻[0, 1] due the first part of assumption A.7 (Fernholz 1983, prop. 7.2.1). Therefore, from the von Mises expansion,

$$n^{1/2}[T_1(F_n)-T_1(F)]=n^{1/2}[\tau (U_n)-\tau (U)]=n^{1/2}{\tau }_U^{\prime }(U_n-U)+n^{1/2}\text{ Rem}(U_n-U).$$ (33)

Since U_n = F_n ◦ F⁻¹, U = F ◦ F⁻¹, and ${\tau }_U^{\prime }$ is linear by definition, we can write:

$${\tau }_U^{\prime }(U_n-U)={\tau }_U^{\prime }[(F_n-F)○F^{-1}]=\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}{\tau }_U^{\prime }[({\delta }_{X_{ij}}-F)○F^{-1}]=\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}IC(X_{ij},F,T_1),$$

where the last equality follows from Fernholz (1983, lem. 4.4.1). Next, a comparison of (29) and (33) shows that n^1/2Δ_1n = n^1/2 Rem(U_n − U). The result now follows from Proposition 4.

To prove Proposition 4, we begin by establishing convergence of the weighted empirical process U_n. Let 𝔾 be a continuous Gaussian process with mean zero and covariance ${\sum }_{k=1}^{k^{*}}\mu (k){\theta }^2(k){\phi }_k^2(x,y)$. Here k*, μ(k) and θ(k) are as defined in assumptions A.3–A.5, and ${\phi }_k^2(x,y)=\text{cov}[{\sum }_{l=1}^{k}I(F({\tilde{X}}_l)\le x),{\sum }_{l=1}^{k}I(F({\tilde{X}}_l)\le y)]$. The following result is proven in Assaad (2012) by essentially proceeding along the lines of Olsson and Rootzen (1996, thm. 3.1).

Lemma 1

Suppose that the assumptions A.1–A.5 hold. Then, for U_n defined in (31), we have: $n^{1/2}(U_n-U)\stackrel{d}{\to }𝔾$ in 𝔻[0, 1].

Next, it is well-known that U_n is not a random element of 𝔻[0, 1] as this space when equipped with ‖·‖_∞ norm is complete but not separable (Fernholz 1983, chap. 4). We deal with this difficulty as in Fernholz by studying a continuous version $U_n^{*}$ of U_n. Let Y₍₀₎ = 0, Y₍₁₎ = F(X₍₁₎), …, Y_(N) = F(X_(N)), Y_(N+1) = 1. The intervals [Y_(i−1), Y_(i)], i = 1, …, N + 1, form a partition of [0, 1]. Next, let p_i−1 be an arbitrary probability mass that is less than the weight of X_(i), i = 1, …, N, and take p_N = 1 − (p₀ + … + p_N−1) so that ${\sum }_{i=0}^N{p}_i=1$. Define

$$U_n^{*}(x)=({\overline{p}}_{i-2}+p_{i-1}\frac{x-Y_{(i-1)}}{Y_{(i)-}{Y}_{(i-1)}})I_{[Y_{(i-1)},Y_{(i)}]}(x),$$ (34)

with ${\overline{p}}_j={\sum }_{i=0}^j{p}_i$. This $U_n^{*}$ is continuous since Y_(i) ≠ Y_(j) for i ≠ j (with probability 1). Essentially this $U_n^{*}$ distributes the probability mass p_i−1 uniformly in interval i for each i. The way p_i−1 are defined ensures:

$${\Vert U_n^{*}-U_n\Vert }_{\infty }\le \underset{i=1,\dots ,n}{\text{max}}{w}_i=\underset{k=1,\dots ,k^{*}}{\text{max}}w(k)\hspace{1em}(a.s.).$$ (35)

Let ℂ[0, 1] denote the space of continuous functions on [0, 1] equipped with the sup-norm ‖·‖_∞. Since this space is complete and separable, Billingsley (1968, p. 84) and (34) imply that $U_n^{*}$ is a random element of ℂ[0, 1]. In addition, from (35) and the assumption A.5, it can be seen that:

$$n^{1/2}{\Vert U_n-U_n^{*}\Vert }_{\infty }=o_{p_{*}}(1).$$ (36)

Here we use the inner probability measure P_* corresponding to P instead of P as U_n and hence $n^{1/2}(U_n-U_n^{*})$ is not a random element of 𝔻[0, 1]. We can now state the following results.

Lemma 2

The random element $n^{1/2}(U_n^{*}-U)$ is tight in ℂ[0, 1].

Proof

The fact that $n^{1/2}(U_n-U)\stackrel{d}{\to }𝔾$ in 𝔻[0, 1] (by Lemma 1) implies from (36) and van der Vaart (1998, thm 18.10 (iv)) that $n^{1/2}(U_n^{*}-U)\stackrel{d}{\to }𝔾$ in ℂ[0, 1]. Therefore, $n^{1/2}(U_n^{*}-U)$ is relatively compact. Now the tightness follows from Prohorov’s theorem as ℂ[0, 1] is separable and complete.

Lemma 3

∀ε > 0, ∃ a compact set K ⊂ 𝔻[0, 1], M > 0 and n₀ ∈ ℕ such that ∀n ≥ n₀, we have:

$$P_{*}\{d_{\infty }(\sqrt{n}(U_n-U),K)\le M/n^{1/2}\}>1-\epsilon ,$$

where d_∞(H, K) = inf_E∈K ‖H − E‖_∞ for H ∈ 𝔻[0, 1] and K ⊂ 𝔻[0, 1].

Proof

From (35) and A.5, ∃ M > 0 and n₀ ∈ ℕ such that ${\Vert U_n-U_n^{*}\Vert }_{\infty }<M/n$, almost everywhere ∀n ≥ n₀. Further, by Lemma 2, ∃ a compact set K ⊂ ℂ[0, 1] such that ∀n:

$$P_{*}[n^{1/2}(U_n^{*}-U)\in K]>1-\epsilon .$$ (37)

This K is also compact in 𝔻[0, 1] as ℂ[0, 1] ⊂ 𝔻[0, 1]. Now, define the events $A=\{n^{1/2}(U_n^{*}-U)\in K\},B=\{{\Vert U_n^{*}-U_n\Vert }_{\infty }<M/n\}$ and C = {d_∞(n^1/2[U_n − U], K) ≤ M/n^1/2}. The event A ∩ B is a subset of the event C because if A and B occur, then

$$d_{\infty }(n^{1/2}[U_n-U],K)\le d_{\infty }(n^{1/2}[U_n-U],n^{1/2}[U_n^{*}-U])=n^{1/2}{\Vert U_n^{*}-U_n\Vert }_{\infty }<M/n^{1/2}.$$

The result now follows from (37) by noticing that P_* (A ∩ B) = P_*(A) for all n ≥ n₀.

Next, we state a result of Fernholz (1983) after making minor modifications to it to suit our purpose.

Lemma 4

[Fernholz (1983, lem. 4.3.1)] Let Q : 𝔻[0, 1] × ℝ ↦ ℝ and suppose that for any compact set K ⊂ 𝔻[0, 1], lim_t→0 Q(H, t) = 0 uniformly for H ∈ K. Let ε > 0, and let δ_n be a sequence of numbers with δ_n ↓ 0. Then for any compact set K ⊂ 𝔻[0, 1], ∃n₀ such that ∀n ≥ n₀, if d_∞(H, K) ≤ δ_n then |Q(H, rδ_n)| < ε, for any constant r ∈ ℝ.

We are now ready to prove Proposition 4.

Proof of Proposition 4

Let ε > 0, and C_n be the event {d_∞(n^1/2[U_n − U], K) ≤ M/n^1/2}. From Lemma 3, we can choose a compact set K ⊂ 𝔻[0, 1] and M > 0 such that P_*(C_n) > 1 − ε/2, ∀n ≥ n₀. Further, since P_* is an inner probability measure, we can find measurable sets E_n such that E_n ⊂ C_n and P(E_n) > P_*(C_n) − ε/2. Thus, we have:

$$P(E_n)>P_{*}(C_n)-\epsilon /2>1-\epsilon ,\forall n\ge n_0.$$ (38)

Next, let $\text{Rem}(H)=\tau (U+H)-\tau (U)-{\tau }_U^{\prime }(H)$. The Hadamard differentiability of τ at U implies that Rem(tH)/t → 0 as t → 0, uniformly for H ∈ K found earlier. Now, upon applying Lemma 4 with

$$Q(H,t)=\text{Rem}(tH)/t,{\delta }_n=M/n^{1/2},r=1/M\text{ and }H=n^{1/2}(U_n-U),$$

we can find n₁ such that ∀n > n₁, d_∞(n^1/2[U_n − U], K) ≤ M/n^1/2 implies |Q(n^1/2[U_n − U], 1/n^1/2)| < ε. Therefore, for n > n₂ = max{n₀, n₁}, we have:

$$P_{*}\{|Q(n^{1/2}[U_n-U],1/n^{1/2})|<\epsilon \}=P_{*}\{n^{1/2}|\text{Rem}(U_n-U)|<\epsilon \}=P\{n^{1/2}|\text{Rem}(U_n-U)|<\epsilon \}\ge P_{*}(C_n)>P(E_n)>1-\epsilon ,$$

where the second equality is due to the fact that Rem(U_n − U) is a random element of 𝔻[0, 1] even though U_n is not (see Fernholz 1983, p. 40), and the last inequality is from (38). Hence, $n^{1/2}\text{Rem}(U_n-U)\stackrel{p}{\to }0$.

8.3 Proof of Proposition 3

As seen next, the result in Proposition 3 readily follows from the Bahadur representation in Theorem 3.

Proof of Proposition 3

For l = 1, …, r, define:

$${\mathrm{\Delta }}_{2n,l}={\hat{Q}}_{p_l}-Q_{p_l}-\{p_l-F_n(Q_{p_l})\}/f(Q_{p_l})={\hat{Q}}_{p_l}-Q_{p_l}-\sum _{i=1}^n{w}_i\sum _{j=1}^{k_i}IC(X_{ij},F,Q_{p_l}),$$

where the last equality follows from (16). Using (29), we can write, ${\mathrm{\Delta }}_{2n}={\sum }_{l=1}^r{a}_l{\mathrm{\Delta }}_{2n,l}$. Next, for each l, taking the constant sequence p⁽ⁿ⁾ = p_l ∀n in Theorem 3 yields n^1/2Δ_2n,l = o_p(1). This implies n^1/2Δ_2n = o_p(1) and hence the result holds.

8.4 Proof of Theorem 2

We first present two results that are needed for proving Theorem 2.

Lemma 5

Under the assumptions of Theorem 2, ${\hat{r}}_n^2\stackrel{p}{\to }{\sigma }^2{f}^2(Q_p)$ as n → ∞, where σ² is given by (17).

Proof

It suffices to show that $\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)\stackrel{p}{\to }\rho (Q_p,Q_p)$, defined by (4), since then

$${\hat{r}}_n^2=p(1-p)\sum _{k=1}^{k^{*}}k\{1+(k-1)\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)\}{\mu }_n(k){\{nw(k)\}}^2\stackrel{p}{\to }{\sigma }^2{f}^2(Q_p).$$

Under the assumptions, ρ(Q_p,Q_p) is continuous at (Q_p,Q_p). In addition, as $n^{1/2}({\hat{Q}}_p-Q_p)\stackrel{d}{\to }N(0,{\sigma }^2)$ from Theorem 1, we have Q̂_p = O_p(1), implying that for a given ε > 0, ∃M_ε > 0 such that

$$\underset{n\to \infty }{\text{lim}}P(|{\hat{Q}}_p|\le M_{\epsilon })\to 1.$$ (39)

To prove the convergence of ρ̂, note that

$$|\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)-\rho (Q_p,Q_p)|\le |\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)-\rho ({\hat{Q}}_p,{\hat{Q}}_p)|+|\rho ({\hat{Q}}_p,{\hat{Q}}_p)-\rho (Q_p,Q_p)|.$$

The second term on the right goes to zero in probability due to the continuity of ρ. Thus it just remains to show that the first term also goes to zero in probability. To see this, we have for ε > 0,

$$P(|\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)-\rho ({\hat{Q}}_p,{\hat{Q}}_p)|>\epsilon )=P(|\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)-\rho ({\hat{Q}}_p,{\hat{Q}}_p)|>\epsilon ,|{\hat{Q}}_p|\le M_{\epsilon })+P(|\hat{\rho }({\hat{Q}}_p,{\hat{Q}}_p)-\rho ({\hat{Q}}_p,{\hat{Q}}_p)|>\epsilon ,|{\hat{Q}}_p|>M_{\epsilon })\le P(\underset{|x|\le M_{\epsilon }}{\text{sup}}|\hat{\rho }(x,x)-\rho (x,x)|>\epsilon )+P(|{\hat{Q}}_p|>M_{\epsilon }).$$

The first term on the right goes to zero from Olsson and Rootzen (1996, p. 1563). The second term goes to zero from (39). This establishes the result.

Lemma 6

Suppose the assumptions of Theorem 2 hold.

1. Let p⁽ⁿ⁾ be a sequence of probabilities such that p⁽ⁿ⁾ = p + c/n^1/2 + o(1/n^1/2). Then as n → ∞, $n^{1/2}({\hat{Q}}_{p^{(n)}}-{\hat{Q}}_p)\stackrel{p}{\to }c/f(Q_p)$.

2. Let p⁽ⁿ⁾ be a sequence of probabilities such that p̂⁽ⁿ⁾ = p + ĉ_n/n^1/2, where ${ĉ}_n\stackrel{p}{\to }c$. Then as n → ∞, $n^{1/2}({\hat{Q}}_{{\hat{p}}^{(n)}}-{\hat{Q}}_p)\stackrel{p}{\to }c/f(Q_p)$.

Proof

The part (a) can be proved by adapting van der Vaart (1998, lem. 21.7) to deal with repeated measurements (see Assaad 2012). Here we focus on using (a) to prove (b). Fix ε > 0 and consider,

$$P(n^{1/2}|{\hat{p}}^{(n)}-p^{(n)}|\le \epsilon )=P(p^{(n)}-\epsilon /n^{1/2}\le {\hat{p}}^{(n)}\le p^{(n)}+\epsilon /n^{1/2})\le P({\hat{Q}}_{p^{(n)}-\epsilon /n^{1/2}}\le {\hat{Q}}_{{\hat{p}}^{(n)}}\le {\hat{Q}}_{p^{(n)}+\epsilon /n^{1/2}}).$$

The probabilities above go to one since $n^{1/2}({\hat{p}}^{(n)}-p^{(n)})={ĉ}_n-c+o(1)\stackrel{p}{\to }0$. As a result,

$$\underset{n\to \infty }{\text{lim}}P\{n^{1/2}({\hat{Q}}_{p^{(n)}-\epsilon /n^{1/2}}-{\hat{Q}}_p)\le n^{1/2}({\hat{Q}}_{{\hat{p}}^{(n)}}-{\hat{Q}}_p)\le n^{1/2}({\hat{Q}}_{p^{(n)}+\epsilon /n^{1/2}}-{\hat{Q}}_p)\}=1.$$ (40)

Next, we can deduce from (a) that $n^{1/2}({\hat{Q}}_{p^{(n)}-\epsilon /n^{1/2}}-{\hat{Q}}_p)\stackrel{p}{\to }(c-\epsilon )/f(Q_p)$ and $n^{1/2}({\hat{Q}}_{p^{(n)}+\epsilon /n^{1/2}}-{\hat{Q}}_p)\stackrel{p}{\to }(c+\epsilon )/f(Q_p)$. Therefore,

$$\underset{n\to \infty }{\text{lim}}P(n^{1/2}({\hat{Q}}_{p^{(n)}-\epsilon /n^{1/2}}-{\hat{Q}}_p)-(c-\epsilon )/f(Q_p)\ge -\epsilon )=1,$$ (41)

$$\underset{n\to \infty }{\text{lim}}P(n^{1/2}({\hat{Q}}_{p^{(n)}+\epsilon /n^{1/2}}-{\hat{Q}}_p)-(c+\epsilon )/f(Q_p)\le \epsilon )=1.$$ (42)

Let A_n, B_n and C_n denote the events in (40), (41) and (42), respectively. Notice that the event A_n ∩ B_n ∩ C_n implies the event

$$-\epsilon \{1+1/f(Q_p)\}\le n^{1/2}({\hat{Q}}_{{\hat{p}}^{(n)}}-{\hat{Q}}_p)-c/f(Q_p)\le \epsilon \{1+1/f(Q_p)\}.$$

From Lehmann (1999, lem. 2.1.2), its probability goes to one since each of the three probabilities, P(A_n), P(B_n) and P(C_n), goes to one. This establishes the result as ε > 0 is arbitrary.

We are now ready to prove Theorem 2.

Proof of Theorem 2

We can write the coverage probability as

$$P({\hat{Q}}_{{\hat{l}}_n}\le Q_p\le {\hat{Q}}_{{û}_n})=P\{n^{1/2}({\hat{Q}}_p-{\hat{Q}}_{{û}_n})\le n^{1/2}({\hat{Q}}_p-Q_p)\le n^{1/2}({\hat{Q}}_p-{\hat{Q}}_{{\hat{l}}_n})\}.$$

From Theorem 1, we know that $n^{1/2}({\hat{Q}}_p-Q_p)\stackrel{d}{\to }N(0,{\sigma }^2)$. Therefore, it suffices to show that $n^{1/2}({\hat{Q}}_{{û}_n}-{\hat{Q}}_p)\stackrel{p}{\to }{z}_{1-\beta /2}\sigma$ and $n^{1/2}({\hat{Q}}_{{\hat{l}}_n}-{\hat{Q}}_p)\stackrel{p}{\to }{z}_{1-\beta /2}\sigma$ as then the result follows from Slutsky’s theorem. To get the limits of the Q̂ differences, take ĉ_n = z_1−β/2r̂_n so that l̂_n = p − ĉ_n/n^1/2 and û_n = p + ĉ_n/n^1/2. Next, an application of Lemma 5 gives ${ĉ}_n\stackrel{p}{\to }{z}_{1-\beta /2}\sigma f(Q_p)$. The desired result now follows from part (b) of Lemma 6 upon taking p̂⁽ⁿ⁾ = l̂_n and p̂⁽ⁿ⁾ = û_n.

8.5 Proof of Theorem 3

The following lemma is needed to prove Theorem 3.

Lemma 7

[Ghosh, 1971] Let {V_n} and {W_n} be two sequences of random variables satisfying the following conditions:

$$W_n=O_p(1);\mathit{\text{ and }}\forall t\text{ and }\forall \epsilon >0,\underset{n\to \infty }{\text{lim}}P(V_n\le t,W_n\ge t+\epsilon )=0,\underset{n\to \infty }{\text{lim}}P(W_n\le t,V_n\ge t+\epsilon )=0.$$ (43)

Then V_n − W_n = o_p(1).

Proof of Theorem 3

We proceed along the lines of Ghosh (1971) to get this result. Let γ_n = Q_p+(p⁽ⁿ⁾− p)/f(Q_p), V_n = n^1/2(Q̂_p⁽ⁿ⁾ − γ_n) and W_n = n^1/2{p − F_n(Q_p)}/f(Q_p). Since

$$V_n-W_n=n^{1/2}({\hat{Q}}_{p^{(n)}}-Q_p)-n^{1/2}\{p^{(n)}-F_n(Q_p)\}/f(Q_p),$$

it is enough to verify that V_n and W_n satisfy (43) as then the result is an immediate consequence of Lemma 7.

From (16), we can write $W_n=n^{1/2}{\sum }_{i=1}^n{w}_i{\sum }_{j=1}^{k_i}IC(X_{ij},F,Q_p)$. This W_n can be shown to be asymptotically normal by proceeding as in Proposition 1. Thus, W_n = O_p(1). Next, for a given t, let

$$Z_{t,n}=n^{1/2}\{F({\gamma }_n+t/n^{1/2})-F_n({\gamma }_n+t/n^{1/2})\}/f(Q_p),t_n=n^{1/2}\{F({\gamma }_n+t/n^{1/2})-p^{(n)}\}/f(Q_p).$$

It can be seen that the event {V_n ≤ t} ⊂ {Z_t,n ≤ t_n}, and lim_n→∞ t_n = t as n^1/2(p⁽ⁿ⁾ − p) = O(1). Moreover, the random variable Z_t,n − W_n has mean zero and variance

$$E{[Z_{t,n}-W_n]}^2=\frac{n}{f^2(Q_p)}\text{var}[F_n(Q_p)-F_n({\gamma }_n+t/n^{1/2})]=\frac{n}{f^2(Q_p)}\sum _{i=1}^n{w}_i^2{J}_n(k_i),$$ (44)

where $J_n(k_i)=\text{Var}[{\sum }_{j=1}^{k_i}\{{\delta }_{X_{ij}}(Q_p)-{\delta }_{X_{ij}}({\gamma }_n+t/n^{1/2})\}]$, which, from A.2, can be written as

$$k_i\text{var}[{\delta }_{{\overline{X}}_1}(Q_p)-{\delta }_{{\overline{X}}_1}({\gamma }_n+t/n^{1/2})]+2\left(\begin{array}{c}\hfill k_i\hfill \\ \hfill 2\hfill \end{array}\right)\text{cov}[{\delta }_{{\overline{X}}_1}(Q_p)-{\delta }_{{\overline{X}}_1}({\gamma }_n+t/n^{1/2}),{\delta }_{{\overline{X}}_2}(Q_p)-{\delta }_{{\overline{X}}_2}({\gamma }_n+t/n^{1/2})].$$

Upon simplifying it using the facts that var[δ_X̃1 (a)] = F(a){1 − F(a)} and cov[δ_X̃1 (a), δ_X̃2 (b)] = G(a, b) − F(a)F(b), and applying continuity of G at (Q_p,Q_p), we get lim_n→∞ J_n(k_i) = 0. Next, by writing (44) as

$$E{[Z_{t,n}-W_n]}^2=\frac{1}{f^2(Q_p)}\sum _{k=1}^{k^{*}}{\{nw(k)\}}^2{\mu }_n(k)J_n(k),$$

it follows from A.4 and A.5 that lim_n→∞ E[Z_t,n − W_n]² = 0. Therefore, Z_t,n − W_n = o_p(1). This together with t_n → t imply that P(Z_t,n ≤ t_n, W_n ≥ t+ε) → 0, where ε > 0. Further, since {V_n ≤ t} ⊂ {Z_t,n ≤ t_n}, we can deduce that P(V_n ≤ t, W_n ≥ t+ε) → 0, ∀t, ε > 0. A similar argument shows that P(W_n ≤ t, V_n ≥ t+ε) → 0, ∀t, ε > 0. Thus, V_n and W_n satisfy conditions (43) of Lemma 7, which completes the proof.

8.6 Proof of Theorem 4

To prove (a), take r = 2 in the general L-statistic formula (7) and set the continuous part T₁ equal to zero to get T(F_n) = a₁Q̂_p1 + a₂ Q̂_p2, a₁, a₂ ∈ ℝ, (a₁, a₂) ≠ (0, 0). In this case, T(F) = a₁Q_p1 + a₂Q_p2. From Theorem 1, $n^{1/2}[T(F_n)-T(F)]\stackrel{d}{\to }N(0,{\sigma }^2)$, where σ², obtained using (11), (12) and (13), can be written as

$${\sigma }^2=a_1^2\frac{{\nu }_1^2(p_1)}{f^2(Q_{p_1})}+2a_1{a}_2\frac{{\nu }_{12}(p_1{p}_2)}{f(Q_{p_1})f(Q_{p_2})}+a_2^2\frac{{\nu }_2^2(p_2)}{f^2(Q_{p_2})},$$

with ν₁, ν₂ and ν₁₂ as defined in (22). Since this result holds for any (a₁, a₂) ≠ (0, 0), we can deduce from the Cramer-Wold device (van der Vaart 1998, p. 16) that n^1/2(Q̂_p1 − Q_p1, Q̂_p2 − Q_p2) jointly converges in distribution to a bivariate normal distribution with mean (0, 0), variance $({\nu }_1^2(p_1)/f^2(Q_{p_1}),{\nu }_2^2(p_2)/f^2(Q_{p_2}))$ and covariance ν₁₂(p₁, p₂)/{f(Q_p1)f(Q_p2)}. Next, take h(Q_p1, Q_p2) = F(Q_p2) − F(Q_p1) so that n^1/2[F(Q̂_p2) − F(Q̂_p1) − (p₂ − p₁)] = n^1/2[h(Q̂_p1, Q̂_p2) − h(Q_p1, Q_p2)]. From the bivariate delta method (Lehmann 1999, p. 295), this quantity converges in distribution to N(0, ν²(p₁, p₂)), completing the proof.

To prove (b), note that from Theorem 1, $n^{1/2}({\hat{Q}}_{p_l}-Q_{p_l})\stackrel{d}{\to }N(0,{\nu }_l^2(p_l)/f^2(Q_{p_l}))$, l = 1, 2. It now follows from the usual delta method that $n^{1/2}(F({\hat{Q}}_{p_l})-p_l)=n^{1/2}(F({\hat{Q}}_{p_l})-F(Q_{p_l}))\stackrel{d}{\to }N(0,{\nu }_l^2(p_l))$, l = 1, 2.