Efficiency of the u,v method of estimation

Abstract

Given a pool of motorists, how do we estimate the total intensity of those who had a prespecified number of traffic accidents in the past year? We previously have proposed the u,v method as a solution to estimation problems of this type. In this paper, we prove that the u,v method provides asymptotically efficient estimators in an important special case.

1. The u,v Method

Given a pool of motorists, how do we estimate the total intensity of those in the pool who had a prespecified number of traffic accidents in a given time period? We may also consider patients with a prespecified number of heart attacks, or salesmen with a prespecified number of disgruntled customers, etc. In general, let θ_i be the intensity and X_i the number of occurrences of certain type of events of the ith individual in a pool of size n. Suppose that for 1 ≤ i ≤ n conditionally on θ_i, X_i has the Poisson distribution with . We are interested in estimating the sum

1.1

where u(x) is a known “utility function” dictated by practical considerations. In the examples above, S_n is the sum of the intensity θ_i for those individuals with X_i = a traffic accidents (heart attacks, disgruntled customers, etc.) for a prespecified integer a, if

1.2

Robbins (1) considered estimation of the sum in 1.1 and certain other related quantities for general, but known, conditional distributions F(x|y) of X_i given θ_i = y. The solution he proposed, called the u,v method, estimates S_n by

1.3

if there exists a function v(x) such that

1.4

In the Poisson case, Eq. 1.4 has the unique solution

1.5

provided that Σ_x=0^∞|u(x)|y^x/x! < ∞ for all y > 0.

In this paper, we consider the asymptotic efficiency of the u,v method. We prove the asymptotic efficiency of 1.3 for the estimation of 1.1 in the special case of Eq. 1.2 in the Poisson setting in Section 2. In Section 3, we discuss related problems and extensions to the estimation of the sums of u(X_i, θ_i) for general utility functions u(x, y) and general conditional distributions F(x|y).

2. The Poisson Case

Let f(x|y) ≡ e^−yy^x/x!, x = 0, 1, 2, … , be the Poisson probability mass function with intensity y > 0 and 𝒢 be a known family of probability distributions with support (0, ∞). Suppose (X, θ), (X_i, θ_i), are independent identically distributed random vectors such that

2.1

where G ∈ 𝒢 is an unknown distribution. We consider in this section estimation of

2.2

with the u_a in Eq. 1.2 for a given a. By the u,v method, 2.2 should be estimated by

2.3

as in 1.3 and Eq. 1.5. For example, according to 2.3, the total intensity of those motorists with no traffic accidents in the past year is estimated by the total number of motorists with exactly one accident in the past year.

The estimator 2.3 also can be derived from an empirical Bayes point of view. If the distribution G in 2.1 is known, then the Bayes estimator of 2.2 under the squared error loss is the conditional expectation

which can be written as

2.4

with

2.5

where f_G(x) ≡ ∫ f(x|y)dG(y) is the marginal probability mass function of X. An empirical Bayes estimator of 2.2 can be obtained by substituting the conditional expectation τ_a(G) with a suitable estimator, say τ̃_a,n, in the Bayes estimator S_n,G in 2.4; i.e.

2.6

If G is completely unknown, we may estimate f_G(x) by its empirical version and consequently estimate τ_a(G) by

2.7

This leads to the estimator 2.3 via

The relationship 2.6 can be reversed to derive estimates of τ_a(G) from those of 2.2, say S̃_n ≡ S̃_n(X₁, … , X_n); i.e.

2.8

This provides a vehicle for the investigation of the efficiency of S̃_n via the efficiency of τ̃_a,n. Let H_* ≡ H_∗,G be the tangent space of the family {f_G∶G ∈ 𝒢} at G,

2.9

where 𝒞_G is the collection of all “differentiable” paths η∶[0, 1] → 𝒢 satisfying

2.10

with the f_G in Eq. 2.5, and

2.11

is the score function for the path η in the parameter space 𝒢. See Bickel et al. (2). Define

2.12

with the u_a in Eq. 1.2. It will be shown in the proof of Theorem 2.1 that at each G ∈ 𝒢 the efficient influence function for the estimation of τ_a(·) is

2.13

where H_* is the tangent space given in 2.9.

Theorem 2.1 (i) A sequence {S̃_n ≡ S̃_n(X₁, … , X_n)} is asymptotically efficient for the estimation of the {S_n} in 2.2 if and only if {τ̃_a,n} in 2.8 is asymptotically efficient for the estimation of the functional τ_a(G) in 2.5. In this case,

2.14

where σ₁²(G) ≡ E_Gψ_*²(X; G) with the ψ_* in 2.13, σ₂²(G) ≡ E_Gu_a(X)Var_G(θ|X) with the u_a in Eq. 1.2, and f_G is the marginal probability mass function of X. (ii) If G is completely unknown, i.e., 𝒢 = {all distributions in (0, ∞)}, then {V_n} in 2.3 is asymptotically efficient for the estimation of 2.2 and

2.15

Proof: The proof has three parts.

Step 1. Decomposition of (S̃_n − S_n)/: By 2.8 and 2.4

2.16

where f̂_n(a) is as in 2.7, ξ_n,1 ≡ {τ̃_a,n − τ_a(G)} and ξ_n,2 ≡ {S_n,G − S_n}/. Conditionally on {X_i, i ≥ 1}, S_n,G − S_n are sums of independent (not identically distributed) random variables with mean zero, so that by the Lindeberg central limit theorem and the law of large numbers

2.17

almost surely for all {X_i, i ≥ 1}. The Lindeberg condition can be verified by the law of large numbers, but we shall omit the details. Because the limiting distribution in Eq. 2.17 does not depend on {X_i, i ≥ 1} and f̂_n(a) → f_G(a), by Eq. 2.16

2.18

provided that either (S̃_n − S_n)/ or ξ_1,n ≡ {τ̃_a,n − τ_a(G)} are stochastically bounded, where ℒ(Z; P) is the distribution of Z under probability P and ★ stands for convolution. Thus, {S̃_n} is asymptotically efficient for the estimation of S_n if and only if {τ̃_a,n} is asymptotically efficient for the estimation of τ_a(G).

Step 2. Efficient influence function for the estimation of τ_a(G): It follows from the information bound in standard semiparametric estimation theory that the limiting distribution of asymptotically efficient {τ̃_a,n} is

2.19

provided that ψ_* is the efficient influence function for the estimation of τ_a(G). By 2.13, this is the case if for all η ∈ 𝒞_G

2.20

where ρ_η is as in Eq. 2.11. See ref. 2. Thus, it suffices to verify Eq. 2.20 for the proof of Theorem 2.1 part i.

Because f_G(x) > 0 for all x ≥ 0, by (2.11) t⁻¹{f_η(t)(x) − f_G(x)} → f_G(x)ρ_η(x), so that by Eq. 2.5 and 2.12

Therefore, Eq. 2.20 holds.

Step 3. Asymptotic efficiency of the u,v method: Let ψ be as in 2.12 and τ̂_a,n be as in 2.7. By the central limit theorem and the strong law of large numbers, (τ̂_a,n − τ_a(G)) converges in distribution to N(0, E_Gψ²(X; G)). Because V_n is the estimator of S_n corresponding to τ̂_a,n by 2.8, it suffices to show ψ = ψ_* in view of Theorem 2.1 part i and its proof.

For y > 0 define η(t) ≡ (1 − t)G + tδ_y, where δ_y puts the whole mass at y. Set ρ_(y)(x) ≡ {f(x|y) − f_G(x)}/f_G(x). Then, E_Gρ_(y)²(X) < ∞ by the Poisson assumption, and the left-hand side of Eq. 2.10 is

Thus, ρ_(y) ≡ f(x|y)/f_G(x) − 1 is in the tangent space H_* for all y > 0 by 2.9. If h is orthogonal to H_* in L₂(f_G), then

for all y > 0, so that h(x) = E_Gh(X) for all x ≥ 0 by the completeness of the Poisson family. This implies H_* = L₂(f_G) ∩ {h∶E_Gh(X) = 0}. Hence, ψ_* = ψ by 2.13 and the proof is complete.

3. Discussion

3.1. Related Problems.

Let Y_i be random variables such that E[Y_i|θ_i, X_i] = λθ_i. Suppose Y_i are unobservable and λ is known. Consider the prediction of

3.1

based on observations X₁, … , X_n. For example, we may want to predict the total number of accidents in the coming year for the group of motorists with no accidents in the past year, with λ = 1.02 due to 2% growth of drivers in the region of concern. By the u,v method, 3.1 can be predicted by λV_n if Eq. 1.4 holds, with the V_n in 1.3. The argument in Section 2 still applies here in the Poisson case with u(x) = u_a(x) in Eq. 1.2: {λV_n}, with the V_n in 2.3, is asymptotically efficient for the prediction of 3.1 with

3.2

where σ₁²(G) is as in Theorem 2.1.

In many applications, Y_i are observable and the problem is to estimate λ. In this case, the u,v methodology provides the estimator

3.3

The u,v method also produces estimates of variances. For example, if Eq. 1.4 holds, the variance E_G(V_n − S_n)² = nE_G{v(X) − u(X)θ}² can be estimated by

3.4

with two applications of the u,v method, first to u₁ ≡ u² and then to u₂ ≡ v₁ − 2uv, where v_j satisfy ∫ (v_j(x) − u_j(x)y)F(dx|y) = 0, ∀y.

The u,v method can further be extended to obtain unbiased estimation of

3.5

If there exist functions v_i(x₁, … , x_n) satisfying

for all y, i ≤ n and {x_j, j ≠ i}, then we may estimate 3.5 by

3.6

For example, in the exponential case f(x|y) ≡ y⁻¹e^−x/y1{x > 0}, the θ_i associated with the largest observation can be written as 3.5,

and its unbiased estimation 3.6 is V_n = X_Rn − X_Rn−1 with

where R_i are the antiranks of the observations defined by X_R1 < ⋯ < X_Rn.

The related problems mentioned here and their applications were considered in refs. 1 and 3–5.

3.2. Extensions.

The applicability of our methodology is not limited to the sum of u(X_i)θ_i in 1.1. In general, 1.3 can be used to estimate

3.7

for any integrable functions u(x, y), as long as

3.8

In fact, for the estimation of variance in 3.4, Eq. 3.8 holds for the pair ũ(x, y) and ṽ(x), with ũ(x, y) ≡ {v(x) − u(x)y}² and ṽ(x) ≡ v²(x) + v₂(x).

The asymptotic theory for the estimation of 3.7 is more complicated and will be studied elsewhere. Define

3.9

The asymptotic independence of (S̃_n − S_n,G)/ and (S_n,G − S_n)/ can still be derived from the Lindeberg central limit theorem and the strong law of large numbers as in Section 2, but the rest of the argument there does not directly apply without the one-to-one linear mappings between estimates of S_n and τ_a(G) in 2.6 and 2.8.