Goodness of fit test for uniform distribution with censored observation

Abstract

We develop new goodness of fit test for uniform distribution based on a conditional moment characterization. We study the asymptotic properties of the proposed test statistic. We also present a goodness of fit test for uniform distribution to incorporate the right censored observations and studied its properties. A Monte Carlo simulation study is carried out to evaluate the finite sample performance of the proposed tests. We illustrate the test procedures using real data sets.

Introduction

Uniform distribution is a widely used statistical model in various fields of applied as well as theoretical statistics. Most of the Monte Carlo simulation algorithms use a sample from a uniform distribution to generate random samples from other distributions. In view of the probability integral transformation, the goodness of fit test for any particular continuous distribution is equivalent to the test that the transformed sample is from a uniform distribution in the interval [0, 1].

Tests for uniform distribution have great significance in all disciplines due to the property that the distribution with maximum entropy is standard uniform. Various tests were developed in the literature to test the hypothesis that the data comes from uniform distribution. Interested readers may refer to Kimball (1947), Sherman (1950), Quesenberry and Miller Jr. (1977), Hegazy and Green (1975), Young (1982), Cheng and Spiring (1987) and Frosini et al. (1987). Under the uniform distribution assumption, Asgharian et al. (2002) estimated the survival of dementia patients and de Una-Alvarez (2004) estimated the distribution of unemployment spells in women. Qiu and Jia (2018) and Noughabi (2021) proposed tests for uniformity, based on measures of extropy. Cho et al. (2021) developed a test for the uniformity of exchangeable random variables on the circle.

Uniform distribution is also used to model lifetime data. Weissman and Cohen (1996) discussed the estimation of uniformly distributed censored failure times. Liang (2008) considered the empirical Bayes testing and Liang and Huang (2009) studied the empirical Bayes estimation of the parameter $\theta$ based on randomly right censored data from $U(0,\theta )$. The maximum likelihood estimation of the uniformly distributed lifetimes with type I hybrid censoring is addressed by Górny and Cramer (2019). Yu (2021) discussed the maximum likelihood estimation of the uniformly distributed lifetime data under right censoring and illustrated the method using breast cancer data set. Another important application of uniform distribution in lifetime data models arises from the fact that the ordered lifetime from a particular distribution can be written as a quantile transformation of the corresponding data from a standard uniform distribution (Kamps & Cramer, 2001; Balakrishnan & Dembińska, 2008; Balakrishnan & Cramer, 2014).

Even though several goodness of fit tests have been proposed in the literature for uniform distribution, to the best of our knowledge these tests do not accommodate censored lifetimes. This motivates us to develop new goodness of fit test for uniform distribution for complete data as well as censored data based on a conditional moment characterization [see Ahmed (1991) and Ebner and Liebenberg (2021)].

The rest of the paper is organised as follows. In Sect. 2, we develop a new non-parametric test for uniform distribution. We study the asymptotic properties of the test statistic. In Sect. 3, we discuss how to incorporate the right censored observations in the proposed methodology. The results of a Monte Carlo simulation study are reported in Sect. 4. In Sect. 5, the test procedures are illustrated with application on real data sets. Finally, in Sect. 6, we give some concluding remarks.

Test statistic

In this section, we develop a goodness of fit test for the uniform distribution for complete data. We use a conditional moment characterization for developing the test and we state the result for sake of completeness.

Theorem 1

(Ebner & Liebenberg, 2021) Let X be a random variable taking values in [0, 1]. Then $X\sim B(\alpha ,\beta )$ for $\alpha ,\beta >0$ if and only if

$$\begin{array}{c}\hfill (\alpha +\beta )E(XI(X>t))=\alpha E(I(X>t))+\frac{t^{\alpha }{(1-t)}^{\beta }}{B(\alpha ,\beta )},0\le t\le 1,\end{array}$$

where I(A) denotes the indicator function of a set A.

Corollary 1

Let X be a random variable taking values in [0, 1]. Then $X\sim U[0,1]$ if and only if

$$\begin{array}{c}\hfill 2E(XI(X>t))=E(I(X>t))+t(1-t),0\le t\le 1.\end{array}$$

Next, we discuss the proposed method. Based on a random sample $X_1,\dots ,X_n$ from F, we are interested in testing the null hypothesis

$$\begin{array}{c}\hfill H_0:F\in U[0,1]\end{array}$$

against the alternative

$$\begin{array}{c}\hfill H_1:F\notin U[0,1].\end{array}$$

To develop the test, we define a departure measure which discriminate between null and alternative hypothesis. Consider $\mathrm{\Delta }(F)$ given by

$$\begin{array}{cc}\hfill \mathrm{\Delta }(F)=& {\int }_0^1\left(2,E,(,X,I,(,X,>,t,),),-,E,(,I,(,X,>,t,),),-,t(1-t)\right)dF(t).\hfill \end{array}$$

In view of Corollary 1, $\mathrm{\Delta }(F)$ is zero under $H_0$ and non zero under $H_1$. Hence $\mathrm{\Delta }(F)$ can be considered as a measure of departure from $H_0$ towards $H_1$. Next, we express $\mathrm{\Delta }(F)$ in a simple form as

$$\begin{array}{cc}\hfill \mathrm{\Delta }(F)=& {\int }_0^1\left(2,E,(,X,I,(,X,>,t,),),-,E,(,I,(,X,>,t,),),-,t,(,1,-,t,)\right)dF(t)\hfill \\ \hfill =& 2{\int }_0^1{\int }_0^{1}yI(y>t)dF(y)dF(t)-E(X)-E(X(1-X))\hfill \\ \hfill =& 2{\int }_0^1{\int }_t^{1}ydF(y)dF(t)-2E(X)+E(X^2)\hfill \\ \hfill =& 2{\int }_0^{1}y{\int }_0^{y}dF(t)dF(y)-2E(X)+E(X^2)\hfill \\ \hfill =& {\int }_0^{1}2yF(y)dF(y)-2E(X)+E(X^2)\hfill \\ \hfill =& E(max(X_1,X_2)-2X+X^2).\hfill \end{array}$$ 1

Consider a symmetric kernel $h_1(X_1,X_2)=\frac{1}{2}\left(2,max,(X_1,X_2),-,2,X_1,-,2,X_2,+,X_1^2,+,X_2^2\right)$, then $E(h_1(X_1,X_2))=\mathrm{\Delta }(F)$. Hence we propose a U-statistic based test statistic given by

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }}=\frac{2}{n(n-1)}\sum _{i=1}^n\sum _{j=1,j<i}^n{h}_1(X_i,X_j),\end{array}$$

for testing uniform distribution. The $\widehat{\mathrm{\Delta }}$ is an unbiased and consistent estimator of $\mathrm{\Delta }(F)$ (Lehmann, 1951). Let $X_{(i)},i=1,\dots ,n$ be the order statistics based on a random sample of size n from F. Then we can represent $\widehat{\mathrm{\Delta }}$ in a simple form as

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }}=\frac{1}{n(n-1)}\sum _{i=1}^n\left(2,(i-n),+,(n-1),X_{(i)}\right)X_{(i)}.\end{array}$$

We reject the null hypothesis $H_0$ against the alternative for a large value of $|\widehat{\mathrm{\Delta }}|$. Next, we find an asymptotic critical region of the test using the normal approximation. The following result is useful in this direction.

Theorem 2

As $n\to \infty$, $\sqrt{n}(\widehat{\mathrm{\Delta }}-\mathrm{\Delta })$ converges in distribution to a Gaussian random variable with mean zero and variance ${\sigma }^2$, where ${\sigma }^2$ is given by

$$\begin{array}{c}\hfill {\sigma }^2=Var(2XF(X)+2{\int }_X^{1}ydF(y)-2X+X^2).\end{array}$$ 2

Proof

Asymptotic normality of $\widehat{\mathrm{\Delta }}$ follows from the central limit theorem for U-statistics. The asymptotic variance is $4{\sigma }_1^2$ where ${\sigma }_1^2$ is given by Lee (2019)

$$\begin{array}{c}\hfill {\sigma }_1^2=Var\left[E,\left(h_1,(X_1,X_2),|,X_1\right)\right].\end{array}$$ 3

Consider

$$\begin{array}{cc}\hfill E[max(X_1,X_2)|X_1=x]=& E[xI(X_2\le x)+X_{2}I(X_2>x)]\hfill \\ \hfill =& xP(X_2\le x)+{\int }_0^{1}yI(x<y)dF(y)\hfill \\ \hfill =& xF(x)+{\int }_x^{1}ydF(y).\hfill \end{array}$$

Therefore, from (3) we obtain the variance given in (2) and this completes the proof. $\square$

Under the null hypothesis $H_0$, we know that $\mathrm{\Delta }(F)=0$ and so we have the following result.

Corollary 2

Under $H_0$, as $n\to \infty$, $\sqrt{n}\widehat{\mathrm{\Delta }}$ converges in distribution to a Gaussian random variable with mean zero and variance $\frac{1}{45}$.

Proof

Under $H_0$, we have

$$\begin{array}{cc}\hfill {\sigma }^2=& Var(2XF(X)+2{\int }_X^{1}ydF(y)-2X+X^2)\hfill \\ \hfill =& Var(2X^2+2{\int }_X^{1}ydy-2X+X^2)\hfill \\ \hfill =& 4Var(X(X-1))=\frac{1}{45}.\hfill \end{array}$$

$\square$

A distribution-free test for testing uniform distribution can be constructed using Corollary 2. We reject $H_0$ against $H_1$ at a significance level $\alpha$, if

$$\begin{array}{c}\hfill \sqrt{45n}|\widehat{\mathrm{\Delta }}|>Z_{\alpha /2},\end{array}$$

where $Z_{\alpha }$ is the upper $\alpha$-percentile point of the standard normal distribution. The performance of the test is evaluated through a Monte Carlo simulation study and the result of the same is reported in Sect. 4.

Remark 2.1

When $X\sim U[0,\theta ]$, we can implement the test using the integral transformation $Y=X/\theta$. If $\theta$ is unknown it can be estimated from the given sample $X_1,\dots ,X_n$. Maximum likelihood estimator of $\theta$ is given by $\widehat{\theta }=max(X_1,\dots ,X_n)$.

Using Monte Carlo simulation, we checked whether the empirical type I error of the proposed test is maintainable. The result of the simulation study is reported in Table 1. For finding the empirical type I error, we generate samples of sizes $n=25,50,75$ and 100 from $U[0,\theta ]$ for different values of $\theta$.

Table 1.

Empirical type I error of the test

	U[0, 3]		U[0, 5]		U[0, 8]		U[0, 10]		U[0, 20]
$n/\alpha$	0.01	0.05	0.01	0.05	0.01	0.05	0.01	0.05	0.01	0.05
25	0.0135	0.0524	0.0129	0.0523	0.0124	0.0526	0.0124	0.0530	0.0128	0.0532
50	0.0117	0.0517	0.0118	0.0517	0.0117	0.0519	0.0116	0.0512	0.0119	0.0515
75	0.0113	0.0514	0.0112	0.0512	0.0111	0.0512	0.0111	0.0511	0.0109	0.0510
100	0.0105	0.0509	0.0098	0.0508	0.0105	0.0504	0.0101	0.0509	0.0104	0.0507

Test statistic for right censored case

Next, we discuss how to incorporate the randomly right censored observations in the proposed testing procedure. Consider the right-censored data $(Y,\delta )$, with $Y=min(X,C)$ and $\delta =I(X\le C)$, where C is the censoring time. We assume censoring times and lifetimes are independent. Now we need to address the testing problem discussed in Sect. 2 based on n independent and identical observations $\{(Y_i,{\delta }_i),1\le i\le n\}$. Let ${\widehat{K}}_c$ be the Kaplan-Meier of $K_c$, the survival function of C. We develop the test using the same departure measure $\mathrm{\Delta }(F)$ given in (1). Consider a symmetric kernel defined by $h(Y_1,Y_2)=\frac{1}{2}(2max(Y_1,Y_2)-2Y_1-2Y_2+Y_1^2+Y_2^2)$. In the right censored case, a U statistics-based test is given by Datta et al. (2010)

$$\begin{array}{c}\hfill {\widehat{\mathrm{\Delta }}}_c=\frac{2}{n(n-1)}\sum _{i=1}^n\sum _{j<i;j=1}^n\frac{h(Y_1,Y_2){\delta }_i{\delta }_j}{{\widehat{K}}_c(Y_i){\widehat{K}}_c(Y_j)},\end{array}$$

provided ${\widehat{K}}_c(Y_i)>0$ and ${\widehat{K}}_c(Y_j)>0$ for all $i<j$, $i=1,2,\dots ,n$ with probability 1. We reject $H_0$ in favour of $H_1$ for large values of $|{\widehat{\mathrm{\Delta }}}_c|$.

Now, we obtain the limiting distribution of ${\widehat{\mathrm{\Delta }}}_c$. Let $N_i^c(t)=I(Y_i\le t,{\delta }_i=0)$ be the counting process corresponds to the censoring variable $C_i$, $R_i(t)=I(Y_i\ge t)$. Also let ${\lambda }_c$ be the hazard rate of C. The martingale associated with counting process $N_i^c(t)$ is given by

$$\begin{array}{c}\hfill M_i^c(t)=N_i^c(t)-{\int }_0^t{R}_i(u){\lambda }_c(u)du.\end{array}$$

Denote $h_1(x)=E(h(X_1,X_2|X_1=x))$, $G(x,y)=P(X_1\le x,Y_1\le y,\delta =1),x\in [0,1]$, $\overline{H}(t)=P(Y_1>t)$ and

$$\begin{array}{c}\hfill w(t)=\frac{1}{\overline{H}(t)}{\int }_{{[0,1]}^2}\frac{h_1(x)}{K_c(y-)}I(y>t)dG(x,y).\end{array}$$

The proof of next theorem follows from Theorem 1 of Datta et al. (2010) for the choice of the kernel $h(Y_1,Y_2).$

Theorem 3

If the integrals ${\int }_{{[0,1]}^2}\frac{h_1^2(x)}{K_c^2(y)}dG(x,y)$ and ${\int }_0^1{w}^2(t){\lambda }_c(t)dt$ are finite, then as $n\to \infty$, $\sqrt{n}({\widehat{\mathrm{\Delta }}}_c-\mathrm{\Delta })$ converges in distribution to Gaussian with mean zero and variance $4{\sigma }_c^2$, where ${\sigma }_c^2$ is given by

$$\begin{array}{c}\hfill {\sigma }_c^2=Var\left(\frac{(2XF(X)+2{\int }_X^{1}ydF(y)-2X+X^2){\delta }_1}{2K_c(Y_1-)},+,\int ,w,(t),d,M_1^c,(t)\right).\end{array}$$

Corollary 3

Assume that the condition stated in Theorem 3 holds. Under $H_0$, as $n\to \infty$, the distribution of $\sqrt{n}\widehat{\mathrm{\Delta }}$ is Gaussian with mean zero and variance $4{\sigma }_{c0}^2$, where ${\sigma }_{c0}^2$ is given by

$$\begin{array}{c}\hfill {\sigma }_{c0}^2=Var(\frac{X(X-1){\delta }_1}{K_c(Y_1-)}+\int w(t)d{M}_1^c(t)).\end{array}$$

We, now find an estimator of ${\sigma }_{c0}^2$ using the reweighted techniques. An estimator of ${\sigma }_{c0}^2$ is given by

$$\begin{array}{c}\hfill {\widehat{\sigma }}_{c0}^2=\frac{4}{(n-1)}\sum _{i=1}^n{(V_i-\overline{V})}^2,\end{array}$$

where

$$\begin{array}{cc}\hfill V_i=& \frac{{\widehat{h}}_1(X_i){\delta }_i}{{\widehat{K}}_c(Y_i)}+\widehat{w}(X_i)(1-{\delta }_i)-\sum _{j=1}^n\frac{\widehat{w}(X_i)I(X_i>X_j)(1-{\delta }_i)}{{\sum }_{i=1}^{n}I(X_i>X_j)},\hfill \\ \hfill \overline{V}=& \frac{1}{n}\sum _{i=1}^n{V}_i,{\widehat{h}}_1(X)=\frac{1}{n}\sum _{i=1}^n\frac{h(X,Y_i){\delta }_i}{{\widehat{K}}_c(Y_i-)},R(t)=\frac{1}{n}\sum _{i=1}^{n}I(Y_i>t)\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \widehat{w}(t)=\frac{1}{R(t)}\sum _{i=1}^n\frac{{\widehat{h}}_1(X_i){\delta }_i}{{\widehat{K}}_c(Y_i)}I(X_i>t).\end{array}$$

Under right censored situation, we reject the null hypothesis $H_0$ against the alternative hypothesis $H_1$ at a significance level $\alpha$, if

$$\begin{array}{c}\hfill \frac{\sqrt{n}|{\widehat{\mathrm{\Delta }}}_c|}{{\widehat{\sigma }}_{0c}}>Z_{\alpha /2}.\end{array}$$

The results of a Monte Carlo Simulation which asses the finite sample performance of ${\widehat{\mathrm{\Delta }}}_c$ is also reported in Sect. 4.

Empirical evidence

To evaluate the finite sample performance of our test procedures, we conduct a Monte Carlo simulation study using R software. The simulation is repeated ten thousand times. For complete data, to show the competitiveness of our test, we compare the empirical power of our test with the existing test procedures.

Complete data

We find the empirical type I error and power of the proposed test for different choices of alternatives. We compare our test (SS) with Kolmogorov-Smirnov test (KS) as well as with the widely used tests for uniformity; Neyman-Barton test (1937) (NB), Sherman’s test (1950) (S), Kuiper test (1960) (K), Hegazy-Green test (1975) (HG), Quesenberry and Miller test (1977) (QM) and Frosini test (1987) (F). The R code for these tests are available in the R package ‘uniftest’. For finding the empirical type I error, we generate samples of sizes $n=25,50,75$ and 100 from U[0, 1]. We consider different choices of alternatives for finding the empirical power of the tests. The results of the simulation study are summarized in Table 1.

The comparative study shows the competitiveness of the newly proposed test to classical procedures. The empirical type I error of the test attains the nominal level and the test shows very good power. For the choice of alternative distribution U(0, 1.2) the test has good power, which shows that our test captures even a slight deviation from the null distribution. Even for small sample size $n=25$, we obtain high power which reaches to unity by $n=50$ for other choices of distributions.

Censored data

The performance of proposed test procedure incorporating right censored observations is also validated through a Monte Carlo simulation study. We consider a mild censoring situation where 20% of lifetimes are censored and heavy censored situation where 40% of the lifetimes are censored (Table 3). In censored case, we choose the sample sizes $n=50,75,100$ and 200 and simulation is repeated ten thousand times.

Table 3.

Empirical type I error and power the test with 20% and 40% censoring

	U(0, 1)		U(0, 1.2)		Exp(1)		Weibull(1, 2)		Pareto(1, 1)
$n/\alpha$	0.01	0.05	0.01	0.05	0.01	0.05	0.01	0.05	0.01	0.05
20% censoring
50	0.0088	0.0474	0.6993	0.8303	0.9999	1.0000	1.0000	1.0000	1.0000	1.0000
75	0.0096	0.0506	0.8635	0.9387	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
100	0.0106	0.0489	0.9397	0.9761	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
200	0.0106	0.0499	0.9984	0.9995	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
40% censoring
50	0.0079	0.0469	0.5174	0.6892	0.9771	0.9856	1.0000	1.0000	1.0000	1.0000
75	0.0091	0.0488	0.6537	0.8112	0.9972	0.9982	1.0000	1.0000	1.0000	1.0000
100	0.0109	0.0508	0.8025	0.9209	0.9996	0.9998	1.0000	1.0000	1.0000	1.0000
200	0.0103	0.0502	0.9839	0.9963	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

For finding the empirical type I error we simulate observations from U[0, 1]. Various choices of alternatives are considered for finding empirical power. In all these cases, we generate censored observations from U[0, c] where c is chosen in such way that Results of the simulation study with the right censored observations are given in Table 2. The empirical type I error reaches the chosen significance level in both mild and heavy censored situations. As in the case of complete data, the test statistic detects even a small deviation (U[0, 1.2]) from the null hypothesis. The empirical power of the test reaches one for other choices of the alternative distributions we considered even for sample size $n=50$.

Table 2.

Empirical type I error and power of the proposed test with other tests for uniformity

	n	$\alpha$	SS	KS	NB	S	K	HG	QM	F
U(0, 1)	25	0.01	0.0079	0.0086	0.0074	0.0077	0.0068	0.0056	0.0076	0.0082
	50		0.0082	0.0127	0.0085	0.009	0.0092	0.0077	0.0083	0.0087
	75		0.0094	0.0121	0.0115	0.011	0.0094	0.0086	0.0114	0.0092
	100		0.0107	0.0112	0.0109	0.011	0.0106	0.0111	0.0110	0.0108
	25	0.05	0.0398	0.0382	0.0428	0.0446	0.0416	0.0385	0.0535	0.0382
	50		0.0427	0.0429	0.0436	0.0512	0.0457	0.0536	0.0516	0.0525
	75		0.0478	0.0524	0.0473	0.0419	0.0538	0.0452	0.0418	0.0468
	100		0.0512	0.0516	0.0552	0.0493	0.0515	0.0525	0.0516	0.0489
U(0, 1.2)	25	0.01	0.5364	0.4562	0.9551	0.6465	0.1851	0.2576	0.6718	0.2459
	50		0.8212	0.6275	0.9669	0.9469	0.3165	0.4884	0.9488	0.4689
	75		0.9373	0.7864	0.9784	0.9892	0.4577	0.6826	0.9653	0.6781
	100		0.9764	0.8513	0.9886	0.9965	0.5287	0.7954	0.9899	0.7981
	25	0.05	0.6978	0.5074	0.9655	0.8135	0.3792	0.4467	0.8743	0.4354
	50		0.9156	0.7025	0.9799	0.9849	0.4697	0.7215	0.9697	0.7567
	75		0.9767	0.8257	0.9891	0.9979	0.5981	0.8655	0.9789	0.8539
	100		0.9999	0.8894	0.9978	0.9999	0.6754	0.9428	0.9982	0.9433
U(0, 1.5)	25	0.01	0.9910	0.9320	1.0000	0.9960	0.6180	0.9140	0.9980	0.8950
	50		1.0000	1.0000	1.0000	0.9950	0.8770	0.9990	1.0000	0.9960
	75		1.0000	1.0000	1.0000	1.0000	0.9610	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	0.9990	1.0000	1.0000	1.0000
	25	0.05	0.9972	0.9390	1.0000	0.9980	0.7370	0.9700	0.9990	0.9680
	50		1.0000	0.9970	1.0000	1.0000	0.9310	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	0.9770	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
Exp(1)	25	0.01	1.0000	0.9991	1.0000	1.0000	0.9916	0.9927	1.0000	0.9927
	50		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	25	0.05	1.0000	0.9994	1.0000	1.0000	0.9954	0.9966	1.0000	0.9974
	50		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
Gamma(1, 2)	25	0.01	1.0000	1.0000	1.0000	1.0000	0.9923	1.0000	1.0000	1.0000
	50		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	25	0.05	1.0000	1.0000	1.0000	.0000	0.9818	1.0000	1.0000	1.0000
	50		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
Weibull(1, 2)	25	0.01	0.9943	0.9893	1.0000	1.0000	0.7722	0.9952	1.0000	0.9945
	50		1.0000	1.0000	1.0000	1.0000	0.9419	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	0.9940	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	0.9990	1.0000	1.0000	1.0000
	25	0.05	0.9975	0.9984	1.0000	1.0000	0.8341	1.0000	1.0000	1.0000
	50		1.0000	1.0000	1.0000	1.0000	0.9639	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	0.9948	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
Pareto(1, 2)	25	0.01	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	50		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	25	0.05	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	50		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	75		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	100		1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Data analysis

We illustrate the test procedure using complete and censored data sets.

Complete data

We illustrate the proposed test procedures using two real data sets. First, we consider the data set given in Table 5 of Illowsky and Dean (2018). The data set consists of 55 smiling times of an eight-week-old baby measured in seconds, which can be treated as independent observations. The data follows a U[0, 23] distribution and we use the proposed test procedure to verify this. We implement the test using the integral transformation discussed in Remark 2.1. For the transformed data $Y=X/23$, the value of the test statistic is obtained as $\widehat{\mathrm{\Delta }}=-1.2523$, which belongs to the acceptance region. Hence we accept the null hypothesis that the data follows U[0, 23] distribution.

Next, we consider the study of the incubation period of COVID-19 by Lauer et al. (2020). The data is available at the link https://github.com/HopkinsIDD/ ncov_incubation. The number of days from exposure date to the COVID virus to the hospital presentation is calculated for 89 patients. We fit several lifetime distributions to this data using ‘fitdist’ available in the R package. Among the well-known models, we observe that the Weibull distribution with scale parameter $\alpha$ = 4.6885 and shape parameter $\beta$ = 1.5820 fit the data well. The data is transformed using integral transformation $1-exp{(-\frac{x}{4.6885})}^{1.5820}$, and for the transformed data we obtain the value of the test statistic as 1.0407. Hence we accept the hypothesis that the transformed data follows U[0, 1] distribution. This asserts that the original data follows the Weibull distribution.

Censored data

We consider the breast cancer data analysed in Yu (2021) to illustrate our test procedure. The data are obtained from a long-term clinical follow-up study on 371 women with stages I–III unilateral invasive breast cancer. The patients were treated by surgery at Memorial Sloan-Kettering Cancer Center in New York City between 1985 and 2001. For the illustration, we use a subset of the data with 50 lifetimes which is given by Wong et al. (2017). Yu (2021) fitted a uniform distribution to this data. The data is heavily censored with 26 censoring times. The proposed method is applied to the data with the transformation specified in Remark 2.1. The value of the test statistic is 0.8490, which suggests that the data follows U[0, 1]. Our result affirms the claim of Yu (2021) that the data fits uniform distribution.

Concluding remarks

Using the characterization of beta families by Ahmed (1991), we developed a simple non-parametric test based on U-statistic theory for testing uniform distribution. We study the asymptotic properties of the test statistic. The test is distribution-free. Even though several tests are available in the literature to test uniformity, these tests cannot accommodate censored lifetimes. Motivated by this, we discussed how to incorporate the right censored data in our test procedure. An extensive Monte Carlo simulation study is carried out to validate the finite sample performance of the tests. In the complete case, the test has well-controlled error rates even for small sample sizes. Power comparisons show that the performance of our test is competent with the existing tests. In the censored case, even with a high percentage of censored data (40%) our test performs well in terms of empirical power and attains the size of the test. Finally, we illustrated the test procedures using real data sets for complete as well as “censored cases".

Lifetime data often endure various forms of censoring apart from right censoring, which include left censoring, interval censoring, middle censoring etc. and different forms of truncation. The proposed test can be modified to incorporate these situations. The goodness of fit tests for other lifetime distributions using Stein’s type characterization can be developed.