Abstract

The exponential distribution is considered in situtations where intervals between events are considered as well as where a skewed distribution is appropriate. The exponential distribution also plays key role in survival analysis. Goodness-of-fit for exponentiality is crucial as, in the natural sciences, some of the commonly used distributions such as gamma and Weibull distributions are just translated versions of the exponential distributions. Several well known exponentiality tests are reviewed. A power comparison is performed using simulation.

1. Introduction

Assessing whether a data set is in compliance with a postulated distribution is termed as goodness-of-fit of a distribution. Tests for exponentiality is the general term for goodness-of-fit tests for exponential distributions. An exhaustive list of references for goodness-of-fit tests for exponential distributions are given in N. Balakrishnan and Asit P. Basu ¹, K. Yu. Volkova ², A. P. Rogozhnikov and B. Yu. Lemeshko ³, and the references there in.

In this paper, we implement several exponentiality tests, such as, Test for exponentiality based on Ahsanullah characterization ^{4, 5}, Atkinson test for exponentiality ⁶, Cox and Oakes test for exponentiality ⁷, Cramervon Mises test for exponentiality ⁷, Deshpande test for exponentiality ⁸, Test for exponentiality of Epps and Pulley ( ⁷ section 2.8.1), Epstein test for exponentiality ⁹, Frozini test for exponentiality ¹⁰, test for exponentiality based on the Gini statistic ¹¹, Gnedenko F-test of exponentiality ⁹, Harris modification of Gnedenko F-test ⁹, Hegazy-Green test for exponentiality ¹², Hollander-Proshan test for exponentiality ¹³, Kimber-Michael test ¹⁴ and ¹⁵, Kochar test for exponentiality ¹⁶, Kolmogorov-Smirnov test ( ⁷ section 2.1) forexponentiality, Lorenz test for exponentiality ¹⁷, Moran test for exponentiality ^{18, 19}, Pietra statistic ⁹, exponentiality based on Rossberg characterization ², Shapiro-Wilk test for exponentiality ²⁰, WE test for exponentiality ⁹, Wong and Wong test for exponentiality ⁹, and Anderson-Darling test for exponentiality ²¹.

2. Tests Descriptions

The test is based on the following statistic by Alexey Novikov and Ruslan Pusev,

where is the empirical distribution function,

Under exponentiality, one has

^{4, 5}. This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as AHTT and AHTS, respectively.

The Atkinson test for exponentiality is based on the following statistic ⁶,

The statistic is asymptotically (absolute) normal,

where,

This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as AKTT and AKTS, respectively.

The Cox and Oakes test is a test for the composite hypothesis of exponentiality ⁷. The test statistic is,

where is asymptotically standard normal ⁷. This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as COTT and COTS, respectively.

The Cramer-von Mises test for exponentiality (Henze and Meintanis ⁷ is based on the following statistic,

where is the empirical distribution function of the scaled data The p-value is computed by Monte Carlo simulation. Here we represent the test as CMTS.

Deshpande test for the composite hypothesis of exponentiality ⁸, is based on the following statistic,

Under exponentiality, one has

where

This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as DPTT and DPTS, respectively.

The test statistic is

where is asymptotically standard normal ( ⁷ section 2.8.1). This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as EPTT and EPTS, respectively.

The test ⁹ is based on the following statistic:

where

are order statistics. Under exponentiality, EPS_n is approximately distributed as a chi-square with degrees of freedom. This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as ESTT and ESTS, respectively.

The Frozini test for exponentiality is based on the following statistic ¹⁰,

The p-value is computed by Monte Carlo simulation. Here we represent the test as FRTS.

The test is based on the Gini statistic,

Under exponentiality, the normalized statistic is asymptotically standard normal ¹¹. This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as GSTT and GSTS, respectively.

The test ⁹ is based on the following statistic,

where

are order statistics. Under exponentiality, follows an F distribution with and degrees of freedom. Here we represent the test as GNTT.

The test ⁹ is based on the following statistic,

where

are order statistics. Under exponentiality, follows an F distribution with and degrees of freedom. Here we represent the test as HGTT.

The Hegazy-Green test for exponentiality ¹² is based on the following statistic,

The p-value is computed by Monte Carlo simulation. Here we represent the test as HFTS.

The Hegazy-Green test for exponentiality ¹² is based on the following statistic,

The p-value is computed by Monte Carlo simulation. Here we represent the test as HRTS.

The test ¹³ is based on the following statistic,

Under exponentiality, one has

This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as HPTT and HPTS, respectively.

The Kimber-Michael test ^{14, 15} for exponentiality is based on the following statistic,

where

The p-value is computed by Monte Carlo simulation. Here we represent the test as KMTS.

The Kochar test for exponentiality ¹⁶ is based on the following statistic,

where

The statistic T is asymptotically standard normal. This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as KCTT and KCTS, respectively.

The Kolmogorov-Smirnov test ( ⁷ section 2.1) for exponentiality is based on the following statistic,

where, is the empirical distribution function of the scaled data The p-value is computed by Monte Carlo simulation. Here we represent the test as KSTS.

The Lorenz test for exponentiality ¹⁷ for exponentiality is based on the following statistic,

The statistic is asymptotically standard normal. This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as LZTT and LZTS, respectively.

The Moran test for exponentiality ^{18, 19} is based on the following statistic,

where, where is Euler-Mascheroni constant. The statistic is asymptotically normal,

This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as MRTT and MRTS, respectively.

The test is based on the following Pietra statistic ⁹,

The p-value is computed by Monte Carlo simulation. Here we represent the test as PSTS.

The test is based on the following statistic ²,

where is the empirical distribution function,

Here denotes the order statistic of The p-value is computed from the limit null distribution. Under exponentiality, one has

This test can be implemented either using the asymptotic normal distribution or using the simulated distribution under the exponentiality assumption. Here we represent them as RCTT and RCTS, respectively.

The Shapiro-Wilk test for exponentiality ²⁰ is based on the following statistic,

The p-value is computed by Monte Carlo simulation. Here we represent the test as SWTS.

The Wong and Wong test for the composite hypothesis of exponentiality ⁹ is based on the following statistic,

where and are the smallest and the largest order statistics respectively. The p-value is computed by Monte Carlo simulation. Here we represent the test as WWTS.

Anderson-Darling upper tail test is based on the statistic ²¹

where and is the ordered measurement in the sample. The p-value is computed by Monte Carlo simulation. Here we represent the test as ADTS.

3. Power Comparison

Samples are generated from standard exponential distribution to check whether the proportions of rejections match with the respective levels of siginificances. And samples are generated from some selected alternative distributions, such as, standard Uniform (0, 1), standard half-normal, Weibull (1.0, 1.4), Gamma (2.0, 1.0), Beta (2.0, 1.0), Gamma (0.5, 1.0), and Log Normal (0.0, 0.8), distributions to investigate how powerful the tests are. We have considered sample sizes, 12, 20, and 28. Levels of significances considered are 1%, 5%, and 10%. In each cases, 10,000 samples are considered.

In Table 3, when samples are taken from the standard exponential distribution (null distribution), the levels of significances are closely estimated for the tests AKTS, COTS, CMTS, DPTS, EPTT, EPTS, ESTT, ESTS, FRTS, GSTT, GSTS, GNTT, HGTT, HFTS, HRTS, HPTS, KMTS, KCTS, KSTS, LZTS, MRTS, SWTS, WETS, WWTS, and ADTS. The tests PSTS and RCTS are also closely estimating the levels of significances with exceptions for smaller samples. Tests AHTT, AKTT, DPTT, KCTT, and MRTT are not estimating the levels of significances closely. Tests AHTS, HPTT, LZTT, and RCTT are wrongly estimating the levels of significances always as zeros or very close to zeros.

In Table 4, when samples are taken from standard uniform distribution, the worst performing tests are, AHTS, HFTS, HRTS, HPTT, LZTT, RCTT, and SWTS, their rejection proportions are zeros or close to zeros. Tests AKTT, AKTS, COTT, COTS, DPTT, DPTS, EPTT, EPTS, FRTS, GSTT, GSTS, GNTT, HPTS, KMTS, PSTS, and WETS, have some decent powers at least for higher samples. Tests AHTT, KCTT, and ADTS have high powers, among them KCTT has highest powers for all sample sizes and for all levels considered.

In Table 5, when samples are taken from standard half normal distribution, the worst performing tests are, AHTS, HFTS, HRTS, LZTT, RCTT, and SWTS, their rejection proportions are zeros. Tests AHTT, and KCTT have high powers, between them KCTT has highest powers for higher sample sizes and for higher levels of significances.

In Table 6, when samples are taken from Weibull (1.0,1.4) distribution, the worst performing tests are, AHTS, HFTS, HRTS, HPTT, LZTT, RCTT, SWTS, and WWTS, their rejection proportions are zeros or close to zeros. Tests AKTT, COTT, COTS, DPTT, DPTS, EPTT, EPTS, FRTS, GSTT, GSTS, GNTT, HPTS, KCTS, LZTS, MRTT, MRTS, PSTS, WETS, and ADTS have some decent powers at least for higher samples and higher levels of significances. Tests AHTT, and KCTT have high powers, between them KCTT has highest powers for higher sample sizes and for higher levels of significances.

In Table 7, when samples are taken from Gamma (2.0,1.0) distribution, the worst performing tests are, AHTS, ESTT, ESTS, HPTT, LZTT, RCTT, SWTS, and WWTS, their rejection proportions are zeros or close to zeros. Tests AKTT, AKTS, COTT, COTS, DPTS, EPTT, EPTS, FRTS, GSTT, GSTS, GNTT, HPTS, KCTS, LZTS, MRTT, MRTS, RCTS, and WETS have some decent powers at least for higher samples and higher levels of significances. Tests DPTT, KCTT, PSTS, and ADTS have some good powers at least for higher samples and higher levels of significances. Tests AHTT, HFTS, and HRTS have high powers, among them HFTS has highest powers for higher sample sizes and for higher levels of significances.

In Table 8, when samples are taken from Beta(2.0,1.0) distribution, the worst performing tests are, AHTS, HFTS, HRTS, LZTT, RCTT, SWTS, and WWTS, their rejection proportions are zeros or close to zeros. Tests HGTT and HPTT have some decent powers at least for higher samples and higher levels of significances. Tests AKTT, AKTS, COTT, CMTS, ESTT, ESTS, KMTS, and KSTS, have high powers at least for higher samples and higher levels of significances. Tests AHTT, COTS, DPTT, DPTS, EPTT, EPTS, FRTS, GSTT, GSTS, GNTT, HPTS, KCTT, KCTS, LZTS, MRTS, PSTS, RCTS, WETS, and ADTS have very high powers, among them AHTT, DPTT, EPTS, FRTS, GSTT, GSTS, HPTS, KCTT, PSTS, WETS, and ADTS have highest powers irrespective of sample sizes and levels of significances.

In Table 9, when samples are taken from Gamma (0.5,1.0) distribution, the worst performing tests are, AHTS, GNTT, HGTT, HFTS, HRTS, HPTT, LZTT, PSTS, RCTT, and ADTS, their rejection proportions are zeros or close to zeros. Tests AKTT, AKTS, DPTT, DPTS, EPTT, EPTS, FRTS, GSTT, GSTS, HPTS, KCTS, KSTS, RCTS, SWTS, and WETS have some good powers at least for higher samples and higher levels of significances. Tests CMTS, KMTS, LZTS, and WWTS have good powers at least for higher samples and higher levels of significances. Tests COTT, COTS, MRTT, and MRTS, have high powers, among them MRTT and MRTS have highest powers irrespective of sample sizes and levels of significances.

In Table 10, when samples are taken from Log Normal(0,0.8) distribution, the worst performing tests are, AHTS, ESTT, ESTS, HPTT, KMTS, LZTT, RCTT, SWTS, and WWTS, their rejection proportions are zeros or close to zeros. Tests DPTS, FRTS, HFTS, MRTS, and RCTS have some decent powers at least for higher samples and higher levels of significances. Tests DPTT and KCTT have good powers at least for higher samples and higher levels of significances. Only AHTT have some high powers irrespective of sample sizes and levels of significances.

4. Concluding Remarks

Overall worst performing tests are Ahsanullah characterization test using simulation (AHTS), Hegazy-Green test using simulation (HFTS), Hegazy-Green alternative test using simulation (HRTS), Lorenz test using normal approximation (LZTT), Rossberg characterization test using normal approximation (RCTT), Shapiro-Wilk test using simulation (SWTS), and Wong and Wong test using simulation (WWTS). These tests do not estimate the levels of significances correctly and have very low powers against the alternatives considered here.

Ahsanullah characterization test using normal approximation (AHTT) and Kochar test using normal approximation (KCTT) have high powers except for Gamma distribution with parameters 0.5 and 1.0. A few other tests also show higher powers but for some selected alternatives.

References