Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

Goodness-of-Fit Tests Based on Correcting Moments of Entropy Estimators

In this article, we consider the entropy estimator introduced by Alizadeh Noughabi and Arghami (2010) and derive the nonparametric distribution function corresponding to our estimator as a piece-wise uniform distribution. We use the results to introduce goodness-of-fit tests for the normal and the exponential distributions. The critical values and powers for some alternatives are obtained by simulation. The powers of the proposed tests under various alternatives are compared with the competitors.     

Recent tests of exponentiality against IFR alternatives: a survey

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study, we restrict ourselves to the problem of testing exponentiality (which essentially means no aging) against positive aging, which is captured by the class of increasing failure rate alternatives. Recent tests are discussed and compared. The empirical size of the tests is obtained by simulation. Power computations, using simulations, are done for each test procedure. These comparisons are done both for small and large sample sizes. Suggestions are made regarding the choice of the test when a particular alternative is suspected.     

Testing exponentiality using transformed data

In this paper, we introduce a test for uniformity and use it as the second stage of an exact goodness-of-fit test of exponentiality. By simulation, the powers of the proposed test under various alternatives are compared with exponentiality test based on Kullback–Leibler information proposed by Ebrahimi et al. [N. Ebrahimi, M. Habibullah, and E.S. Soofi, Testing exponentiality based on Kullback–Leiber information, J. R. Statist. Soc. Ser. B 54 (1992), pp. 739–748]. The results are impressive, i.e. the proposed test has higher power than the test based on entropy.     

Testing exponentiality based on Kullback—Leibler information for progressively Type II censored data

In many life-testing and reliability experiments, data are often censored in order to reduce the cost and time associated with testing and since the conventional Type-I and Type-II censoring schemes are not flexible enough, progressive censoring is developed by researchers. In this article, we develop a general goodness of fit test by using a new estimate of Kullback–Leibler information based on progressively Type-II censored data. Consistency and other properties of the proposed test are shown. Then, we use the proposed test statistic to test for exponentiality based on progressively Type-II censored data. The power values of the proposed test under different progressively Type-II censoring schemes are computed, through Monte Carlo simulations. It is observed that the proposed test is quite powerful in compared with the test proposed by Balakrishnan et al. (2007 Balakrishnan, N., Habibi Rad, A., and Arghami, N. R. (2007). Testing exponentiality based on Kullback–Leibler information with progressively type-II censored data. IEEE Transactions on Reliability 56:301–307. [Google Scholar]). Two real datasets from progressive censoring literature are finally presented for illustrative purpose.     

Testing dominance of biometric functions

The hazard rate (HR) and mean residual lifetime are two of the most practical and best-known functions in biometry, reliability, statistics and life testing. Recently, the reversed HR function is found to have interesting properties useful in additional areas such as censored data and forensic science. For these three biometric functions, we propose testing methods that they take on a known functional form against that they dominate or are dominated by this known form. This goodness-of-fit-type testing is wider in applications and more interesting than the long-standing testing procedures for exponentiality against the monotonicity of these functions or even the change point problems. This is so since we can test against any choice of the survival distribution and not just exponentiality. For this general testing, we present easy to implement tests and generalize them into classes of statistics that could lead to more powerful and efficient testing.     

AN EXACT TEST FOR HAZARD SIMILARITY

An exact test is developed for hazard similarity and in particular for exponentiality. This test is distinct from more common goodness-of-fit tests such as the Kolmogorov–Smirnov goodness-of-fit test, as it does not require full specification of the null distribution. This test is obtained through a characterization of hazard-similar distributions and a generalization of Fisher's test for association.     

A K-S Type Test for Exponentiality Based on Empirical Hankel Transforms

A uniqueness theorem for a recently introduced special Hankel transform of probability distributions on the non negative half-line motivates a K-S type test statistic based on empirical Hankel transforms for testing the hypothesis of exponentiality. This article deals with the asymptotic behavior of the new test.     

Tests for exponentiality against NBUE alternatives: a Monte Carlo comparison

Testing of various classes of life distributions has been addressed in the literature for more than 45 years. In this paper, we consider the problem of testing exponentiality (which essentially implies no ageing) against positive ageing which is captured by the fairly large class of new better than used in expectation (NBUE) distributions. These tests of exponentiality against NBUE alternatives are discussed and compared. The empirical size of the tests is obtained by simulations. Power comparisons for different popular alternatives are done using Monte Carlo simulations. These comparisons are made for both small and large sample sizes. The paper concludes with a discussion in which suggestions are made regarding the choices of the test when a particular alternative is suspected.     

Small sample equivalence tests for exponentiality

We consider small sample equivalence tests for exponentialy. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require much larger sample sizes, in comparison with classical “difference tests,” to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of small sample equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. We perform small sample simulation studies to assess the bias of our various tests and show that all of the Bayes posteriors we consider are integrable. Our simulation studies show that the saddlepoint-approximated UMPU method performs remarkably well for small sample sizes and is the only method that consistently exhibits an empirical significance level close to the nominal 5% level.     

Monte Carlo comparison of five exponentiality tests using different entropy estimates

The paper studies five entropy tests of exponentiality using five statistics based on different entropy estimates. Critical values for various sample sizes determined by means of Monte Carlo simulations are presented for each of the test statistics. By simulation, we compare the power of these five tests for various alternatives and sample sizes.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

Locally best invariant test for outliers in a gamma type distribution

It is shown that the locally best invariant test for the existence of outliers for scale parameters of the gamma distribution is given by Bartholomew's test for exponentiality which is the ratio of the sum of squares of the data to the square of the sample mean. The optimality robustness, including null and nonnull robustness of the test is shown. A small simulation study to compare the power among the other eight competitive tests for testing exponentiality is performed. It is seen that the locally best invariant test is not always best but is reasonably good. It is slightly better than Cochran's test and suffers less from the limiting masking effect.     

An adjusted cumulative Kullback-Leibler information with application to test of exponentiality

Abstract

In order to discriminate between two probability distributions extensions of Kullback–Leibler (KL) information have been proposed in the literature. In recent years, an extension called cumulative Kullback–Leibler (CKL) information is considered by authors which is closely related to equilibrium distributions. In this paper, we propose an adjusted version of CKL based on equilibrium distributions. Some properties of the proposed measure of divergence are investigated. A test of exponentiality based on the adjusted measure, is proposed. The empirical power of the presented test is calculated and compared with some existing standard tests of exponentiality. The results show that our proposed test, for some important alternative distributions, has better performance than some of the existing tests.     

Exponentiality Test Based on the Progressive Type II Censoring via Cumulative Entropy

In this article, we use cumulative residual Kullback-Leibler information (CRKL) and cumulative Kullback-Leibler information (CKL) to construct two goodness-of-fit test statistics for testing exponentiality with progressively Type-II censored data. The power of the proposed tests are compared with the power of goodness-of-fit test for exponentiality introduced by Balakrishnan et al. (2007 Balakrishnan, N., Habibi Rad, A., Arghami, N.R. (2007). Testing exponentiality based on Kullback-Leibler information with progressively type-II censored data. IEEE Transactions on Reliability 56(2):301–307.[Crossref], [Web of Science ®] , [Google Scholar]). We show that when the hazard function of the alternative is monotone decreasing, the test based on CRKL has higher power and when the hazard function of the alternative is non-monotone, the test based on CKL has higher power. But, when it is monotone increasing the power difference between test based on CKL and their proposed test is not so remarkable. The use of the proposed tests is shown in an illustrative example.     

Comments on “Testing for NBUL using goodness of fit approach with applications” (Statistical Papers (2010) 51: 27–40, DOI 10.1007/s00362-007-0113-0)

We comment on a new testing procedure for testing exponentiality against NBUL alternatives. We show that the proposed test is inappropriate and point out the subtle flaw in the argument. The other deficiencies in the paper are also highlighted.     

Goodness-of-fit tests for lifetime distributions based on Type II censored data

In this article, the general test statistic introduced by Alizadeh Noughabi and Balakrishnan [Goodness of fit using a new estimate of Kullback-Leibler information based on Type II censored data. IEEE Trans Reliab. 2015;64:627–635.] is applied for testing goodness of fit of lifetime distributions based on Type II censored data. The test statistic is constructed based on an estimate of Kullback–Leibler (KL) information. We investigate the properties of the proposed test statistic such as the test statistic is nonnegative, just like KL information. We apply this test statistic to following distributions: Exponential, Weibull, Log-normal and Pareto. The critical values and Type I error of the proposed tests are obtained. It is shown that the proposed tests have an excellent Type I error and hence can be used confidently in practice. Then, by Monte Carlo simulations, the power values of the proposed tests are computed against several alternatives and compared with those of the existing tests. Finally, some real-world reliability data are used for illustrative purpose.     

Correlation type gogdness-of-fit statistics with censored sampling

Some comments are made concerning the possible forms of a correlation coefficient type goodness-of-fit statistic, and their relationship with other goodness-of-fit statistics, Critical values for a correlation goodness-of-fit statistic and for the Cramer-von Mises statistic are provided for testing a completely-specified null hypothesis for both complete and censored sampling, Critical values for a correlation test statistic are provided for complete and censored sampling for testing the hypothesis of normality, two parameter exponentiality, Weibull (or, extreme value) and an exponential-power distribution, respectively. Critical values are also provided for a test of one-parameter exponentiality based on the Cramer-von Mises statistic     

On Testing Exponentiality against NBAFR Alternatives

In this article, we propose an interesting approach for testing exponentiality against NBAFR alternatives. A measure of deviation from exponentiality has been derived on the basis of an inequality which we have proved. A test statistic has been constructed using density estimators and its asymptotic normality established. The consistency of the said test is also proved.     

Tests of non-monotonic stochastic aging notions in reliability theory

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study we restrict ourselves to the problem of testing exponentiality against non-monotonic aging notions. We model non-monotonic aging using the notions of bathtub failure rate, increasing and then decreasing mean residual life and new worse then better than used in expectation classes. The different tests of exponentiality against these alternatives are discussed in detail.