Goodness-of-fit tests for Pareto distribution based on a characterization and their asymptotics

In this paper we present a new characterization of the Pareto distribution and consider goodness-of-fit tests based on it. We provide an integral and Kolmogorov–Smirnov-type statistics based on U-statistics and we calculate Bahadur efficiency for various alternatives. We find locally optimal alternatives for those tests. For small sample sizes, we compare the power of those tests with some common goodness-of-fit tests.     

Goodness-of-fit tests for progressively Type-II censored data from location–scale distributions

In this article, we propose several goodness-of-fit methods for location–scale families of distributions under progressively Type-II censored data. The new tests are based on order statistics and sample spacings. We assess the performance of the proposed tests for the normal and Gumbel models against several alternatives by means of Monte Carlo simulations. It has been observed that the proposed tests are quite powerful in comparison with an existing goodness-of-fit test proposed for progressively Type-II censored data by Balakrishnan et al. [Goodness-of-fit tests based on spacings for progressively Type-II censored data from a general location–scale distribution, IEEE Trans. Reliab. 53 (2004), pp. 349–356]. Finally, we illustrate the proposed goodness-of-fit tests using two real data from reliability literature.     

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 for one-shot device accelerated life testing data

In this article, we develop a formal goodness-of-fit testing procedure for one-shot device testing data, in which each observation in the sample is either left censored or right censored. Such data are also called current status data. We provide an algorithm for calculating the nonparametric maximum likelihood estimate (NPMLE) of the unknown lifetime distribution based on such data. Then, we consider four different test statistics that can be used for testing the goodness-of-fit of accelerated failure time (AFT) model by the use of samples of residuals: a chi-square-type statistic based on the difference between the empirical and expected numbers of failures at each inspection time; two other statistics based on the difference between the NPMLE of the lifetime distribution obtained from one-shot device testing data and the distribution specified under the null hypothesis; as a final statistic, we use White's idea of comparing two estimators of the Fisher Information (FI) to propose a test statistic. We then compare these tests in terms of power, and draw some conclusions. Finally, we present an example to illustrate the proposed tests.     

Goodness-of-fit tests for location–scale families based on random distance

For location–scale families, we consider a random distance between the sample order statistics and the quasi sample order statistics derived from the null distribution as a measure of discrepancy. The conditional qth quantile and expectation of the random discrepancy on the given sample are chosen as test statistics. Simulation results of powers against various alternatives are illustrated under the normal and exponential hypotheses for moderate sample size. The proposed tests, especially the qth quantile tests with a small or large q, are shown to be more powerful than other prominent goodness-of-fit tests in most cases.     

Testing Exponentiality Based on Type II Censored Data and a New cdf Estimator

In this article, we use a new cdf estimator to obtain a nanparametric entropy estimate and use it for testing exponentiality and normality. We also use the new cdf estimator to estimate the joint entropy of the Type II censored data which we use for some goodness-of-fit tests based on Kullback–Leibler information and show, by simulation, that it compares favorably with the leading competitor.     

A Comparative Study of Some Modified Chi-Squared Tests

Some recent results in the theory and applications of modified chi-squared goodness-of-fit tests are briefly discussed. It seems that for the first time power of modified chi-squared type tests for the logistic and three-parameter Weibull distributions based on moment type estimators is studied. Power of different modified tests against some alternatives for equiprobable fixed or random grouping intervals, and for Neyman–Pearson classes is investigated. It is shown that power of test statistic essentially depends on the quantity of Fisher's sample information this statistic uses. Some recommendations on implementing modified chi-squared type tests are given.     

On the goodness-of-fit test based on the ratio of cumulative hazard functions with the Type II censored data

For comparing two cumulative hazard functions, we consider an extension of the Kullback–Leibler information to the cumulative hazard function, which is concerning the ratio of cumulative hazard functions. Then we consider its estimate as a goodness-of-fit test with the Type II censored data. For an exponential null distribution, the proposed test statistic is shown to outperform other test statistics based on the empirical distribution function in the heavy censoring case against the increasing hazard alternatives.     

Non Null Asymptotic Distributions of Some Criteria in Censored Exponential Regression

In this article, we consider the class of censored exponential regression models which is very useful for modeling lifetime data. Under a sequence of Pitman alternatives, the asymptotic expansions up to order n^{? 1/2} of the non null distribution functions of the likelihood ratio, Wald, Rao score, and gradient statistics are derive in this class of models. The non null asymptotic distribution functions of these statistics are obtained for testing a composite null hypothesis in the presence of nuisance parameters. The power of all four tests, which are equivalent to first order, are compared based on these non null asymptotic expansions. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, we consider Monte Carlo simulations. We also present an empirical application for illustrative purposes.     

Kolmogorov–Smirnov test for life test data with hybrid censoring

This work considers goodness-of-fit for the life test data with hybrid censoring. An alternative representation of the Kolmogorov–Smirnov (KS) statistics is provided under Type-I censoring. The alternative representation leads us to approximate the limiting distributions of the KS statistic as a functional of the Brownian bridge for Type-II, Type-I hybrid, and Type-II hybrid censored data. The approximated distributions are used to obtain the critical values of the tests in this context. We found that the proposed KS test procedure for Type-II censoring has more power than the available one(s) in literature.     

A tool for systematically comparing the power of tests for normality

In this article, we describe a new approach to compare the power of different tests for normality. This approach provides the researcher with a practical tool for evaluating which test at their disposal is the most appropriate for their sampling problem. Using the Johnson systems of distribution, we estimate the power of a test for normality for any mean, variance, skewness, and kurtosis. Using this characterization and an innovative graphical representation, we validate our method by comparing three well-known tests for normality: the Pearson χ² test, the Kolmogorov–Smirnov test, and the D'Agostino–Pearson K ² test. We obtain such comparison for a broad range of skewness, kurtosis, and sample sizes. We demonstrate that the D'Agostino–Pearson test gives greater power than the others against most of the alternative distributions and at most sample sizes. We also find that the Pearson χ² test gives greater power than Kolmogorov–Smirnov against most of the alternative distributions for sample sizes between 18 and 330.     

A density-based empirical likelihood ratio goodness-of-fit test for the Rayleigh distribution and power comparison

The Rayleigh distribution has been used to model right skewed data. Rayleigh [On the resultant of a large number of vibrations of the some pitch and of arbitrary phase. Philos Mag. 1880;10:73–78] derived it from the amplitude of sound resulting from many important sources. In this paper, a new goodness-of-fit test for the Rayleigh distribution is proposed. This test is based on the empirical likelihood ratio methodology proposed by Vexler and Gurevich [Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy. Comput Stat Data Anal. 2010;54:531–545]. Consistency of the proposed test is derived. It is shown that the distribution of the proposed test does not depend on scale parameter. Critical values of the test statistic are computed, through a simulation study. A Monte Carlo study for the power of the proposed test is carried out under various alternatives. The performance of the test is compared with some well-known competing tests. Finally, an illustrative example is presented and analysed.     

Performances of some goodness-of-fit tests for sampling designs in ranked set sampling

In this study, we consider different sampling designs of ranked set sampling (RSS) and give empirical distribution function (EDF) estimators for each sampling designs. We provide comparative graphs for the EDFs. Using these EDFs, power of five goodness-of-fit tests are obtained by Monte Carlo simulations for Tukey's g–h distributions under RSS and simple random sampling (SRS). Performances of these tests are compared with the tests based on the SRS. Also, critical values belong to these tests are obtained for different set and cycle sizes.     

Power transformations to induce normality and their applications

Summary.  Random variables which are positive linear combinations of positive independent random variables can have heavily right-skewed finite sample distributions even though they might be asymptotically normally distributed. We provide a simple method of determining an appropriate power transformation to improve the normal approximation in small samples. Our method contains the Wilson–Hilferty cube root transformation for χ ² random variables as a special case. We also provide some important examples, including test statistics of goodness-of-fit and tail index estimators, where such power transformations can be applied. In particular, we study the small sample behaviour of two goodness-of-fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box–Ljung–Pierce portmanteau test, one in the time domain and the other in the frequency domain. A power transformation with a finite sample mean and variance correction is proposed, which ameliorates the small sample effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant under certain alternatives. Furthermore, the corrected tests also have better power than the Box–Ljung–Pierce portmanteau test, unlike the uncorrected versions.     

A New Goodness-of-Fit Test for Censored Data with an Application in Monitoring Processes

In this article, we propose a new goodness-of-fit test for Type I or Type II censored samples from a completely specified distribution. This test is a generalization of Michael's test for censored data, which is based on the empirical distribution and a variance stabilizing transformation. Using Monte Carlo methods, the distributions of the test statistics are analyzed under the null hypothesis. Tables of quantiles of these statistics are also provided. The power of the proposed test is studied and compared to that of other well-known tests also using simulation. The proposed test is more powerful in most of the considered cases. Acceptance regions for the PP, QQ, and Michael's stabilized probability plots are derived, which enable one to visualize which data contribute to the decision of rejecting the null hypothesis. Finally, an application in quality control is presented as illustration.     

A Nonparametric Test Procedure Based on Quantiles for the Right Censored Data

In this study, we propose nonparametric tests using the several quantile statistics simultaneously for the right censored data. First of all, we consider statistics of the quadratic form with estimated covariance matrices. Then we derive the limiting distribution using the large sample approximation theory. Also we consider different forms of statistics such as the maximal and summing types with their limiting distributions. Then we illustrate our procedure with examples and compare performance among tests with empirical powers through a simulation study. Also we comment briefly on some interesting features including re-sampling methods as concluding remarks. Finally in Appendices, we provide proofs for the theoretic results needed for the derivation of the limiting distributions of the proposed test statistics.     

A Cautionary Note on the Use of the Grønnesby and Borgan Goodness-of-Fit Test for the Cox Proportional Hazards Model

Grønnesby and Borgan (1996, Lifetime Data Analysis 2, 315–328) propose an omnibus goodness-of-fit test for the Cox proportional hazards model. The test is based on grouping the subjects by their estimated risk score and comparing the number of observed and a model based estimated number of expected events within each group. We show, using extensive simulations, that even for moderate sample sizes the choice of number of groups is critical for the test to attain the specified size. In light of these results we suggest a grouping strategy under which the test attains the correct size even for small samples. The power of the test statistic seems to be acceptable when compared to other goodness-of-fit tests.     

On a goodness-of-fit test for normality with unknown parameters and type-II censored data

We propose a new goodness-of-fit test for normal and lognormal distributions with unknown parameters and type-II censored data. This test is a generalization of Michael's test for censored samples, which is based on the empirical distribution and a variance stabilizing transformation. We estimate the parameters of the model by using maximum likelihood and Gupta's methods. The quantiles of the distribution of the test statistic under the null hypothesis are obtained through Monte Carlo simulations. The power of the proposed test is estimated and compared to that of the Kolmogorov–Smirnov test also using simulations. The new test is more powerful than the Kolmogorov–Smirnov test in most of the studied cases. Acceptance regions for the PP, QQ and Michael's stabilized probability plots are derived, making it possible to visualize which data contribute to the decision of rejecting the null hypothesis. Finally, an illustrative example is presented.     

Goodness-of-fit tests for perturbed dynamical systems

We consider the goodness-of-fit testing problem for stochastic differential equation with small diffusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramér–von Mises, Kolmogorov–Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We discuss the construction of the goodness-of-fit test based on the local time.     

On the sample breakdown robustness of some nonparametric tests

In this paper, relying on the sample breakdown points, we investigate the sample breakdown properties of some nonparametric tests. It is shown that the sample breakdown points of the sign test asymptotically dominate those of the Wilcoxon test for one–sided hypotheses, However, the different conclusion is derived in the case of testing some shrinking neighborhood hypotheses. The breakdown behaviors of the Kolmogorov test and X²–test are also explored. These studies unify or refine some existing breakdown analyses of tests.