Goodness-of-Fit Tests for the Gamma Distribution Based on the Empirical Laplace Transform

We propose a class of goodness-of-fit tests for the gamma distribution that utilizes the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity, the test statistics approach limit values related to the first non zero component of Neyman's smooth test for the gamma law. The new tests are compared with other omnibus tests for the gamma distribution.     

On the distribution of a test for exponentiality based on progressively type-II right censored spacings

A test for exponentiality based on progressively Type-II right censored spacings has been proposed recently by Balakrishnan et al. (2002). They derived the asymptotic null distribution of the test statistic. In this work, we utilize the algorithm of Huffer and Lin (2001) to evaluate the exact null probabilities and the exact critical values of this test statistic.     

Tests of Fit for Exponentiality based on the Empirical Laplace Transform

The Laplace transform \psi(t)=E[{\rm exp}(-tX)] of a random variable X with exponential density u exp( m u x ), x S 0, satisfies the equation (\lambda+t)\psi(t)-\lambda=0 , t S 0. We study the behavior of a class of consistent tests for exponentiality based on a suitably weighted integral of [({\hat\lambda}_n+t)\psi_n(t)-{\hat\lambda}_n]^2 , where {\hat\lambda}_n is the maximum-likelihood estimate of u , and é n is the empirical Laplace transform, each based on an i.i.d. sample X 1 , …, X n . As the decay of the weight function tends to infinity, the test statistic approaches the square of the first nonzero component of Neyman's smooth test for exponentiality. The new tests are compared with other omnibus tests for exponentiality.     

Test of fit for Marshall–Olkin distributions with applications

Marshall and Olkin [1967. A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44], introduced a bivariate distribution with exponential marginals, which generalizes the simple case of a bivariate random variable with independent exponential components. The distribution is popular under the name ‘Marshall–Olkin distribution’, and has been extended to the multivariate case. L2-type statistics are constructed for testing the composite null hypothesis of the Marshall–Olkin distribution with unspecified parameters. The test statistics utilize the empirical Laplace transform with consistently estimated parameters. Asymptotic properties pertaining to the null distribution of the test statistic and the consistency of the test are investigated. Theoretical results are accompanied by a simulation study, and real-data applications.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

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.     

Data-Transformation and Test of Fit for the Generalized Pareto Hypothesis

In this article, we consider Crámer–von Mises type goodness-of-fit statistics for the Generalized Pareto law. The tests involve a certain transformation of the original observations, which, at least in the case of completely specified null distribution, may be viewed as transforming to uniformity and comparing the resulting moments of arbitrary positive order to those of a uniform distribution. The method is shown to be consistent, and the asymptotic null distribution of the test statistic is derived. Simulation results indicate that the proposed test compares well with standard methods based on the empirical distribution function.     

A new test for exponentiality against HNBUE alternatives

Abstract

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.     

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.     

Goodness-of-fit test for exponentiality based on spacings for general progressive Type-II censored data

There have been numerous tests proposed to determine whether or not the exponential model is suitable for a given data set. In this article, we propose a new test statistic based on spacings to test whether the general progressive Type-II censored samples are from exponential distribution. The null distribution of the test statistic is discussed and it could be approximated by the standard normal distribution. Meanwhile, we propose an approximate method for calculating the expectation and variance of samples under null hypothesis and corresponding power function is also given. Then, a simulation study is conducted. We calculate the approximation of the power based on normality and compare the results with those obtained by Monte Carlo simulation under different alternatives with distinct types of hazard function. Results of simulation study disclose that the power properties of this statistic by using Monte Carlo simulation are better for the alternatives with monotone increasing hazard function, and otherwise, normal approximation simulation results are relatively better. Finally, two illustrative examples are presented.     

Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation

Three goodness-of-fit tests for exponentiality based on the functional equation characterization 1?F(2x)=[1?F(x)]² for every x?0 are proposed and shown to compare well to several popular tests against common alternative cdf's. Small-sample critical values for α=0.10,0.05 are developed for the two superior test statistics and the asymptotic null-distributions are characterized.     

On the existence of transformations preserving the structure of order statistics in lower dimensions

In a Type-II right censored sample from the standard uniform distribution, several transformations of respective order statistics are examined, which transform the censored sample into a complete sample in a lower dimension. Such transformations have been considered by Lin et al. (2008), Michael and Schucany (1979) and O’Reilly and Stephens (1988) in the context of goodness-of-fit tests. It is shown that by dropping the assumption of an underlying uniform distribution, these transformed random variables can no longer be considered themselves as order statistics, in general. In the case of the transformation of Michael and Schucany, it is shown that the uniform distribution is the only one possessing this property.     

Quantile based tests for exponentiality against DMRQ and NBUE alternatives

Quantile-based reliability analysis has received much attention recently. We propose new quantile-based tests for exponentiality against decreasing mean residual quantile function (DMRQ) and new better than used in expectation (NBUE) classes of alternatives. The exact null distribution of the test statistic is derived when the alternative class is DMRQ. The asymptotic properties of both the test statistics are studied. The performance of the proposed tests with other existing tests in the literature is evaluated through simulation study. Finally, we illustrate our test procedure using real data sets.     

Estimation for Stochastic Models Driven by Laplace Motion

Laplace motion is a Lévy process built upon Laplace distributions. Non Gaussian stochastic fields that are integrals with respect to this process are considered and methods for their model fitting are discussed. The proposed procedures allow for inference about the parameters of the underlying Laplace distributions. A fit of dependence structure is also addressed. The importance of a convenient parameterization that admits natural and consistent estimation for this class of models is emphasized. Several parameterizations are introduced and their advantages over one another discussed. The proposed estimation method targets the standard characteristics: mean, variance, skewness and kurtosis. Their sample equivalents are matched in the closest possible way as allowed by natural constraints within this class. A simulation study and an example of potential applications conclude the article.     

A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively censored data

This paper proposes an approximation to the distribution of a goodness-of-fit statistic proposed recently by Balakrishnan et al. [Balakrishnan, N., Ng, H.K.T. and Kannan, N., 2002, A test of exponentiality based on spacings for progressively Type-II censored data. In: C. Huber-Carol et al. (Eds.), Goodness-of-Fit Tests and Model Validity (Boston: Birkhäuser), pp. 89–111.] for testing exponentiality based on progressively Type-II right censored data. The moments of this statistic can be easily calculated, but its distribution is not known in an explicit form. We first obtain the exact moments of the statistic using Basu's theorem and then the density approximants based on these exact moments of the statistic, expressed in terms of Laguerre polynomials, are proposed. A comparative study of the proposed approximation to the exact critical values, computed by Balakrishnan and Lin [Balakrishnan, N. and Lin, C.T., 2003, On the distribution of a test for exponentiality based on progressively Type-II right censored spacings. Journal of Statistical Computation and Simulation, 73 (4), 277–283.], is carried out. This reveals that the proposed approximation is very accurate.     

On entropy goodness-of-fit test based on integrated distribution function

In this study, we consider an entropy-type goodness-of-fit (GOF) test based on integrated distribution functions. We first construct the test for the simple vs. simple hypothesis and then extend it to the composite hypothesis case. It is shown that under regularity conditions, the null limiting distribution of the proposed test is a function of a Brownian bridge. A bootstrap method is also considered and is shown to be weakly consistent. A simulation study and real data analysis are conducted for illustration.     

On the sample redundancy and a test for exponentiality

In this paper we obtain the. sampling properties of a scale-free estimator of "redundancy," an information theoretic measure, which is used by economists and communications engineers. We then propose a new test of exponentiality based upon the sample redun- dancy. This test is asymptotically unbiased against a large class of alternatives, and it performs a t least as well as a recently proposed test with respect to power and asymptotic relative effi- ciency.     

A test of fit for a continuous distribution based on the empirical convex conditional mean function

Gupta and Kirmani (2008 Gupta, R.C., Kirmani, S.N.U.A. (2008). Characterization based on convex conditional mean function. J. Stat. Plann Inference. 138:964–970.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the convex conditional mean function (CCMF) characterizes the distribution function completely. In this paper, we introduce a consistent estimator of CCMF and call it empirical convex conditional mean function (ECCMF). Then we construct a simple consistent test of fit based on the integrated squared difference between ECCMF and CCMF. The theoretical and asymptotic properties of the estimator ECCMF and the proposed test statistic are studied. The performance of the constructed test is investigated under different distributions using simulations.     

Exponentiality test based on Renyi distance between equilibrium distributions

ABSTRACT

On the basis of Csiszar's φ-divergence discrimination information, we propose a measure of discrepancy between equilibriums associated with two distributions. Proving that a distribution can be characterized by associated equilibrium distribution, a Renyi distance of the equilibrium distributions is constructed that made us to propose an EDF-based goodness-of-fit test for exponential distribution. For comparing the performance of the proposed test, some well-known EDF-based tests and some entropy-based tests are considered. Based on the simulation results, the proposed test has better powers than those of competing entropy-based tests for the alternatives with decreasing hazard rate function. The use of the proposed test is evaluated in an illustrative example.     

A circular-cone test for testing homogeneity against a simple tree order

Several methods exist for the problem of testing the equality of several treatments against the one-sided alternative that the treatments are better than the control. These methods include Dunnett's test, Bartholomew's likelihood-ratio test, the Abelson-Tukey-Schaafsma-Smid optimal-contrast test, and the multiple-contrast test of Mukerjee, Robertson, and Wright. A new test is proposed based on an approximation of the likelihood-ratio test of Bartholomew. This test involves using a circular cone in place of the alternative-hypothesis cone. The circular-cone test has excellent power characteristics similar to those of Bartholomew's test. Moreover, it has the advantages of being simpler to compute and may be used with unequal sample sizes.