New tests based on Rubin's empirical distribution function

In this paper we derive some new tests for goodness-of-fit based on Rubin's empirical distribution function (EDF). Substituting Rubin's EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling statistics, since Rubin's EDF for a given sample is a randomized distribution function, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. We show that the new tests are consistent under simple hypothesis. Several power comparisons are also performed to show that the new tests are generally more powerful than the classical ones.     

A two-sample distribution-free scale test of the smirnov type

The two-sample, distribution-free statistics of Smirnov (1939) are used to define a new statistic. While the Smirnov statistics are used as a general goodness-of-fit test, a distribution-free scale test based on this new statistic is developed. It is shown that this new test has higher power than the two-sided Smirnov statistic in detecting differences in scale for some symmetric distributions with equal means/medians. The critical values of the proposed test statistic and its limiting distribution are given     

Goodness-of-fit tests for a heavy tailed distribution

We study the Kolmogorov–Smirnov test, Berk–Jones test, score test and their integrated versions in the context of testing the goodness-of-fit of a heavy tailed distribution function. A comparison of these tests is conducted via Bahadur efficiency and simulations.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

Empirical process of long-range dependent sequences when parameters are estimated

In this paper, we study the asymptotic behavior of empirical processes when parameters are estimated, assuming that the underlying sequence of random variables is long-range dependent. We show completely different phenomena compared to i.i.d. situation, as well as compared to ordinary empirical processes of long-range dependent sequences. Applications include Kolmogorov–Smirnov and Cramer–Smirnov–von Mises goodness-of-fit statistics.     

An Empirical Analysis of Some Nonparametric Goodness-of-Fit Tests for Censored Data

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ² tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.     

Optimal distribution-free confidence bands for a distribution function

Distribution-free confidence bands for a distribution function are typically obtained by inverting a distribution-free hypothesis test. We propose an alternate strategy in which the upper and lower bounds of the confidence band are chosen to minimize a narrowness criterion. We derive necessary and sufficient conditions for optimality with respect to such a criterion, and we use these conditions to construct an algorithm for finding optimal bands. We also derive uniqueness results, with the Brunn–Minkowski Inequality from the theory of convex bodies playing a key role in this work. We illustrate the optimal confidence bands using some galaxy velocity data, and we also show that the optimal bands compare favorably to other bands both in terms of power and in terms of area enclosed.     

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.     

Powerful goodness-of-fit tests based on the likelihood ratio

Summary. A new approach of parameterization is proposed to construct a general goodness-of-fit test. It can not only generate traditional tests (including the Kolmogorov–Smirnov, Cramér–von Mises and Anderson–Darling tests) but also produce new types of omnibus tests, which are generally much more powerful than the old ones.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

Power comparison of the Kolmogorov–Smirnov test under ranked set sampling and simple random sampling

Many studies have been used to compare the power of several goodness-of-fit (GOF) tests under simple random sampling (SRS) and ranked set sampling (RSS). In our study, a different design procedure and ranking process in RSS are thoroughly investigated. A simulation study is conducted to compare the power of the Kolmogorov–Smirnov test under SRS and RSS with different sets and cycle sizes for several distributions. Level-2 sampling design and partially rank-ordered sets are used. Also, we benefited from auxiliary variables in the ranking process. Finally, results are presented with tables and figures. Under these conditions we show that the RSS has better performance against the SRS in finite population.     

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.     

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.     

Testing the Martingale Hypothesis

We propose new tests of the martingale hypothesis based on generalized versions of the Kolmogorov–Smirnov and Cramér–von Mises tests. The tests are distribution-free and allow for a weak drift in the null model. The methods do not require either smoothing parameters or bootstrap resampling for their implementation and so are well suited to practical work. The article develops limit theory for the tests under the null and shows that the tests are consistent against a wide class of nonlinear, nonmartingale processes. Simulations show that the tests have good finite sample properties in comparison with other tests particularly under conditional heteroscedasticity and mildly explosive alternatives. An empirical application to major exchange rate data finds strong evidence in favor of the martingale hypothesis, confirming much earlier research.     

Inference after separated hypotheses testing: an empirical investigation for linear models

Model selection problems arise while constructing unbiased or asymptotically unbiased estimators of measures known as discrepancies to find the best model. Most of the usual criteria are based on goodness-of-fit and parsimony. They aim to maximize a transformed version of likelihood. For linear regression models with normally distributed error, the situation is less clear when two models are equivalent: are they close to or far from the unknown true model? In this work, based on stochastic simulation and parametric simulation, we study the results of Vuong's test, Cox's test, Akaike's information criterion, Bayesian information criterion, Kullback information criterion and bias corrected Kullback information criterion and the ability of these tests to discriminate between non-nested linear models.     

Goodness of Fit Tests for Multivariate Counting Process Models with Applications

In this paper, we develop some distribution-free tests for checking the adequacy of the parametric forms of the intensity processes of a multivariate counting process model. The proposed tests, based in Khmaladze's transformations, are derived from the transforms of weighted aggregated martingale residual processes. The transformed processes converge weakly to independent Gaussian martingales under the assumed model. The distribution-free tests, such as Kolmogorov–Smirnov and Cramer–von Mises type tests, are appropriately defined to account for deviations in each of the transformed aggregated martingale residual processes. Consistency of the tests are discussed. The tests are applicable to multiplicative intensity models such as a competing risks model as well as to non-multiplicative intensity models such as a constant relative or excess mortality model. A small simulation study is conducted and an illustration to a real data example is provided.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.     

Multivariate goodness-of-fit tests based on statistically equivalent blocks

Various nonparametric procedures are known for the goodness-of-fit test in the univariate case. The distribution-free nature of these procedures does not extend to the multivariate case. In this paper, we consider an application of the theory of statistically equivalent blocks(SEB)to obtain distribution-free procedures for the multivariate case. The sample values are transformed to random variables which are distributed as sample spacings from a uniform distribution on [0, 1], under the null hypothesis. Various test statistics are known, based on the spacings, which are used for testing uniformity in the univariate case. Any of these statistics can be used in the multivariate situation, based on the spacings generated from the SEB. This paper gives an expository development of the theory of SEB and a review of tests for goodness-of-fit, based on sample spacings. To show an application of the SEB, we consider a test of bivariate normality.     

An entropy test for the Rayleigh distribution and power comparison

In this paper, a goodness-of-fit test is proposed for the Rayleigh distribution. This test is based on the Kullback–Leibler discrimination methodology proposed by Song [2002, Goodness of fit tests based on Kullback–Leibler discrimination, IEEE Trans. Inf. Theory 48(5), pp. 1103–1117]. The critical values and powers for some alternatives are obtained by simulation. The proposed test is compared with other tests, namely Kolmogorov–Smirnov, Kuiper, Cramer–von Mises, Watson and Anderson–Darling. The use of the proposed test is shown in a real example.