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Recently Khmaladze has shown how to ‘rotate’ one empirical process to another. We apply this methodology to goodness of fit tests for Bernoulli trials, generated by a single distributional family, but with covariates varying over the sample. Grouping the data, we demonstrate that goodness of fit tests after rotation to distribution free processes are easily computed, and exhibit high power to reject incorrect null hypotheses.

This paper proposes a new procedure to test uniformity on the Stiefel manifold. The theoretical analysis of the test procedure, and numerical experiments are conducted to illustrate the usage and the efficiencies through the power under alternative hypotheses.

In this paper we derive a formula for the sample size of a nonparametric test based on weighted L 2 -Wasserstein distance with a known alternative. As an example we apply the formula to determine the sample size for a hypothesis testing from the Tukey’s contaminated distribution model. The reliability of the sample size formula is evaluated in terms of type I errors and the powers of the test. The Monte Carlo simulation results show that the test based on the weighted L 2 -Wasserstein distance performs equally well as the Anderson–Darling test and is superior to the Kolmogorov–Smirnov and the Cramer–von Mises tests.

We discuss a goodness-of-fit test arising from information-theoretical considerations. We show that, for a simple null hypothesis, our test has superior asymptotic power compared to the Anderson–Darling test when the alternative lies in a certain large class of distribution functions.

A test based on the ratio of the sample mean absolute deviation and the sample standard deviation is proposed for testing the Laplace distribution hypothesis. The asymptotic null distribution for this test statistic is found to be normal. The use of Anderson–Darling test based on a data transformation is also discussed.

We develop an algorithm for finding exact critical values for the two-sample Kolmogorov–Smirnov test in the finite-population case. We then compare these exact critical values to the asymptotic values that are available in the literature. The asymptotic critical values work well for equal sample sizes, but can be excessively conservative when the sample sizes differ.

We propose two versions of asymptotically distribution free empirical processes. When a composite null hypothesis contains a family of distributions indexed by a one dimensional parameter space, and when that single parameter is estimated by maximum likelihood, the resulting distribution free goodness of fit tests are simpler than tests applying the Khmaladze transformation. For the Pareto distribution, the process we advocate is especially simple. The theory is illustrated by fitting the Pareto distribution to threshold exceedances of stock returns, and the Weibull distribution to fibre strength data.

Yitzhaki (1996) showed that the OLS estimator of the slope coefficient in a simple regression is a weighted average of the slopes delineated by adjacent observations. The weights depend only on the distribution of the independent variable. In this paper I demonstrate that equal weights can only be obtained if and only if the independent variable is normally distributed. This necessary and sufficient condition is used to develop a new test for normality which is distribution free and not sensitive to outliers. The test is compared with standard normality tests, in particular, the popular Jarque-Bera test. It is shown that the new test provides a better power for testing normality against all classes of alternative distributions. Finally, the test is applied to check normality in time-series data from major international financial markets.

The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality says that if F n is an empirical distribution function for variables i.i.d. with a distribution function F , and K n is the Kolmogorov statistic n sup x | ( F n − F ) ( x ) | , then there is a constant C such that for any M > 0 , Pr ( K n > M ) ≤ C exp ( − 2 M 2 ) . Massart proved that one can take C = 2 (DKWM inequality), which is sharp for F continuous. We consider the analogous Kolmogorov–Smirnov statistic for the two-sample case and show that for m = n , the DKW inequality holds for n ≥ n 0 for some C depending on n 0 , with C = 2 if and only if n 0 ≥ 458 . The DKWM inequality fails for the three pairs ( m , n ) with 1 ≤ m n ≤ 3 . We found by computer search that the inequality always holds for n ≥ 4 if 1 ≤ m n ≤ 200 , and further for n = 2 m if 101 ≤ m ≤ 300 . We conjecture that the DKWM inequality holds for all pairs m ≤ n with the 457 + 3 = 460 exceptions mentioned.

We consider nonparametric regression models with multivariate covariates and estimate the regression curve by an undersmoothed local polynomial smoother. The resulting residual-based empirical distribution function is shown to differ from the error-based empirical distribution function by the density times the average of the errors, up to a uniformly negligible remainder term. This result implies a functional central limit theorem for the residual-based empirical distribution function.

A distribution free goodness of fit test based on empirical distribution function is proposed for a one-sample and two-sample scenario under the alternative hypothesis of stochastic ordering. The exact and asymptotic null distribution of the proposed test statistic is derived. A Monte Carlo simulation study has been made to compare the power of the proposed test and other competitive test procedures. The proposed test performs better in the comparative study.

In this paper a procedure to test the equality of error distributions in several nonparametric regression models is introduced. Kolmogorov–Smirnov and Cramer–von Mises-type statistics are proposed and their asymptotic distributions are obtained. A bootstrap mechanism is used to approximate the critical values in practice.

The Watson distribution is frequently used for modeling axial data. In this paper, we present goodness-of-fit methods for the bipolar Watson distribution defined on the hypersphere. We analyze by simulation some questions concerning these tests: the adequacy of the asymptotic chi-square distribution used in the tests and the adequacy of using the tabulated critical values for the Kolmogorov–Smirnov statistic when the parameters of the bipolar Watson distribution are unknown. We illustrate these techniques with simulated data from this distribution.

It is well known that the tests based on the residual empirical process for fitting an error distribution in regression models are not asymptotically distribution free. One either uses a Monte-Carlo method or a bootstrap method to implement them. Another option is to base tests on the Khmaladze transformation of these processes because it renders them asymptotically distribution free. This note compares Monte-Carlo, naive bootstrap, and the smooth bootstrap methods of implementing the Kolmogorov–Smirnov test with the Khmaladze transformed test. We find that the transformed test outperforms the naive and smooth bootstrap methods in preserving the level. The note also includes a power comparison of these tests.

We describe how to test whether the distribution functions of errors from two linear regression models are the same, with statistics based on empirical distribution functions constructed with residuals. A smooth bootstrap method is used to approximate critical values. Simulations show that the procedure works well in practice.

Given a Brownian bridge B0 with trend g:[0,1]→[0,∞), (1) Y(z)=g(z)+B 0 (z),z∈[0,1], we are interested in testing H0:g≡0 against the alternative K:g>0. For this test problem we study weighted Kolmogorov tests reject H 0 ⇔ sup z∈[0,1] w(z)Y(z)>c, where c>0 is a suitable constant and w:[0,1]→[0,∞) is a weight function. To do such an investigation a recent result of the authors on a boundary crossing probability of the Brownian bridge is useful. In case the trend is large enough we show an optimality property for weighted Kolmogorov tests. Furthermore, an additional property for weighted Kolmogorov tests is shown which is useful to find the more favourable weight for specific test problems. Finally, we transfer our results to the change-point problem whether a regression function is or is not constant during a certain period.

If X and Y are continuous random variables with joint distribution function H, then the Kendall distribution function of (X,Y) is the distribution function of the random variable H(X,Y). Kendall distribution functions arise in the study of stochastic orderings of random vectors. In this paper we study various properties of Kendall distribution functions for both populations and samples.

We consider the efficient estimation of a bivariate distribution function (DF) under the class of radially symmetric distributions and propose an estimator based on the mean of the empirical distribution and survival functions. We obtain the mean and variance of the estimator and show that it has an asymptotic normal distribution. We also show that the nonparametric maximum likelihood estimator of the bivariate DF coincides with the new estimator under radial symmetry. We study the asymptotic relative efficiency of this estimator and show that it results in a minimum of 50% reduction in sample size over the empirical DF at any point (x,y) in R 2 . A bootstrap procedure to test whether the data support a radially symmetric model is examined. A simulation study compares the size and power of this test under bivariate normality, against alternatives in the Plackett's family of bivariate distributions, to two other procedures based on Kolmogorov–Smirnov distance.

This paper deals with an asymptotic local test for linearity of nonlinear dynamical systems. The aim of the test is to compare two estimators of the leading function of the dynamical system, built with the observations contained in a fixed domain D . The first one is naturally local since it is a kernel-based estimator. The second one is a D -localized version of the least squares estimator. We prove a convergence result, including rate, for the latter estimator and deduce a central limit theorem leading to an asymptotic test. Some simulations illustrate the need of such a local procedure and investigate the finite sample case.

This article concerns with the problem of testing whether a mixture of two normal distributions with bounded means and specific variance is simply a pure normal. The large sample behavior of the likelihood ratio test for the problem is carefully investigated. In the case of one mean parameter, it is shown that the large sample null distribution of the likelihood ratio test statistic is the squared supremum of a Gaussian process with zero mean and explicitly given covariances. In the case of two mean parameters, both the simple and composite hypotheses of normality are considered. Under the simple null hypothesis, the large sample null distribution is found to be an independent sum of a chi-square variable and the squared supremum of another Gaussian process whose covariance structure is slightly different from the one mean parameter case, while under the composite null hypothesis, the chi-square term is absent.

This paper develops a large deviation theorem for families of sample means of U-statistic structure (i.e., U-processes). These results extend the work of Sethuraman (1964) and Wu (1994) on large deviation theory for families of ordinary sample means and the classical empirical process. Along the way we obtain an extension to U-statistics of an important isoperimetric inequality of Talagrand (1994) for ordinary means. Applications include the simplicial depth function of Liu (1990) and sup-norm statistics (e.g., Kolmogorov–Smirnov type goodness-of-fit statistics) defined over U-processes.

To check the validity of a parametric model for survival data, a number of supremum-type tests have been proposed in the literature using Khmaladze's (1993, Ann. Statist. 18, 582–602) transformation of a test process. However, such a transformation is usually very complicated and lacks a clear interpretation. Information could also be lost through transformation. In this note, we propose a simulation-based supremum-type test directly from the original test process using an idea originally introduced by Lin et al. (1993, Biometrika 80, 557–572). The test is developed under the framework of Aalen's (1978, Ann. Statist. 6, 701–726) multiplicative intensity counting process model, and therefore applies to a number of survival models including those with very general forms of censoring and truncation. By comparing the observed test process with a set of simulated realizations of an approximating process, our method can be used as a graphical tool as well as a formal test for checking the adequacy of the assumed parametric model. We establish consistency of the resulting test under any fixed alternative. Its performance is investigated in a simulation study. Illustrations are given using some real data sets.

Sen (1974, Weak convergence of multidimensional empirical processes for stationary φ-mixing processes. Ann. Probab. 2, 147–154, Theorem 3.2) established the tail probability inequalities for Kolmogorov–Smirnov statistics for sequences of φ-mixing random variables. In the present note, Sen's result is improved considerably.

In this note, we discuss the Kolmogrov and Erdos test for self-normalized sums. Some general results are obtained.

A nonparametric test for second-order stochastic dominance is introduced in the framework of the one sample problem. It is based on a supremum statistic which is suitable for second-order problems. Its asymptotic distribution is identified and quantiles of the finite sample and asymptotic distributions are derived by simulations. Its power is compared to those of a test suggested by Deshpande and Singh (1985) which is based on an integral statistic. Finally, it is shown that the new test can also be used for testing first order stochastic dominance. A family of alternatives is given where the second-order test has uniformly higher power than the usual first-order Kolmogorov test.

This paper presents a distribution-free multivariate Kolmogorov-Smirnov goodness-of-fit test. The test uses a statistic which is built using Rosenblatt's transformation and an algorithm is developed to compute it in the bivariate case. An approximate test, that can be easily computed in any dimension, is also presented. The power of these multivariate tests is studied in a simulation study.

In this paper we propose a new test of conditional heteroskedasticity for time series by introducing a Kolmogorov-Smirnov-type test statistic. The asymptotic properties of the new test statistic are established. The results demonstrate that such a test is consistent.

We consider the problem of hypothesis testing about a value of functional. For a given functional T the problem is to test a hypothesis T(P) = 0 versus alternatives T(P) > b0 > 0 where P is an arbitrary probability measure. Under the natural assumptions we show that the test statistics T( P n ) depending on the empirical probability measures P n are asymptotically minimax. Since the sets of alternatives is fixed the asymptotic minimaxity is considered in the senses of Bahadur and Hodges-Lehmann efficiencies. In particular the functional T can be the functional corresponding to the test statistics of Kolmogorov and omega square tests.

For a strictly stationary sequence {Xi} define D n =sup x∈ R m |F n (x)-F(x)| , where Fn and F are the empirical distribution function (d.f.) and common d.f. pertaining to {Xi}. Sen (1974, Theorem 3.2) established the tail probability inequalities for Dn when {Xn} is φ-mixing process. In the present paper, it is proved that Sen's result remains to be true for a class of stro mixing processs also which include φ-mixing and the asymptotically uncorrelated processes as special cases. Some applications are given.

Some order statistics from a univariate random sample of known size, drawn from a random variable X, are censored or lost. We are concerned with the problem of estimating how many observations we have lost within each interval between the remaining (observed) order statistics (including − ∞ and + ∞ among the remaining order statistics), in order to infer the unknown structure of the original sample.

This paper gives asymptotically distribution free tests of the Kolmogorov–Smirnov type based on aligned observations for testing the hypothesis that k distributions differ only in regression lines, parallel or not. The design matrices are allowed to be random.

Although still being recommended as a goodness-of-fit test for normality, we present some motivations to consider D'Agostino's statistic D for testing of uniformity on an unknown interval. As a matter of fact, D turns out to be a linear function of the Cramer-von Mises distance between the empirical distribution function and a rectangular distribution with suitably estimated endpoints. The limiting null distribution of D is found.

A class of linear rank statistics is considered for testing a sequence of independent random variables with common distribution against alternatives involving a change in distribution at an unknown time point. It is shown that, under the null hypothesis and suitably normalized, this class of statistics converges in distribution to the double exponential extreme value distribution.

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n −4 with the second term of the order n −3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n . Also we have shown that F -tables can be used to test the hypothesis when the sample size is moderately large.

The limiting null distribution of Kent's (1982) statistic, to test whether a sample comes from the Fisher distribution is derived when κ, the concentration parameter, goes to ∞. A modification is suggested, the limiting null distribution of which is ξ22 when either κ or n, the sample size, goes to ∞. Tests of Fisherness based on the eigenvalues of the sample cross product matrix are also considered. Numerical examples are presented.

For testing the hypothesis that the two distributions differ only in regression lines, parallel or not, asymptotically distribution free test of Kolmogorov-Smirnov type based on certain aligned observations are proposed.

For testing the hypothesis that two (symmetric) distributions may differ only in locations, a Kolmogorov-Smirnov type test based on the aligned observations is considered and its properties are studied.