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## Goodness-of-fit tests for semiparametric and parametric hypotheses based on the probability weighted empirical characteristic function

### Statistical Papers (2016-12-01) 57: 957-976 , December 01, 2016

We investigate the finite-sample properties of certain procedures which employ the novel notion of the probability weighted empirical characteristic function. The procedures considered are: (1) Testing for symmetry in regression, (2) Testing for multivariate normality with independent observations, and (3) Testing for multivariate normality of random effects in mixed models. Along with the new tests alternative methods based on the ordinary empirical characteristic function as well as other more well known procedures are implemented for the purpose of comparison.

## Selecting the Best Population Using a Test for Equality Based on Minimal Wilcoxon Rank-sum Precedence Statistic

### Methodology and Computing in Applied Probability (2007-06-01) 9: 263-305 , June 01, 2007

In this paper, we first give an overview of the precedence-type test procedures. Then we propose a nonparametric test based on early failures for the equality of two life-time distributions against two alternatives concerning the best population. This procedure utilizes the minimal Wilcoxon rank-sum precedence statistic (Ng and Balakrishnan, 2002, 2004) which can determine the difference between populations based on early (100*q*%) failures. Hence, this procedure can be useful in life-testing experiments in biological as well as industrial settings. After proposing the test procedure, we derive the exact null distribution of the test statistic in the two-sample case with equal or unequal sample sizes. We also present the exact probability of correct selection under the Lehmann alternative. Then, we generalize the test procedure to the *k*-sample situation. Critical values for some sample sizes are presented. Next, we examine the performance of this test procedure under a location-shift alternative through Monte Carlo simulations. Two examples are presented to illustrate our test procedure with selecting the best population as an objective.

## Testing for one-sided alternatives in nonparametric censored regression

### TEST (2012-09-01) 21: 498-518 , September 01, 2012

Assume that we have two populations (*X*_{1},*Y*_{1}) and (*X*_{2},*Y*_{2}) satisfying two general nonparametric regression models *Y*_{j}=*m*_{j}(*X*_{j})+*ε*_{j}, *j*=1,2, where *m*(⋅) is a smooth location function, *ε*_{j} has zero location and the response *Y*_{j} is possibly right-censored. In this paper, we propose to test the null hypothesis *H*_{0}:*m*_{1}=*m*_{2} versus the one-sided alternative *H*_{1}:*m*_{1}<*m*_{2}. We introduce two test statistics for which we obtain the asymptotic normality under the null and the alternative hypotheses. Although the tests are based on nonparametric techniques, they can detect any local alternative converging to the null hypothesis at the parametric rate *n*^{−1/2}. The practical performance of a bootstrap version of the tests is investigated in a simulation study. An application to a data set about unemployment duration times is also included.

## On the optimal choice of the number of empirical Fourier coefficients for comparison of regression curves

### Statistical Papers (2015-11-01) 56: 981-997 , November 01, 2015

The paper is devoted to the elaboration of an efficient approach for comparison of two regression curves based on the empirical Fourier coefficients of regression functions. For the problem of testing for the equality of the two unknown functions in the case of homoscedastic error structure and observation at equidistant points, we derive a new procedure with adaptive choice of the number of the coefficients used in the hypotheses testing. Our approach is based on approximation of the most powerful test using the full knowledge of the regression functions. The results are justified by theoretical arguments and the superiority of the new procedure is also confirmed by a simulation study.

## Recent and classical tests for exponentiality: a partial review with comparisons

### Metrika (2005-02-01) 61: 29-45 , February 01, 2005

### Abstract.

A wide selection of classical and recent tests for exponentiality are discussed and compared. The classical procedures include the statistics of Kolmogorov-Smirnov and Cramér-von Mises, a statistic based on spacings, and a method involving the score function. Among the most recent approaches emphasized are methods based on the empirical Laplace transform and the empirical characteristic function, a method based on entropy as well as tests of the Kolmogorov-Smirnov and Cramér-von Mises type that utilize a characterization of exponentiality via the mean residual life function. We also propose a new goodness-of-fit test utilizing a novel characterization of the exponential distribution through its characteristic function. The finite-sample performance of the tests is investigated in an extensive simulation study.

## Asymptotic efficiency of new exponentiality tests based on a characterization

### Metrika (2016-02-01) 79: 221-236 , February 01, 2016

Two new tests for exponentiality, of integral- and Kolmogorov-type, are proposed. They are based on a recent characterization and formed using appropriate V-statistics. Their asymptotic properties are examined and their local Bahadur efficiencies against some common alternatives are found. A class of locally optimal alternatives for each test is obtained. The powers of these tests, for some small sample sizes, are compared with different exponentiality tests.

## Testing skew normality via the moment generating function

### Mathematical Methods of Statistics (2010-03-01) 19: 64-72 , March 01, 2010

In this paper, goodness-of-fit tests are constructed for the skew normal law. The proposed tests utilize the fact that the moment generating function of the skew normal variable satisfies a simple differential equation. The empirical counterpart of this equation, involving the empiricalmoment generating function, yields appropriate test statistics. The consistency of the tests is investigated under general assumptions, and the finite-sample behavior of the proposed method is investigated via a parametric bootstrap procedure.

## Copula-based measures of reflection and permutation asymmetry and statistical tests

### Statistical Papers (2016-01-16): 1-23 , January 16, 2016

We propose measures of copula reflection and permutation asymmetry for data with positive quadrant dependence. We first define the measures of reflection asymmetry using a weighting function and then extend this approach to construct measures of permutation asymmetry for bivariate data. We define the corresponding statistical tests based on these measures and find that the proposed tests have higher statistical power comparing to some other tests for permutation and reflection symmetry studied in the literature. In addition, the measures can be used to summarize dependence structure of a multivariate data set in a few numbers and to select a more appropriate copula in the model.

## A note on testing independence by a copula-based order selection approach

### TEST (2013-03-01) 22: 62-82 , March 01, 2013

We suggest a new consistent asymptotically distribution-free test for independence of the components of bivariate random variables. The approach combines methods of order-selection tests with nonparametric copula density estimation. We deduce the asymptotic distribution of the test statistic and investigate the small sample performance by means of a simulation study and a data application.

## Tail fitting for truncated and non-truncated Pareto-type distributions

### Extremes (2016-09-01) 19: 429-462 , September 01, 2016

In extreme value analysis, natural upper bounds can appear that truncate the probability tail. At other instances ultimately at the largest data, deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. We make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (J. Amer. Statist. Assoc. *101*(473), 270–277, 2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a non-truncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point *T* that is large with respect to the available data. In the mathematical modelling we hence let *T*→*∞* at different speeds compared to the limiting fraction (*k*/*n*→0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields, simulation results, and some asymptotic results.