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We study a circular-circular multiplicative regression model, characterized by an angular error distribution assumed to be wrapped Cauchy. We propose a specification procedure for this model, focusing on adapting a recently proposed goodness-of-fit test for circular distributions. We derive its limiting properties and study the power performance of the test through extensive simulations, including the adaptation of some other well-known goodness-of-fit tests for this type of data. To emphasize the practical relevance of our methodology, we apply it to several small real-world datasets and wind direction measurements in the Black Forest region of southwestern Germany, demonstrating the power and versatility of the presented approach.

This paper presents a comprehensive study of nonparametric estimation techniques on the circle using Fej\'er polynomials, which are analogues of Bernstein polynomials for periodic functions. Building upon Fej\'er's uniform approximation theorem, the paper introduces circular density and distribution function estimators based on Fej\'er kernels. It establishes their theoretical properties, including uniform strong consistency and asymptotic expansions. The proposed methods are extended to account for measurement errors by incorporating classical and Berkson error models, adjusting the Fej\'er estimator to mitigate their effects. Simulation studies analyze the finite-sample performance of these estimators under various scenarios, including mixtures of circular distributions and measurement error models. An application to rainfall data demonstrates the practical application of the proposed estimators, demonstrating their robustness and effectiveness in the presence of rounding-induced Berkson errors.

In this paper, we focus on testing multivariate normality using the BHEP test with data that are missing completely at random. Our objective is twofold: first, to gain insight into the asymptotic behavior of BHEP test statistics under two widely used approaches for handling missing data, namely complete-case analysis and imputation, and second, to compare the power performance of test statistic under these approaches. It is observed that under the imputation approach, the affine invariance of test statistics is not preserved. To address this issue, we propose an appropriate bootstrap algorithm for approximating p-values. Extensive simulation studies demonstrate that both mean and median approaches exhibit greater power compared to testing with complete-case analysis, and open some questions for further research., Comment: 15 pages, 17 figures

Spherical and hyperspherical data are commonly encountered in diverse applied research domains, underscoring the vital task of assessing independence within such data structures. In this context, we investigate the properties of test statistics relying on distance correlation measures originally introduced for the energy distance, and generalize the concept to strongly negative definite kernel-based distances. An important benefit of employing this method lies in its versatility across diverse forms of directional data, enabling the examination of independence among vectors of varying types. The applicability of tests is demonstrated on several real datasets.

In this study, we introduce the first-of-its-kind class of tests for detecting change points in the distribution of a sequence of independent matrix-valued random variables. The tests are constructed using the weighted square integral difference of the empirical orthogonal Hankel transforms. The test statistics have a convenient closed-form expression, making them easy to implement in practice. We present their limiting properties and demonstrate their quality through an extensive simulation study. We utilize these tests for change point detection in cryptocurrency markets to showcase their practical use. The detection of change points in this context can have various applications in constructing and analyzing novel trading systems., Comment: 13 pages

This paper introduces a novel two-sample test for a broad class of orthogonally equivalent positive definite symmetric matrix distributions. Our test is the first of its kind and we derive its asymptotic distribution. To estimate the test power, we use a warp-speed bootstrap method and consider the most common matrix distributions. We provide several real data examples, including the data for main cryptocurrencies and stock data of major US companies. The real data examples demonstrate the applicability of our test in the context closely related to algorithmic trading. The popularity of matrix distributions in many applications and the need for such a test in the literature are reconciled by our findings., Comment: 13 pages, 2 figures

The L\'evy distribution, alongside the Normal and Cauchy distributions, is one of the only three stable distributions whose density can be obtained in a closed form. However, there are only a few specific goodness-of-fit tests for the L\'evy distribution. In this paper, two novel classes of goodness-of-fit tests for the L\'evy distribution are proposed. Both tests are based on V-empirical Laplace transforms. New tests are scale free under the null hypothesis, which makes them suitable for testing the composite hypothesis. The finite sample and limiting properties of test statistics are obtained. In addition, a generalization of the recent Bhati-Kattumannil goodness-of-fit test to the L\'evy distribution is considered. For assessing the quality of novel and competitor tests, the local Bahadur efficiencies are computed, and a wide power study is conducted. Both criteria clearly demonstrate the quality of the new tests. The applicability of the novel tests is demonstrated with two real-data examples., Comment: 28 pages, 4 figures

We study the Bahadur efficiency of several weighted L2--type goodness--of--fit tests based on the empirical characteristic function. The methods considered are for normality and exponentiality testing, and for testing goodness--of--fit to the logistic distribution. Our results are helpful in deciding which specific test a potential practitioner should apply. For the celebrated BHEP and energy tests for normality we obtain novel efficiency results, with some of them in the multivariate case, while in the case of the logistic distribution this is the first time that efficiencies are computed for any composite goodness--of--fit test.

In this article we prove a generalization of the Ejsmont characterization of the multivariate normal distribution. Based on it, we propose a new test for independence and normality. The test uses an integral of the squared modulus of the difference between the product of empirical characteristic functions and some constant. Special attention is given to the case of testing univariate normality in which we derive the test statistic explicitly in terms of Bessel function, and the case of testing bivariate normality and independence. The tests show quality performance in comparison to some popular powerful competitors.

In this paper some well-known tests based on empirical distribution functions (EDF) with estimated parameters for testing composite normality hypothesis are revisited, and some new results on asymptotic properties are provided. In particular, the approximate Bahadur slopes are obtained -- in the case of close alternatives -- for the EDF-based tests as well as the likelihood ratio test. The local approximate efficiencies are calculated for several close alternatives. The obtained results could serve as a benchmark for evaluation of the quality of recent and future normality tests.

Recently, the characterization based approach for the construction of goodness of fit tests has become popular. Most of the proposed tests have been designed for complete i.i.d. samples. Here we present the adaptation of the recently proposed exponentiality tests based on equidistribution-type characterizations for the case of randomly censored data. Their asymptotic properties are provided. Besides, we present the results of wide empirical power study including the powers of several recent competitors. This study can be used as a benchmark for future tests proposed for this kind of data.

We present new consistent goodness-of-fit tests for exponential distribution, based on the Desu characterization. The test statistics represent the weighted $L^2$ and $L^{\infty}$ distances between appropriate V-empirical Laplace transforms of random variables that appear in the characterization. In addition, we perform an extensive comparison of Bahadur efficiencies of different recent and classical exponentiality tests. We also present the empirical powers of new tests.

We introduce new consistent and scale-free goodness-of-fit tests for the exponential distribution based on Puri-Rubin characterization. For the construction of test statistics we employ weighted $L^2$ distance between $V$-empirical Laplace transforms of random variables that appear in the characterization. The resulting test statistics are degenerate V-statistics with estimated parameters. We compare our tests, in terms of the Bahadur efficiency, to the likelihood ratio test, as well as some recent characterization based goodness-of-fit tests for the exponential distribution. We also compare the powers of our tests to the powers of some recent and classical exponentiality tests. In both criteria, our tests are shown to be strong and outperform most of their competitors.

In this paper, some recent and classical tests of symmetry are modified for the case of an unknown center. The unknown center is estimated with its $\alpha$-trimmed mean estimator. The asymptotic behavior of the new tests is explored. The local approximate Bahadur efficiency is used to compare the tests to each other as well as to some other tests., Comment: The final version will appear in Statistics

In this paper a new class of uniformity tests is proposed. It is shown that those tests are applicable to the cases of any simple null hypothesis as well as for the composite null hypothesis of rectangular distributions on arbitrary support. The asymptotic properties of test statistics are examined. The tests are compared with some standard and some recent uniformity tests. For each test the Bahadur efficiencies against some common local alternatives are calculated. A class of locally optimal alternatives is found for each proposed test. The power study is also provided. Some applications in time series analysis are presented.

In this paper new two-dimensional goodness of fit tests are proposed. They are of supremum-type and are based on different types of characterizations. For the first time a characterization based on independence of two statistics is used for goodness-of-fit testing. The asymptotics of the statistics is studied and Bahadur efficiencies of the tests against some close alternatives are calculated. In the process a theorem on large deviations of Kolmogorov-type statistics has been extended to the multidimensional case.

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.