The non-null limiting distribution of the generalized Baumgartner statistic based on the Fourier series approximation
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The non-null limiting distribution of the generalized Baumgartner statistic based on the Fourier series approximation Ryo Miyazaki2 · Hidetoshi Murakami1
Received: 12 September 2017 / Revised: 5 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract The non-null limiting distribution of the generalized Baumgartner statistic is approximated by applying the Fourier series approximation. Due to the development of computational power, the Fourier series approximation is readily utilized to approximate its probability density function. The infinite product part for a noncentral parameter in the characteristic function is re-formulated by using a formula of the trigonometric function. The non-central parameter of the generalized Baumgartner statistic is formulated by the first moment of the generalized Baumgartner statistic under the alternative hypothesis. The non-central parameter is used to calculate the power of the generalized Baumgartner statistic. Keywords Fourier series approximation · Generalized Baumgartner statistic · Non-central weighted χ 2 distribution Mathematics Subject Classification 62G10
1 Introduction Testing hypotheses is one of the most important challenges in nonparametric statistics. Various nonparametric tests have been proposed for one-sample, two-sample and multisample testing problems involving the location, scale, location-scale and other parameters. We consider testing the equality of several parameters, i.e. the one-way
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Hidetoshi Murakami [email protected]
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Department of Applied Mathematics, Tokyo University of Science, Tokyo, Japan
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Department of Industrial and Systems Engineering, Graduate School of Science and Engineering, Chuo University, Tokyo, Japan
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R. Miyazaki, H. Murakami
layouts analysis of variance, which is one of the most important types of statistical procedures in biometry. Then the one-way layouts analysis of variance is used in many practices. Let {X pq | p = 1, . . . , k, q = 1, . . . , n p } be k samples of size n p of each observations. Note that independence between the samples is assumed, and the observations within each sample are assumed to be independent and identically distributed. Suppose that the observation X pq is obtained from a continuous distribution function F p (x). Then, we are interested in testing the following hypothesis: H0 : F1 = F2 = · · · = Fk against H1 : not H0 . However, in many applications, the underlying distribution is not adequately understood to assume normality or some other specific distribution. Hence, nonparametric hypothesis testing must be used in these circumstance. Since the means and variances of the each group differ, the tests for the shifted location parameter and the tests for the changed scale parameter may not be appropriate in this situation. Therefore, it is preferable to jointly test for the location and scale differences for a continuous distribution functions. Suppose that the location parameter is a main effect with small scale changes. Under these circumstance
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