Test for Uniformity of Exchangeable Random Variables on the Circle
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Online ISSN 2005-2863 Print ISSN 1226-3192
SHORT COMMUNICATION
Test for Uniformity of Exchangeable Random Variables on the Circle Seonghun Cho1 · Young‑Geun Choi2 · Johan Lim1 · Won Jun Lee3 · Hyun‑Jeong Bai3 · Sungwon Kwon3 Received: 27 May 2020 / Accepted: 30 October 2020 © Korean Statistical Society 2020
Abstract We are motivated by our laboratory experiment on the flocking behavior of termites. To test for the existence of flocking behavior, we revisit the problem to test uniform samples (with the samples uniformly distributed) on the circle. Unlike most existing works, we assume that the samples are exchangeably dependent. We consider the class of normalized infinitely divisible distributions for the spacings of the samples, which form uniform samples on the circle. To test the uniformity, we study a test (Kuiper’s test) based on spacings of the samples and compute the asymptotic null distribution of the test statistic as the scaled Kolmogorov distribution. We apply the procedure to our experimental data and justify the flocking behavior of termites. Keywords Exchangeability · Flocking behavior · Kuiper’s test · Normalized infinitely divisible distribution · Termite · Uniformity on the circle
1 Introduction In this paper, we revisit the problem to test whether samples are uniformly distributed on the circle. Suppose that U1 , U2 , … , UK ∈ (0, 1] (with multiplication 2𝜋 ) are observations on the unit circle in the form of angles. We are interested in testing their uniformity on the circle. Regarding this problem, a significant number of references can be found in the literature. To list a few, Rayleigh (1919) uses the mean direction. Kuiper (1960) and Watson (1961) test the deviation between the empirical and hypothesized uniform cumulative distribution function. Ajne (1968) tests the observed number of points of a sample that fall within a suitable * Johan Lim [email protected] 1
Department of Statistics, Seoul National University, Seoul 08826, Republic of Korea
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Department of Statistics, Sookmyung Women’s University, Seoul 04310, Republic of Korea
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School of Pharmacy, Seoul National University, Seoul 08826, Republic of Korea
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Journal of the Korean Statistical Society
set. Rao (1972) tests the spacings between points. The Sobolev test for uniformity on the circle has recently been proposed by Hermans and Rasson (1985). The readers can find more references from Fisher (1995, Section 4.3), along with a good review from García-Portugués and Verdebout (2018). Despite this large amount of research, we find that most studies assume that U1 , U2 , … , UK (with multiplication 2𝜋 ) are independently distributed on the unit circle, while little research is conducted for the case of dependent samples. In this paper, we assume that the spacings of the samples, defined as Wk = U(k+1) − U(k) for k = 1, 2, … , K − 1 and ( WK = 1 + U(1) )− U(K) , are exchangeW1 , W2 , … , WK is called exchangeably distributed. The vector of spacings ( ) able if ( W𝜋(1) , W𝜋(2) , … ), W𝜋(K) has th
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