A test procedure for distinguishing logarithmically decaying tail from polynomially decaying tail
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
A test procedure for distinguishing logarithmically decaying tail from polynomially decaying tail Deepesh Bhati1,2 Received: 31 May 2019 / Accepted: 7 November 2019 © Korean Statistical Society 2020
Abstract A new statistical test is proposed to distinguish between the distributions having logarithmically decaying upper tail from polynomially decaying upper tail. The empirical size and power of the proposed test are computed through simulation. The proposed test is applied to two real world data sets. Keywords Australian city sizes data set · French Fire losses data set · Logarithmically decaying tail · Polynomially decaying tail
1 Introduction Limiting behaviour of linear/power normalized partial maxima plays an important role in identifying the behaviour of the right tail of the governing distribution. Fisher and Tippett (1928) and Gnedenko (1943) show that the limit in distribution n , with Mn = max{X 1 , . . . , X n }, X 1 , . . . , X n iid and normalizing constants of Mna−b n an > 0 and bn ∈ R, has to be one of a type of Fréchet or Weibull or Gumbel distribution. Pancheva (1984) showed that the limit in distribution of power normal 1 n βn ized maxima, M sgn(Mn ), where αn , βn > 0 are normalizing constants and αn sgn(x) = −1, 0, 1 for x < 0, x = 0 and x > 0, has to be a p-type of one of log Fréchet, log Weibull, standard Fréchet, negative log Fréchet, negative log Weibull or standard Weibull laws. These laws are by now classical and are well known. Throughout this article, we consider the class of distributions belonging to the max domain of attraction of the Fréchet law under linear normalization and those that belong to the max domain of attraction of the log-Fréchet law under power normalization. This is because of the fact that distributions with polynomially decaying (regularly varying)
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Deepesh Bhati [email protected]
1
Department of Statistics, Central University of Rajasthan, Ajmer, India
2
Department of Studies in Statistics, University of Mysore, Mysuru, India
123
Journal of the Korean Statistical Society
upper tail belong to max domain of attraction of the Fréchet law under linear normalization (see Theorem 3.3.7, Embrechts et al. 1997) whereas the distributions having logarithmically decaying (log-regularly varying) upper tail belong to the max domain of attraction of the log-Fréchet law under power normalization (see Theorem 2.1, Mohan and Ravi 1993 and Definition 2 in Desgagné 2013). It is known that, various phenomena in the field of hydrology (see Anderson and Meerschaert 1998; Tessier et al. 1996; Hosking and Wallis 1987), in Insurance (see McNeil 1997; Resnick 1997), in Finance (see Mandelbrot 1963; Fama 1965; Jansen and De Vries 1991; Rachev and Mittnik 2000), in Telecommunication and Web-traffic (see Markovitch and Krieger 2000), in geophysics and earth sciences (see Zaliapin et al. 2005), etc are modelled with distributions having polynomially decaying upper tail or distributions. But in the last d
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