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On testing the log-gamma distribution hypothesis by bootstrap Eduardo Gutiérrez González · José A. Villaseñor Alva · Olga Vladimirovna Panteleeva · Humberto Vaquera Huerta

Received: 21 August 2011 / Accepted: 23 May 2013 / Published online: 11 June 2013 © Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper we propose two bootstrap goodness of fit tests for the loggamma distribution with three parameters, location, scale and shape. These tests are built using the properties of this distribution family and are based on the sample correlation coefficient which has the property of invariance with respect to location and scale transformations. Two estimators are proposed for the shape parameter and show that both are asymptotically unbiased and consistent in mean-squared error. The test size and power is estimated by simulation. The power of the two proposed tests against several alternative distributions is compared to that of the KolmogorovSmirnov, Anderson-Darling, and chi-square tests. Finally, an application to data from a production process of carbon fibers is presented. Keywords Shape parameter · Sample correlation coefficient · Location-scale invariant statistic · Goodness of fit test · Parametric bootstrap 1 Introduction The generalized gamma distribution was first introduced by Amoroso (see Crooks 2007), and was studied by Stacy (1962). This distribution had been used in different applications; for instance, the works of Cantú et al. (2001), Mees and Gerard (1984), and Kaneko (2003). The parameter estimation problem had been studied by Gomes et

E. Gutiérrez González (B) · O. V. Panteleeva UPIICSA—Instituto Politécnico Nacional, Avenida Té 950 Colonia Granjas México, delegación Iztacalco, CP 08400 Mexico, Mexico e-mail: [email protected] J. A. Villaseñor Alva · H. Vaquera Huerta Colegio de Postgraduados, Programa de Estadística, Texcoco Estado de México, 56230 Montecillo, Mexico

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al. (2008), Tsionas (2005), Rao et al. (1991), DiCiccio (1987), Hager and Bain (1970), Harter (1967), Sreekumar and Thomas (2007), and Stacy and Mihran (1965). The work by Tsionas (2005) and Gomes et al. (2008) presented results from a computational approach, while the work by Sreekumar and Thomas (2007) used order statistics. In recent years researchers have sought distributions to model the actual data in a better classical way. For this reason distributions which are flexible enough to allow the possibility of skewness and excess kurtosis are needed, for example Meintanis and Tsionas (2010), Meintanis (2008, 2010). A disadvantage of these tests for goodness of fit lies in the proposed test statistic. In this article two goodness of fit tests are proposed for the log-gamma distribution, which can also be applied to test the Weibull and generalized gamma hypothesis after an appropriate transformation of the data. In the two proposed tests the statistic is very simple and is based on the sample correlation coefficient which has the property of invariance with respect to location and