Estimation in the three-parameter gamma distribution based on the empirical moment generation function

Goodness-of-Fit Tests for the Gamma Distribution Based on the Empirical Laplace Transform

“…Then the KKC-statistic, with r ≥ 3 and t = t 1 t r , is a quadratic form KKC = n r n t − r t ˆ n ĉ n t ˆ n ĉ n −1 r n t − r t ˆ n ĉ n where the vectors r n and r are given by r n t = ln L n −t 1 ln L n −t r with Y i replaced by X i in the ELT, r t ˆ n ĉ n = ln M X t 1 ˆ n ĉ n ln M X t r ˆ n ĉ n and whereĉ n is a consistent estimator of the scaling factor. The r × r matrix depends on the values of the vector t and is given by t ˆ n ĉ n = k j with k j = 1 + t k t jĉ 2 n / 1 −ĉ n t k + t j ˆ n − 1 for k j = 1 r. To calculate the KKC test statistic, we used r = 7 points for evaluations of the empirical and theoretical moment generating functions and a dynamic value of the standard spacings < 0 involved in the components t j = j /ĉ n , j = 1 r, of the vector t as suggested in Koutrouvelis and Canavos (1997) and in Kallioras et al (2006). We noticed that severe numerical problems occurred with some alternative distributions because the matrix is too ill-conditioned to be inverted accurately.…”

“…There exists a wealth of material for testing via the ELT including, among others, Csörgő and Teugels (1990), Baringhaus and Henze (1991), Henze (1993), Henze and Meintanis (2002), Meintanis and Iliopoulos (2003), Henze and Klar (2002), Castillo and Quiroz (2005), and Kallioras et al (2006). For the gamma law in particular, there is the chi-squared test of Kallioras et al (2006) incorporating the empirical moment generating function which, when computed at negative arguments, is equivalent to the ELT; for an earlier graphical procedure the reader is referred to Koutrouvelis and Canavos (1997). Koutrouvelis and Meintanis (1999) and Towhidi and Salmanpour (2007) proposed similar tests based on the empirical characteristic function, which can be easily adapted to the gamma null hypothesis.…”

Theory & Methods: Estimating the parameters of Poisson‐exponential models

“…The EMGF has been successfully employed in estimation and testing problems by Quandt & Ramsey (1978), Epps & Pulley (1985), Baringhaus & Henze (1991, 1992, Henze (1993) and others. Koutrouvelis & Canavos (1997) proposed a method of generalized least squares for estimation in the three-parameter gamma distribution, which employs the values of lnψ n (t) at appropriately rescaled negative values of t. A computationally simpler procedure, also developed by Koutrouvelis & Canavos (1999), estimates the parameters in a Pearson type 3 distribution by iterating between a linear regression model involving lnψ n (t) and the first moment equation. Such a method can be used in the present situation, since from (2) lnψ(t) is linear with respect to λ.…”

“…To assess the fit of the snowfall data to the hypothesized PE model via the EMGF, a graphical procedure similar to that of Koutrouvelis & Canavos (1997) can be used. We employ the MX1 estimate of β and replace t k by t k /β in (5) (k = 1, 2, .…”

Cumulant plots and goodness-of-fit tests for the inverse Gaussian distribution

“…Cumulant plots have been employed before by Ghosh [20], Koutrouvelis and Canavos [21] and Koutrouvelis et al [22] for exploring the appropriateness of the normal, three-parameter gamma (Pearson type III) and three-parameter inverse Gaussian (IG3) distribution, respectively. Unlike the normal cumulant plot of Ghosh [20], which is furnished with a simultaneous confidence band and is equivalent to a gof test, the plots developed in the last two papers are preliminary graphical tools.…”