Asymptotic approximations for the distributions of the (h_‐, φ_‐)‐divergence goodness‐of‐fit statistics: application to Renyi’s statistic

Abstract

IntroductionThe Kullback-Leibler divergence for arbitrary distributions P = (p 1 , …, p M ) t and Q = (q 1 , …, q M ) t is defined by: This expression is the most famous particular case of the ϕ-divergence defined by Csiszár (1967):for any continuous convex function ϕ:[0, ∞) → IR∪{∞}, where 0 ϕ( 0 0 ) = 0 and, Other important ϕ-divergences are: variational or statistical with ϕ(x) = |x -1|, χ 2 -divergence or Kagan with ϕ(x) = (1 -x) 2 , Matusita with ϕ(x) = |1 -x a | 1/a 0 < a < 1, Balakrishman and Sanghvi with, ϕ(x) = (x -1) 2 (x + 1) -1. Rathie and Kannappan (1972 Cressie and Read (1984) with: ϕ(x) = (x a+1 -x) (a(a + 1) -1 a ≠ 0, a ≠ -1, Rukhin (1994) with ϕ(x) = (a + (1a)x) -1 , 0 ≤ a < 1, Lin (1991) with ϕ(x) = axlogx -(ax + 1 -a) log(ax + 1 -a).

Improved Approximations for the Distributions of Multinomial Goodness-of-fit Statistics Based on φ-divergence under Nonlocal Alternatives

“…Yarnold (1972) obtained an approximation based on asymptotic expansion for the null distribution of Pearson's X 2 statistic. In a similar fashion to X 2 , asymptotic expansions for the null distributions of the log-likelihood ratio statistic and the Freeman-Tukey statistic were obtained by Siotani and Fujikoshi (1984), that of the power divergence statistic was obtained by Read (1984), and that of C φ was obtained by Menéndez et al (1997). Against H 0 , consider the local alternative hypothesis…”

Improved transformed deviance statistic for testing a logistic regression model

“…. , 8) is the incidence of neoplasms. The logistic regression model which we apply is given by (1.1) with p = 2 and x α1 = 1 and x α2 = x α , (α = 1, .…”

“…The expansion consists of a term of multivariate Edgeworth expansion for a continuous distribution and a discontinuous term. In a fashion similar to that for Pearson's X 2 statistic, approximations based on asymptotic expansions for null distributions of some kinds of multinomial goodness-of-fit statistics have been investigated [12,10,8]. Edgeworth approximations of the distributions of some kinds of multinomial goodness-of-fit statistics under alternative hypotheses have also been investigated [13,14,11].…”

Improved transformed statistics for the test of independence in r×s contingency tables

“…The expansion consists of a term of multivariate Edgeworth expansion for a continuous distribution and a discontinuous term. In a fashion similar to that for Pearson's X 2 statistic, approximations based on asymptotic expansions for null distributions of the log likelihood ratio test statistic and the Freeman-Tukey statistic were obtained by Siotani and Fujikoshi [16], an approximation of the power divergence statistics was obtained by Read [12] and an approximation of the -divergence statistics was obtained by Menéndez et al [11]. The numerical accuracy of the approximation was shown by Yarnold [19] for Pearson's X 2 statistic and by Read [13] for power divergence statistics.…”