AN ASYMPTOTIC APPROXIMATION FOR THE DISTRIBUTION OF ^|^phi;-DIVERGENCE MULTINOMIAL GOODNESS-OF-FIT STATISTIC UNDER LOCAL ALTERNATIVES

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

“…The A a approximation is constructed from a linear combination of mutually independent noncentral chi-square random variables, and the B a approximation is constructed from a linear expression of a single noncentral chi-square random variable that has the same first moment as A a . Taneichi et al (2001) derived A φ and B φ approximations on the basis of A a and B a approximations for the φ-divergence family of statistics C φ . Let…”

“…Taneichi et al (2001) derived an N φ approximation based on the method of the NT approximation for the power divergence statistic for the φ-divergence family of statistics C φ . The N φ approximation to the power of C φ against the alternative hypothesis H 1 given by (1.5) is represented as Tables 1-5, we have the following results for the R 0.5 statistic.…”

“…On the other hand, Taneichi et al (2002) proposed the AE approximation, which is based on multivariate Edgeworth expansion assuming a continuous distribution for the distribution of C φ P a under H 1,n given by (1.4). Taneichi et al (2001) derived the AE approximation for the distribution of a φ-divergence statistic. The expression of the AE approximation for the power of C φ against 2.…”

“…It is known that C φ are asymptotically distributed according to a noncentral chi-square distribution with k − 1 degrees of freedom and noncentrality parameter Taneichi et al (2001) derived the local Edgeworth expansion under H 1,n and obtained an expression of approximation for the distribution of C φ under H 1,n . That expression consists of a term of multivariate Edgeworth expansion for a continuous distribution and a discontinuous term to account for the discontinuity in X .…”

“…The discontinuous term is expressed in a very complicated form, and it is too complicated to use in practice. Thus, using only the term of the multivariate Edgeworth expansion for a continuous distribution, Taneichi et al (2001) proposed an approximation to the power of C φ . Based on numerical investigation, it was concluded that omission of the discontinuous term does not lead to a serious error in approximating the power of the test based on C φ .…”