# Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Article

## Summary

The members of the power divergence family of statistics$$2\left( {\lambda \left( {\lambda + 1} \right)} \right)^{ - 1} \sum\limits_{j = 1}^k {X_j \left[ {\left( {{{X_j } \mathord{\left/ {\vphantom {{X_j } {n\pi _{0j} }}} \right. \kern-\nulldelimiterspace} {n\pi _{0j} }}} \right)^2 - 1} \right]}$$ all have an asymptotically equivalent χ2 distribution (Cressie and Read [1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX 2 statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.

### Key words

Chi-square statistic Edgeworth expansion PearsonX2 likelihood ratio second order limit distribution

## Preview

### References

1. [1]
Cressie, N. A. C. and Read, T. R. C. (1984). Multinomial goodness-of-fit tests,J. R. Statist. Soc., B, (to appear).Google Scholar
2. [2]
Esséen, C. G. (1945). Fourier analysis of distribution functions: a mathematical study of the Laplace-Gaussian law,Acta Math., Stockholm,77, 1–125.
3. [3]
Hoel, P. G. (1938). On the chi-square distribution for small samples,Ann. Math. Statist.,9, 158–165.
4. [4]
Rao, R. R. (1961). On the central limit theorem inR k,Bull. Amer. Math. Soc.,67, 359–361.
5. [5]
Read, T. R. C. (1984). Small sample comparisons for the power divergence goodness-of-fit statistics, (submitted).Google Scholar
6. [6]
Siotani, M. and Fujikoshi, Y. (1980). Asymptotic approximations for the distributions of multinomial goodness-of-fit tests,Technical report,14, Statistical Research Group, Hiroshima University.Google Scholar
7. [7]
Yarnold, J. K. (1972). Asymptotic approximations for the probability that a sum of lattice random vectors lies in a convex set,Ann. Math. Statist.,43, 1566–1580.