Minimum Φ-divergence Estimator for Homogeneity in Multinomial Populations∗

A problem which is frequently encountered in practice is that of deciding whether some sets of quantitative data are all derived from the same distribution. In this context consider υ independent random samples X = ( X 1 , ..., X (1) n1∗ )t ,...,X = ( X 1 , ..., X (υ) nυ∗ )t , of sizes n1∗, ..., nυ∗ respectively. The question is now to decide if the samples X, ...,X are all derived from the same distribution function F (x) = Q (X ≤ x) , x ∈ R and Q a probability measure on the real line. A way to approach the solution of this problem is to define a partition of the real line into m mutually exclusive and exhaustive intervals, say I1, ..., Im, where Pr ( X k ∈ Ij ) = pij for i = 1, ..., υ, j = 1, ...,m and k = 1, ..., ni∗. If we de-