AISM 54, 918-933

## Independence of likelihood ratio criteria for homogeneity of several populations

### Takesi Hayakawa

Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan

(Received October 17, 2000; revised November 8, 2001)

Abstract.    Let $\Pi_i$ be an $i$-th population with a probability density function $f(\cdot \mid \theta_i)$ with one dimensional unknown parameter $\theta_i, i=1,2,\ldots,k$. Let $n_i$ sample be drawn from each $\Pi_i$. The likelihood ratio criteria $\lambda_{j \vert (j-1)}$ for testing hypothesis that the first $j$ parameters are equal against alternative hypothesis that the first $(j-1)$ parameters are equal and the $j$-th parameter is different with the previous ones are defined, $j=2,3,\ldots,k$. The paper shows the asymptotic independence of $\lambda_{j \vert (j-1)}$'s up to the order $1/n$ under a hypothesis of equality of $k$ parameters, where $n$ is a number of total samples.

Key words and phrases:    Likelihood ratio criterion, asymptotic expansion, homogeneity of parameters, asymptotic independence.

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