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The \(\varphi\)-divergence statistic in bivariate multinomial populations including stratification. (English) Zbl 0783.62044

If \(P=(p_ 1,\dots,p_ M)\) and \(Q= (q_ 1,\dots,q_ M)\) are two discrete distributions then the \(\varphi\)-divergence between \(P\) and \(Q\) is defined by \(I^ C(P\;\|\;Q)= \sum^ M_{i=1} q_ i\varphi(p_ i/q_ i)\). This definition is applied to a two-dimensional discrete distribution \(P_{XY}=\smash{(p_{ij})^ M_{i=1,}}^ K_{j=1}\) with marginals \(P_ X=(p_{1.},\dots,p_{M.})\) and \(P_ Y=(p_{.1},\dots,p_{.K})\). Then \[ I^ C(P_{XY} \|\;P_ X* P_ Y)=\sum\sum p_{i.} p_{.j}\varphi(p_{ij}/p_{i.} p_{.j}). \] The asymptotic distribution of an estimate of \(I^ C(P_{XY} \|\;P_ X* P_ Y)\) is derived and some tests of the hypothesis of independence based on this estimate are presented.
Reviewer: J.Anděl (Praha)

MSC:

62H17 Contingency tables
62E20 Asymptotic distribution theory in statistics
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References:

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