島根大學

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島根大学論集. 自然科学 16

1966-12-25 発行

The estimate Procedures after a Preliminary multivariate nonparametric Two-Sample test

Tamura, Ryoji

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The sometimes pool procedures, which have been developed in the normal theory by Bancroft [3], Kitagawa [5] and many other authors, have been also discussed in the univariate nonparametric cases by the author [7], [8] and [9]. Moreover Asano-Sato [1] and Sato [2] have attempted some multivariate extensions in the normal theory. We are now in the situation to consider the multivariate extensions in the nonparametric case along the line of the previous papers of the author.

Our concern in this paper is the estimate procedure for the location parameters − (A) median vector and (B) shift vector. − The preliminary nonparametric test is performed by a multivariate two-sample statistic of Wilcoxon type which has been proposed by Sugiura [6]. As for the estimates of the location vectors, we use those of Hodges-Lehmann type [4] based on some rank tests.

(A) Let O_<n1> : {(X_<ki>, k=1,. . . .,p)} i=1,. . .,n_1 be a random sample of size n_1 from a continuous p-variate distribution F(x-θ_1) with continuous marginal distribution F_k (x-θ<k_1>) of the k th component X_k and continuous joint marginal distribution F<ij> (x-θ_<i1>, y-θ_<j1>) of the i th and j th components where X' = (X_1,.....,X_p),θ_1= (θ_<11>,....,θ_<p1>). We also assume that F_k (x) with density f_k (x) be symmetrical about origin for all k. We consider the case where there exists another random sample O_<n2> : {(Y_<kj>, k=1,. . . . , p)} j=1, . . . n_2 of size n_2 from the distribution F(y-θ_2) Where Y' = (Y_1,....., Y_p), θ2 = (θ_<12>, . .,θ_<p2>).

We here assumed that the distributions of the random vectors X and Y are of same type except only median vector. We shall apply sometimes pool methods to estimate the median vector θ_1.

(B) Let O_<n1> : {X_<kj>, k=1,. . .,p)}j=1,. . .,n_1, O_<n2> : {(Y_<kj>, k=1,. . .,p)} j=1,. . .,n_2 and O_<n3> : { (Z_<kj>, k=1,. . . . ,p)}j=1,. . . ,n_3 be respectively three random samples of size n_i i=1, 2, 3 from the continuous distributions F(x), F(y-⊿_1) and F(z-⊿_2) where ⊿_j = (⊿_<1j>,. . . , ⊿_<pj>)j=1, 2.

Our purpose is to estimate the value of the shift vector ⊿_1 based on the samples O_<ni>. Sometimes pool procedure is also applied.

Our concern in this paper is the estimate procedure for the location parameters − (A) median vector and (B) shift vector. − The preliminary nonparametric test is performed by a multivariate two-sample statistic of Wilcoxon type which has been proposed by Sugiura [6]. As for the estimates of the location vectors, we use those of Hodges-Lehmann type [4] based on some rank tests.

(A) Let O_<n1> : {(X_<ki>, k=1,. . . .,p)} i=1,. . .,n_1 be a random sample of size n_1 from a continuous p-variate distribution F(x-θ_1) with continuous marginal distribution F_k (x-θ<k_1>) of the k th component X_k and continuous joint marginal distribution F<ij> (x-θ_<i1>, y-θ_<j1>) of the i th and j th components where X' = (X_1,.....,X_p),θ_1= (θ_<11>,....,θ_<p1>). We also assume that F_k (x) with density f_k (x) be symmetrical about origin for all k. We consider the case where there exists another random sample O_<n2> : {(Y_<kj>, k=1,. . . . , p)} j=1, . . . n_2 of size n_2 from the distribution F(y-θ_2) Where Y' = (Y_1,....., Y_p), θ2 = (θ_<12>, . .,θ_<p2>).

We here assumed that the distributions of the random vectors X and Y are of same type except only median vector. We shall apply sometimes pool methods to estimate the median vector θ_1.

(B) Let O_<n1> : {X_<kj>, k=1,. . .,p)}j=1,. . .,n_1, O_<n2> : {(Y_<kj>, k=1,. . .,p)} j=1,. . .,n_2 and O_<n3> : { (Z_<kj>, k=1,. . . . ,p)}j=1,. . . ,n_3 be respectively three random samples of size n_i i=1, 2, 3 from the continuous distributions F(x), F(y-⊿_1) and F(z-⊿_2) where ⊿_j = (⊿_<1j>,. . . , ⊿_<pj>)j=1, 2.

Our purpose is to estimate the value of the shift vector ⊿_1 based on the samples O_<ni>. Sometimes pool procedure is also applied.

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