Statistical interence based on studentized robust statistics

We considered two-sample and multivariate two-way manova model included in Y I D2=h(x i D2，θ)+εD2，i=1，...，n，θ∈ΘwhereεD2，i=1，...，n，are mutually independent and identically distributed with a p-variate continuous distribution function F(x，Σ)having null mean and finite positive definite variance-covariance matrix。In pratical applicational model assumptions，the scale-parameter of the underlying distribution is unknown and Fisher‘s consistency does not hold。We need to construct flexible statistical procedures。So scale invariant statistical procedures based on M-statistics were proposed.Their asymptotic noncentral x y D12 I D 1-distributions for testing homogeneity were drawn under a contiguous sequence of location-alternatives without assuming Fisher consistency：óψD 2 l I D 2(x I D 1(L)I D 1)dF I D 2 l I D 2(x I D 1(L)I D 1)=0。Asymptotic robustness was derived。The permutation tests based on the proposed M-test statistics were considered.Using a Monte Carlo simulation，their power was compared with permutation tests based on parametric test statistics.Next robust estimators for location parameters were proposed，based on studentized M-statistics.The asymptotic normality of these estimators was drawn.After a simple algorithm was studied，the risks of the M-estimators and the least squares estimators were compared due to a simulation。For a univariate case，it was found that(I)the asymptotic relative efficiency(ARE)of the proposed M-procedures relative to parametric procedures agreed with the ARE of one-sample M-estimator proposed by Huber(1964)relative to the sample mean，and that(Ii)for small sample sizes，the M-procedures were mefficient than parametric procedures except the case that underlying distribution normoral more mefficient than parametric procedures except the case that underlying distribution is normore.

We considered two-sample and multivariate two-way manova model included in YィイD2iィエD2=h(xィイD2iィエD2,θ) + εィイD2iィエD2, i=1,・・・, n,θ∈Θ where εィイD2iィエD2, i=1, ・・・, n, are mutually independent and identically distributed with a p-variate continuous distribution function F(x,Σ) having null mean and finite positive definite variance-covariance matrix Σ. In pratical applicational model assumptions, the scale-parameter of the underlying distribution is unknown and Fisher's consistency does not hold. We need to construct flexible statistical procedures. So scale invariant statistical procedures based on M-statistics were proposed. Their asymptotic noncentral xィイD12ィエD1-distributions for testing homogeneity were drawn under a contiguous sequence of location-alternatives without assuming Fisher consistency : ∫ψィイD2lィエD2(xィイD1(l)ィエD1)dFィイD2lィエD2(xィイD1(l)ィエD1)=0. Asymptotic robustness was derived. The permutation tests based on the proposed M-test statistics were considered. Using a Monte Carlo simulation, their power was compared with permutation tests based on parametric test statistics. Next robust estimators for location parameters were proposed, based on studentized M-statistics. The asymptotic normality of these estimators was drawn. After a simple algorithm was studied, the risks of the M-estimators and the least squares estimators were compared due to a simulation. For a univariate case, it was found that (i) the asymptotic relative efficiency (ARE) of the proposed M-procedures relative to parametric procedures agreed with the ARE of one-sample M-estimator proposed by Huber (1964) relative to the sample mean, and that (ii) for small sample sizes, the M-procedures were more efficient than parametric procedures except the case that an underlying distribution is normal.Moreover many computer soft programs were created and shrinkage estimators were discussed.