The Existence Of UMRE Estimators Under Balanced Loss And The Optimality Of Truncated Mean

'Linear model is an important branch of Mathematical Statistics. The parameter estimation on linear model has been an important research subject. Actually, the reason we study the equivariant estimator of parameter comes from a ultimate idea: when we change some unit or the point of reference of some quantity for some reason, the corresponding quantitative value will be changed correspondingly. Does the estimator of parameter in the question change accordingly? A"good"estimator should be consistent, i.e., if the parameter is not changed, the estimator will not change. If the parameter is changed, the estimator will be changed consistently. One of the problems of equivariant estimator of parameter is studying the best equivariant estimator under some meaning. This is a main problem investigated in this paper.Consider the general multivariate linear modelwhere X is a known matrix withΥ(X)= p,θis an unknown parameter, called regression coefficient;Σ≥0 is a parameters matrix, nevertheless parameters matrix V≥0 is assumed to vary in a set of nonnegative definite matrices with a common column space. T= V+XX'. For any given V0∈γ, let T0= V0+XX',(?)= {X'T+X)-1X'T+Y, Px= (X'T+X)-1X'T+For the necessary and sufficient condition on the existence of the uniformly minimum risk equivariant(UMRE) estimator of regression coefficient matrix in mul-tivariate linear model under an affine group or a transitive group of transformations for a matrix balanced loss function. The following theorem was proven:Theorem 3.2.1. In model (1.3) and for balanced loss function (1.7), if V0∈γandμ(V0)=μ(V) for any V∈γ, then the following statements are equivalent:(a) The UMRE estimator of (?) exists. (b) (?)is the UMRE estimator of (?).(c)(X'T0+X)-lX'T0+V［In-X(X'T0+X)-1X'T0+］'= 0 for all V∈Υ.In order to estimate the population mean, we often use sample mean. However, sample mean has a defect:it is too sensitive to exceptional values. In practical problems, it is hard to avoid exceptional values in a sample as all sorts of reasons. In order to keep the optimality of sample mean and improve its disadvantage at the same time, the truncated mean is used instead. We can predict intuitively that if the exceptional value is not quite different with normal values, then the truncated mean should not be better than the sample mean. However, if the exceptional value is quite different from normal values, the truncated mean should be better than the sample mean. So how much is the critical point? This is the second question that this paper will study. We have obtained the following two conclusions by using numerical simulation:Conclusion 4.2.1. In general, if X1,...,Xn come from the same population: N(μ,σ2), the mathematical expectations of the sample mean and the truncated mean are the same, but the variance of the sample mean is less than the truncated mean, it means that the sample mean is better than the truncated mean as the estimator of population mean.Conclusion 4.3.1. If X1,..., X10 come from two population, where X1is from N(μ1,a2), X2,...,X10 is from N(μ2,σ2).If|μ1 -μ2|< 1.667σ, the sample mean is better than the truncated mean as the estimator of population mean. Whereas, if |μ1-μ2|> 1.667σ, the truncated mean is better.