Estimates of MM type for the multivariate linear model.

KUDRASZOW NADIA L.; MARONNA RICARDO A.

Naiguatá

Congreso; Congreso Latinoamericano de Probabilidad y Estadística Matemática, XI CLAPEM.; 2009

Bernoulli Society (Latin American Chapter)

Let y_i=(y_i1,...,y_iq)´ and x_i=(x_i1,...,x_ip)´ with i=1,...,n be the response and predictor vectors, respectively, satisfying the multivariate linear model  y_i=B_0´x_i+u_i, where B_0 is a pxq matrix and u_1,...,u_n are i.i.d. q-dimensional vectors. The x_i are assumed random, i.i.d. and independent of the u_i. If the errors u_i´s have multivariate normal distribution N_q(0,V_0), the classic estimate of this model is the maximum likelihood estimate (MLE) of B_0 that can be obtained by computing the least squares estimate (LSE) for each component of y separately, and the MLE of V_0 is the sample covariance matrix of the corresponding residuals. This estimates are regression equivariant and equivariant under linear transformations of x_i and y_i. However, the lack or robustness of the LSE is well known: a small fraction of bad observations (outliers) may have a large effect on the value of the estimate. In view of this, several robust alternatives have been proposed, although just a few of the proposed estimates conserve the properties of equivariance and have a high efficiency under the normal model. In order to cover both needs we presented MM-estimates for the multivariate linear model. MM-estimates of univariate linear regression were introduced by Yohai (1987) to combine robustness and efficiency. We propose the following extension to the multivariate case. Let (A_n,W_n) (with |W_n|=1) be an initial (possibly inefficient) estimate of (B_0,V_0) with high breakdown point. We define the quadratic Mahalanobis norm d_i^2(B,V)=(y_i-{B´x}_i)´V^{-1}(y_i-{B´x}_i). Let s_n be an M-estimate of scale of d_i(A_n,W_n), with i=1,..,n, based on a "rho-function" r_0.  Let r_1 be another rho-function such that r_1 is less than or equal to r_0. Hence the MM-estimate of multivariate regression (B_n,V_n) is given by sum_{i=1}^{n} r _1( d_i(B,C)/s_n) =min and V_n=(s_n)^{2} C, with B a matrix of pxq and C a positive definite symmetric qxq matrix with |C|=1. We prove that these estimates have breakdown point close to 0.5, are consistent and asymptotically normal, and may attain a high asymptotic efficiency under normal errors. References: [1] Yohai V. (1987). High Breakdown-point and high efficiency estimates for regression, The Annals of Statistics, 15, 642-656.