- مبلغ: ۸۶,۰۰۰ تومان
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The multivariate linear regression model is an important tool for investigating relationships between several response variables and several predictor variables. The primary interest is in inference about the unknown regression coefficient matrix. We propose multivariate bootstrap techniques as a means for making inferences about the unknown regression coefficient matrix. These bootstrapping techniques are extensions of those developed in Freedman (1981), which are only appropriate for univariate responses. Extensions to the multivariate linear regression model are made without proof. We formalize this extension and prove its validity. A real data example and two simulated data examples which offer some finite sample verification of our theoretical results are provided.
In this section we provide two simulated examples which show support for our multivariate bootstrap procedures.
3.1.1. Fixed design
This example illustrates Theorem 1. We generated data according to the multivariate linear regression model (1) where Yi ∈ R 3 , Xi ∈ R 2 , and both β and Σ are prespecified. Our goal is to make inference about vec(β) using confidence intervals. For each component of β, a 95% confidence interval computed using the residual bootstrap in Algorithm 1 is compared with a 95% confidence interval obtained from maximum likelihood estimation of β and Σ in model (1). Four data sets were generated at different sample sizes and the performance of the multivariate residual bootstrap is assessed. The bootstrap is performed B = 5000 times in each dataset. The results are displayed in Table 1. For the first two components of β, we see that the confidence regions obtained from both methods are close to each other and that the distance between the two shrinks as n increases. Similar results are obtained for the other components of β. A separate simulation study shows that the coverage probabilities of our bootstrap are at the nominal level for all of the considered cases.