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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 8

Question

Math 6010 • Linear Models 1994 • Question 8

Consider the linear model: \[ Y_{ij}=\beta _i+\epsilon _{ij},\qquad 1\le i\le 2,\qquad 1\le j\le J. \] Suppose \(\epsilon _{ij}\)’s are independent and for some (known) \(a_1\) and \(a_2\), \(\epsilon _{ij}\sim N(0,a_i^2\sigma ^2)\).

(a) Find the U.M.V.U.E.’s of \(\beta _1\) and \(\beta _2\).

(b) Suppose you know that for some unknown \(\beta \), \(\beta _1=\beta _2\). Find the U.M.V.U.E. of \(\beta \).

Raw LaTeX

Consider the linear model:
\[
Y_{ij}=\beta_i+\epsilon_{ij},\qquad 1\le i\le 2,\qquad 1\le j\le J.
\]
Suppose \(\epsilon_{ij}\)'s are independent and for some (known) \(a_1\) and \(a_2\),
\(\epsilon_{ij}\sim N(0,a_i^2\sigma^2)\).

(a) Find the U.M.V.U.E.'s of \(\beta_1\) and \(\beta_2\).

(b) Suppose you know that for some unknown \(\beta\), \(\beta_1=\beta_2\). Find the
U.M.V.U.E. of \(\beta\).

Topic Tags

Covariance and Independence Estimator Distributions Linear Models Sufficiency, Completeness, and UMVUE