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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 8

Question

Math 6010 • Linear Models 1996 • Question 8

Consider the linear model: \(Y_i=\beta +\epsilon _i\), \(1\le i\le n\). Here, \(\epsilon _i\)’s are i.i.d. \(N(0,\sigma ^2)\), and \(\sigma \) and \(\beta \) are unknown.

(a) Find the least squares estimator of \(\beta \);

(b) Find the UMVU estimator for \(\beta \) (prove the optimality);

(c) Find the UMP test for \(H_0:\beta =\beta _0\) versus \(H_1:\beta =\beta _1\);

(d) Discuss–without proofs–how to find the UMP test for \(H_0:\beta >\beta _0\) vs. \(H_1:\beta \le \beta _0\) from tests of the form in part (c) above.

Raw LaTeX

Consider the linear model: \(Y_i=\beta+\epsilon_i\), \(1\le i\le n\).
Here, \(\epsilon_i\)'s are i.i.d. \(N(0,\sigma^2)\), and \(\sigma\) and \(\beta\) are
unknown.

(a) Find the least squares estimator of \(\beta\);

(b) Find the UMVU estimator for \(\beta\) (prove the optimality);

(c) Find the UMP test for \(H_0:\beta=\beta_0\) versus \(H_1:\beta=\beta_1\);

(d) Discuss--without proofs--how to find the UMP test for
\(H_0:\beta>\beta_0\) vs. \(H_1:\beta\le \beta_0\) from tests of the form in part
(c) above.

Topic Tags

Estimator Distributions F-Tests and t-Tests Generalized Least Squares Linear Models