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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 6

Question

Math 6010 • Linear Models 1996 • Question 6

Suppose \(X_1,\ldots ,X_n\) is an i.i.d. sample from a normal population with \[ \mathbb {E} X_1=\operatorname {Var}(X_1)=\mu >0. \]

(a) Compute the maximum likelihood estimator \(\hat {\mu }\) of \(\mu \);

(b) Is \(\hat {\mu }\) consistent?

(c) Is \(\hat {\mu }\) asymptotically normal?

Raw LaTeX

Suppose \(X_1,\ldots,X_n\) is an i.i.d. sample from a normal population
with
\[
\E X_1=\Var(X_1)=\mu>0.
\]

(a) Compute the maximum likelihood estimator \(\hat{\mu}\) of \(\mu\);

(b) Is \(\hat{\mu}\) consistent?

(c) Is \(\hat{\mu}\) asymptotically normal?

Topic Tags

Estimator Distributions Maximum Likelihood Estimation Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods