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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 \[ EX_1=\operatorname{Var}(X_1)=\mu>0. \] **(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
\[
EX_1=\operatorname{Var}(X_1)=\mu>0.
\]
\begin{parts}
\item Compute the maximum likelihood estimator \(\hat{\mu}\) of \(\mu\);
\item Is \(\hat{\mu}\) consistent?
\item Is \(\hat{\mu}\) asymptotically normal?
\end{parts}

Topic Tags

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