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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2007 • Question 6

Question

Math 6010 • Linear Models Fall 2007 • Question 6

Let Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with distribution function \[ F(t)=\left (\frac {t}{\theta }\right )^2 \quad \text {if } 0\leq t\leq \theta . \]

(i) Find the maximum likelihood estimator for \(\theta \). (7 points)

(ii) Compute the limit distribution of the maximum likelihood estimator. (7 points)

Raw LaTeX

Let Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with distribution function
\[
  F(t)=\left(\frac{t}{\theta}\right)^2
  \quad\text{if } 0\leq t\leq\theta.
\]

(i) Find the maximum likelihood estimator for \(\theta\). (7 points)

(ii) Compute the limit distribution of the maximum likelihood estimator.
(7 points)

Topic Tags

Covariance and Independence Estimator Distributions Maximum Likelihood Estimation Uniform Integrability and Truncation Methods