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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 2

Question

Math 6010 • Linear Models Spring 2015 • Question 2

Find the maximum likelihood estimators of \(\theta \) and \(\alpha \), based on a random sample \(X_1,X_2,\ldots ,X_n\) from the distribution with density \[ f(x;\theta ,\alpha )=\frac {\alpha }{\theta ^\alpha }x^{\alpha -1}, \qquad 0\leq x\leq \theta . \] Here \(\theta \) and \(\alpha \) are positive.

Raw LaTeX

Find the maximum likelihood estimators of \(\theta\) and \(\alpha\),
based on a random sample \(X_1,X_2,\ldots,X_n\) from the distribution with density
\[
  f(x;\theta,\alpha)=\frac{\alpha}{\theta^\alpha}x^{\alpha-1},
  \qquad 0\leq x\leq\theta.
\]
Here \(\theta\) and \(\alpha\) are positive.

Topic Tags

Estimator Distributions Likelihood Ratio Tests Maximum Likelihood Estimation Uniform Integrability and Truncation Methods