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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 5

Question

Math 6010 • Linear Models Fall 2016 • Question 5

If \(X_1,X_2,\ldots ,X_n\) is a random sample from a distribution having pdf of the form \(f(x;\theta )=\theta x^{\theta -1}\), \(0<x<1\) and zero elsewhere, show that a best critical region for testing \(H_0:\theta =1\) against \(H_1:\theta =2\) is \[ C=\left \{(x_1,x_2,\ldots ,x_n): c\leq \prod _{i=1}^n x_i\right \}. \]

Raw LaTeX

If \(X_1,X_2,\ldots,X_n\) is a random sample from a distribution
having pdf of the form \(f(x;\theta)=\theta x^{\theta-1}\), \(0<x<1\) and zero
elsewhere, show that a best critical region for testing \(H_0:\theta=1\) against
\(H_1:\theta=2\) is
\[
  C=\left\{(x_1,x_2,\ldots,x_n): c\leq \prod_{i=1}^n x_i\right\}.
\]

Topic Tags

Estimator Distributions Multivariate Normal Distributions Quadratic Forms and Chi-Squared Distributions Uniform Integrability and Truncation Methods