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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 2

Question

Math 6010 • Linear Models Fall 2018 • Question 2

Let \(X_1,X_2,\ldots ,X_n\) be iid random variables with a Beta\((\theta ,\theta )\) distribution, i.e. their density is \[ f(x;\theta )=\frac {\Gamma (2\theta )}{\Gamma (\theta )^2} x^{\theta -1}(1-x)^{\theta -1}, \qquad 0<x<1. \] Show that a best critical region for testing \(H_0:\theta =1\) versus \(H_A:\theta =2\) is \[ \left \{(x_1,x_2,\ldots ,x_n): c\leq \prod _{i=1}^n x_i\right \}. \]

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid random variables with a
Beta\((\theta,\theta)\) distribution, i.e. their density is
\[
  f(x;\theta)=\frac{\Gamma(2\theta)}{\Gamma(\theta)^2}
  x^{\theta-1}(1-x)^{\theta-1}, \qquad 0<x<1.
\]
Show that a best critical region for testing \(H_0:\theta=1\) versus
\(H_A:\theta=2\) is
\[
  \left\{(x_1,x_2,\ldots,x_n): c\leq\prod_{i=1}^n x_i\right\}.
\]

Topic Tags

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