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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 3

Question

Math 6010 • Linear Models Fall 2015 • Question 3

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ h(t;\sigma )=\frac {1}{(2\pi \sigma ^2)^{1/2}} e^{-(t-1)^2/(2\sigma ^2)}. \] We wish to test \(H_0:\sigma =\sigma _0\) against \(H_A:\sigma \ne \sigma _0\).

(a) Find a test, using the generalized likelihood ratio \(\lambda \).

(b) Provide a large sample approximation for \(-2\log \lambda \) under \(H_0\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed
random variables with density function
\[
h(t;\sigma)=\frac{1}{(2\pi\sigma^2)^{1/2}}
e^{-(t-1)^2/(2\sigma^2)}.
\]
We wish to test \(H_0:\sigma=\sigma_0\) against \(H_A:\sigma\ne\sigma_0\).

\textbf{(a)} Find a test, using the generalized likelihood ratio \(\lambda\).

\textbf{(b)} Provide a large sample approximation for \(-2\log\lambda\) under
\(H_0\).

Topic Tags

Covariance and Independence F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation