Question

4. (20\%) Let \( X_{1}, X_{2}, \ldots, X_{n} \) be an i.i.d. sample from a normal distribution with pdf \[ f(x ; \theta)=\frac{1}{\sqrt{2 \pi \theta}} \exp \left(-\frac{x^{2}}{2 \theta}\right), \quad x \in(-\infty, \infty), \] where \( \theta>0 \) is an unknown parameter. Determine whether there exists a UMP test with significance level \( \alpha \) for testing \( H_{0}: \theta=\theta_{0} \) against \( H_{1}: \theta<\theta_{0} \), where \( \theta_{0}>0 \) is known.

          4. (20\%) Let \( X_{1}, X_{2}, \ldots, X_{n} \) be an i.i.d. sample from a normal distribution with pdf
\[
f(x ; \theta)=\frac{1}{\sqrt{2 \pi \theta}} \exp \left(-\frac{x^{2}}{2 \theta}\right), \quad x \in(-\infty, \infty),
\]
where \( \theta>0 \) is an unknown parameter. Determine whether there exists a UMP test with significance level \( \alpha \) for testing \( H_{0}: \theta=\theta_{0} \) against \( H_{1}: \theta<\theta_{0} \), where \( \theta_{0}>0 \) is known.

4. (20%) Let X1, X2, …, Xn be an i.i.d. sample from a normal distribution with pdf

f(x ; θ)=(1)/(√(2 πθ))(-(x^2)/(2 θ)), x ∈(-∞, ∞),

where θ>0 is an unknown parameter. Determine whether there exists a UMP test with significance level α for testing H0: θ=θ0 against H1: θ<θ0, where θ0>0 is known.

Added by Neeraj S.

Introduction to Mathematical Statistics

Robert V. Hogg, Joseph W. McKean, Allen… 8th Edition

Chapter 4

Instant Answer

Solved by Expert Jerelyn Nevil

Step 1

The hypotheses to test are: \[ H_0: \theta = \theta_0 \quad \text{vs} \quad H_1: \theta < \theta_0 \] where \(\theta_0 > 0\) is a specified value. Show more…

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