Let $\mathbf{X}=(X_1,...,X_n)^T$ is a simple sample where $X$ belongs to exponential distribution family $\mathcal{P}=\{ f(x;\mu,\sigma \}, -\infty<\mu<\infty, 0<\sigma<\infty.$ Density is $$f(x;\mu,\sigma)=\sigma e^{-\sigma(x-\mu)}, \mu<x<\infty.$$

I need to find the uniformly most powerful test for hypothesis $H:\sigma=\sigma_0$ when $\mu$ is known.

So I think alternative hypothesis is $\overline{H}: \sigma>\sigma_0.$

I also know that $\frac{f_1(X)}{f_0(X)}>c.$

And $\varphi(x)= \begin{cases} 1, \text{ when } f_1(X)>cf_0(X) \\ \gamma, \text{ when } f_1(X)=cf_0(X) \\ 0, \text{ when } f_1(X)<cf_0(X) \end{cases}$

In my case I got $$\frac{f_1(X)}{f_0(X)}=\frac{\sigma_1 e^{-\sigma_1(x-\mu)}}{\sigma_0 e^{-\sigma_0(x-\mu)}}$$

$$\frac{\sigma_1 e^{-\sigma_1(x-\mu)}}{\sigma_0 e^{-\sigma_0(x-\mu)}}>c$$

$$\log \frac{\sigma_1 e^{-\sigma_1(x-\mu)}}{\sigma_0 e^{-\sigma_0(x-\mu)}}> \log c$$


Am I right? But what to do next? From theory I know that $\overline{X}>c'$. But how to apply it here?

  • $\begingroup$ If this is some sort of homework, please add the 'self-study' tag. $\endgroup$ – StubbornAtom Mar 20 '19 at 11:38
  • $\begingroup$ You need to replace the unknown parameters with their MLEs under each of the hypotheses $\endgroup$ – Glen_b -Reinstate Monica Mar 20 '19 at 12:03

You should be working with the density of the sample $X_1,X_2,\ldots,X_n$, given by

$$f_{\sigma}(x_1,x_2,\ldots,x_n)=\sigma^n \exp\left[-\sigma\sum_{i=1}^n(x_i-\mu)\right]\mathbf1_{x_{(1)}>\mu}\quad,\sigma>0,\mu\in\mathbb R$$

You are testing the hypothesis $H_0:\sigma=\sigma_0$ against the alternative $H_1:\sigma=\sigma_1(>\sigma_0)$.

The likelihood ratio is therefore

\begin{align} \frac{f_{H_1}(x_1,x_2,\ldots,x_n)}{f_{H_0}(x_1,x_2,\ldots,x_n)}&=\left(\frac{\sigma_1}{\sigma_0}\right)^n e^{-n(\bar x-\mu)(\sigma_1-\sigma_0)} \\&=ke^{-n\bar x(\sigma_1-\sigma_0)}\qquad,\small\text{ for some positive constant }k \end{align}

Since you are to reject $H_0$ for large values of this ratio, the critical region is of the form

\begin{align} ke^{-n\bar x(\sigma_1-\sigma_0)}>\text{ constant }\implies \bar x<\text{ constant } \end{align}


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