Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: \theta \in \Theta_1$, where $\theta_0 \in \Theta$ and $\Theta_1$ is a subset of $\Theta$ such that $\theta_0 \notin \Theta_1$, at size $\alpha$. Suppose that a uniformly most powerful test $\delta^*$ exists and let $\beta^*$ denote its power function.

Assume the model is regular, that all power functions are defined for $\theta \in \Theta$, that $\theta_0$ is an interior point of $\Theta$, and that all power functions are differentiable at $\Theta_0$. Also assume that $\beta^*(\theta)=\alpha$.

Suppose that $\Theta = R$ and that $\Theta_1 =(\theta_0,\infty)$. Find an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$.

Attempted answer:

The probability of a Type I error is given as

$$\beta^*(\theta) \leq \int_k^\infty \frac{df(x;\theta)}{d\theta}dx = \int_k^\infty \frac{f(x;\theta)}{f(x;\theta)}* \frac{df(x;\theta)}{d\theta}dx= E_{\theta}\left[\delta(x)\frac{d\log f(x;\theta)}{d\theta}\right].$$

Here $\delta$ is a function that is 1 when the range is from $k$ to infinity, and zero otherwise. The expression needs to be evaluated at $\theta_0$. The final result is $E_{\theta_0}[\delta \frac{d\log f(x;\theta)}{d\theta}].$


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