# Best approximation for the size of a test

Let $$X \sim \mathrm{Bernoulli}(\vartheta)$$ for some unknown $$\vartheta \in (0,1)$$, and let $$(X_1, …, X_n)$$ be a moderately large IID sample for $$X$$.

Let $$\vartheta_0 \in (0,1)$$. I want to test $$H_0 \colon \vartheta \leq \vartheta_0$$ versus $$H_1 \colon \vartheta > \vartheta_0$$.

An UMP size $$\alpha$$ test for this problem is the one that $$\begin{cases} \text{rejects H_0 if \overline X > c} \\ \text{rejects H_0 with probability \gamma if \overline X = c} \\ \text{does not reject H_0 if \overline X < c} \end{cases}$$ for some constants $$c \in (0,1)$$ and $$\gamma \in [0,1]$$ such that $$\mathbb P_{\vartheta_0}(\overline X > c) + \gamma \, \mathbb P_{\vartheta_0}(\overline X = c) = \alpha$$, where $$\mathbb P_{\vartheta_0}$$ denotes the probability calculated under the assumption that $$\vartheta = \vartheta_0$$.

Now, both $$\dfrac{\overline X - \vartheta_0}{\sqrt{\frac{\vartheta_0(1-\vartheta_0)}{n}}}$$ and $$\dfrac{\overline X - \vartheta_0}{\sqrt{\frac{\overline X(1-\overline X)}{n}}}$$ converge in distribution to a $$N(0,1)$$ as $$n \to +\infty$$.

Therefore we can approximate $$\mathbb P_{\vartheta_0}(\overline X > c)$$ with

1. $$\mathbb P \left( N(0,1) > \dfrac{c - \vartheta_0}{\sqrt{\frac{\vartheta_0(1-\vartheta_0)}{n}}} \right)$$
2. $$\mathbb P \left( N(0,1) > \dfrac{c - \vartheta_0}{\sqrt{\frac{\overline x(1-\overline x)}{n}}} \right)$$ where $$\overline x$$ is the realization of $$\overline X$$

to be able to determine $$c$$, while $$\mathbb P_{\vartheta_0}(\overline X = c)$$ is either exactly $$0$$ or very close to $$0$$.

My question is the following: intuitively I would say that the first one is the best approximation for $$\mathbb P_{\vartheta_0}(\overline X > c)$$, because it contains fewer estimates; but how can I prove that this must be the case?

• Since the mean is equivalent to a regression on a constant and since using null values or estimates relates to LM and Wald statistics, this may be helpful stats.stackexchange.com/questions/449494/… Commented Aug 8, 2022 at 13:00