Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathsf{Ber}(p). We want to test
\begin{array}{rl} H_0: & p\le .5 \\ H_1: & p> .5\\ \end{array}
at asymptotic level 5%. Let \psi be the Wald test.For every \lambda >0 we can show that
\lim _{n \to \infty }\mathbf{P}_{.5-\frac{\lambda }{\sqrt{n}}}[\psi =1]=\mathbf{P}[Z>1.645+ f(\lambda )]
where f(\lambda ) is a function that depends only on \lambda. What is this function?
To find the function f(\lambda), we first need to calculate the probability \mathbf{P}_{.5-\frac{\lambda }{\sqrt{n}}}[\psi =1].
The Wald test statistic for testing p = 0.5 against p > 0.5 is given by:
W = (n * (\hat{p} - 0.5)) / sqrt(n * 0.5 * (1 - 0.5))
where \hat{p} is the sample proportion.
Under the null hypothesis H_0: p <= 0.5, the test statistic W approximately follows a standard normal distribution for large n. Therefore, we can calculate the probability using the standard normal distribution:
\mathbf{P}_{.5-\frac{\lambda }{\sqrt{n}}}[\psi =1] = \mathbf{P}\left[\frac{n(\hat{p} - 0.5)}{\sqrt{n \cdot 0.5 \cdot (1 - 0.5)}} > \frac{n(0.5 - 0.5 - \lambda/\sqrt{n})}{\sqrt{n \cdot 0.5 \cdot (1 - 0.5)}}\right]
Simplifying the expression, we get:
\mathbf{P}_{.5-\frac{\lambda }{\sqrt{n}}}[\psi =1] = \mathbf{P}[Z > -\lambda/\sqrt{0.25}] = \mathbf{P}[Z > -2\lambda]
Since we want the probability in terms of Z > 1.645 + f(\lambda), we rearrange the inequality:
-\lambda > 1.645 + f(\lambda)
Solving for f(\lambda), we have:
f(\lambda) = -\lambda - 1.645
Therefore, the function f(\lambda) is equal to -\lambda - 1.645.
To find the function f(λ), we need to determine the critical value corresponding to the level of significance α = 0.05 in the standard normal distribution.
Since ψ is the Wald test, it follows an asymptotic normal distribution under the null hypothesis. The test statistic Z is given by:
Z = (p_hat - p_0) / sqrt(p_0 * (1 - p_0) / n),
where p_hat is the sample proportion and p_0 is the null hypothesis value.
For this problem, p_0 = 0.5, and since the test is one-sided (p > 0.5), the critical value is given by:
1.645 + f(λ),
where 1.645 corresponds to the 95th percentile of the standard normal distribution.
To find f(λ), we need to solve the following equation:
P(Z > 1.645 + f(λ)) = 0.05,
where Z follows a standard normal distribution.
Using a standard normal distribution table or software, we can find the value of f(λ) that satisfies the equation.