Wald Test

Table of contents

1. Hypothesis
2. Distribution / Test Statistic
3. Asymptotic Test

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Hypothesis

Let $\theta$ be the parameter of interest.

The null hypothesis is:

$$ H_0: \theta = \theta_0 $$

The alternative hypothesis is:

$$ H_1: \theta \neq \theta_0 $$

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Distribution / Test Statistic

Let $\hat{\theta}$ be the MLE of $\theta$.

Using the fact that MLE is asymptomatically normal, and assuming that the null hypothesis is true,

\[\frac{\hat{\theta} - \theta_0}{\hat{\text{se}}(\hat{\theta})} \leadsto N(0, 1)\]

The Wald statistic is:

$$ W = \frac{\hat{\theta} - \theta_0}{\hat{\text{se}}(\hat{\theta})} $$

And at the significance level $\alpha$, we reject the null hypothesis if:

$$ |W| > z_{\alpha/2} $$

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Asymptotic Test

Wald test is an asymptotic test, meaning that it is valid for large sample sizes.

When the sample size is small, consider using other tests like the t-test.