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:

$$H_1:\theta \ne {\theta }_0$$

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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,

$$\frac{\hat{\theta }-{\theta }_0}{\widehat{\text{se}}(\hat{\theta })}⇝N(0,1)$$

The Wald statistic is:

$$W=\frac{\hat{\theta }-{\theta }_0}{\widehat{\text{se}}(\hat{\theta })}$$

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.