Assume that the sample $X_i, i=1,\dots, n$, is iid normally distributed with unknown mean and variance $\theta = (\mu, \sigma^2)$. The Wald test statistic of testing if $\mu = \mu_0$ is
$$W=\frac{n(\hat{\mu} – \mu_0)^2}{\hat{\sigma}^2}$$
where $\hat{\sigma}^2 = \frac{\sum_i (X_i – \hat{\mu})^2}{n}$ is the maximum-likelihood estimator of $\sigma^2$.
- I was wondering if it is correct that $\frac{(n-1)W}{n^2}$ has a F
distribution with $1$ and $n-1$ d.f. under null? -
From a note, it seems that $W$ is an F distribution with $1$
and $n-1$ d.f. under null. Isn't this incorrect? Let $p_2$ be the dimension of $\mu$, and
$p$ be dimension of $X_i$.Under the assumption of normality we have a stronger result. The distribution of $W$ is exactly chi-squared with $p_2$ degrees of
freedom if $σ^2$ is known. In the more general case where $σ^2$ is
estimated using a residual sum of squares based on $n-p$ d.f., the
distribution of $W/p_2$ is an $F$ with $p_2$ and $n-p$ d.f.
Thanks and regards!
Best Answer
To emphasize that the distance between the point estimate $\hat{\mu}$ and the hypothesized value $\mu$ is being scaled by its standard error, I find it easier to write $W$ as
$$ W = \frac{(\hat{\mu} - \mu_0)^2}{\hat{\sigma}^2/n}. $$
Recall the relationship between the $\hat{\sigma}^2$ and the unbiased sample variance $s^2$:
$$ \hat{\sigma}^2 = \frac{n-1}{n} s^2. $$
Now, notice that we can write
$$ \frac{n-1}{n} W = \frac{(\hat{\mu} - \mu_0)^2}{s^2/n}. $$
This is useful for a couple of reasons. First, recognize that the square root of the righthand side is the statistic for the one-sample Student's t-test, which has a Student's t sampling distribution with $n-1$ degrees of freedom under the null hypothesis assuming the sample is iid normally distributed. That is,
$$ \sqrt{\frac{n-1}{n} W} = \frac{\hat{\mu} - \mu_0}{s/\sqrt{n}} \sim t_{n-1}. $$
Next, recall a relationship between the Student's t and (central) F distributions: if $Y \sim t_{\nu}$, then $Y^2 \sim F$ with degrees of freedom 1 and $\nu$. Therefore,
$$ \frac{n-1}{n} W \sim F $$ with degrees of freedom 1 and $n-1$.
The note that you linked in point 2 does not explicitly apply here. First, as you stated, $\sigma^2$ is unknown. Also, you have described the classic one-sample Student's t-test, whereas the link is describing a more general case (i.e., testing regression coefficients). The dimension concepts you quoted are referring to multivariate problems. You can see a connection though by noting that the dimension here is $p = 1$.