Gaussian parameter tests

As the Gaussian distribution is ubiquitous, there exists some hypothesis tests whose entire purpose is to validate the parameters of this kind of distribution. These tests should be used when the sample is composed of iid Gaussian random variables.

Mean test#

Given a random sample $\{ X_{1},\ldots,X_{N} \}$ of Gaussian variables $X_{i}\sim \mathcal{N}(\mu,\sigma^{2})$ , a classical test is to determine whether the mean $\mu$ is a specific value $\mu_{0}$ :

\begin{cases} H_{0}:\mu=\mu_{0} \\ H_{1}:\mu\neq \mu_{0} \end{cases}

The test statistic is given by

T=\frac{\bar{X}-\mu_{0}}{S/\sqrt{ N }}\sim t_{N-1}

where $S$ is the sample standard deviation. This statistic follows a Student's t distribution $t_{N-1}$ with $N-1$ degrees of freedom when $H_{0}$ is true. Because of this, it's also called the t-test.

Alternatively, if $\sigma$ is known, it can be used instead of $S$ . In that case, $T$ is usually called $Z$ and follows a standard normal distribution, $Z\sim \mathcal{N}(0,1)$ :

Z= \frac{\bar{X}-\mu_{0}}{\sigma/\sqrt{ N }}\sim \mathcal{N}(0,1)

This test is known as a Z-test.

Since Gaussian (and Student's t) distributions are symmetrical around the mean, the critical region is chosen to be double-sided, as both too large and too small values of $\mu$ don't satisfy $\mu=\mu_{0}$ . Given an a priori significance level $\alpha$ , in the case $T\sim \mathcal{N}(0,1)$ , the critical region is defined by the integrals

\int_{-\infty}^{-r_{\alpha/2}} \frac{e^{-r^{2}/2}}{\sqrt{ 2\pi }/\sqrt{ N }}dr=\int_{r_{\alpha/2}}^{+\infty} \frac{e^{-r^{2}/2}}{\sqrt{ 2\pi }/\sqrt{ N }}dr=\frac{\alpha}{2}

From this, you can find the critical region bound $r_{\alpha/2}$ for both sides. If $T\sim t_{N-1}$ , simply integrate the Student's t distribution instead.

It is also possible to use $\mu>\mu_{0}$ and $\mu<\mu_{0}$ as alternative hypothesis if you want to know if $\mu$ is an over- or underestimate. In these cases, the critical region is one-sided, since the equality permits one of the two sides.

Variance test#

The premise is the same as above, except we're testing $\sigma^{2}$ instead of $\mu$ :

\begin{cases} H_{0}:\sigma ^{2}=\sigma_{0}^{2} \\ H_{1}:\sigma ^{2}\neq \sigma_{0}^{2} \end{cases}

If we know $\mu$ and $H_{0}$ is true, the test statistic is

T=\sum_{i=1}^{N} \frac{(x_{i}-\mu)^{2}}{\sigma_{0}^{2}}\sim\chi ^{2}_{N}

which has chi-square distribution with $N$ degrees of freedom. The test is also double-tailed, and you integrate the $\chi ^{2}_{N}$ for the critical region. Given an $\alpha$ , you find the values $r_{\alpha/2}^{+}$ and $r_{\alpha/2}^{-}$ for which the integrals are both equal to $\alpha/2$ . However, since the $\chi ^{2}_{N}$ is asymmetrical for low $N$ , these values are going to be different, unlike above where there's a common $r_{\alpha/2}$ .