Background

The Neyman Pearson Theorem provides a method of constructing most powerful tests for simple hypotheses when the distribution of the observation is known except for the value of a single parameter. But in many cases the problem is more complex than this. In this section we will examine a general method that can be used to derive tests of hypotheses. The procedure works for simple or composite hypotheses and whether or not there are `nuisance' parameters with unknown values.

As well as thinking of $H_0$ as being a statement (or assertion) about a parameter $\theta$, it is a set of values taken by $\theta$. Similarly for $H_1$. So it is appropriate to write $\theta \in H_0$, for example, or $\max_{H_0}L(\theta)$. The set of all possible values of $\theta$ is $H_0 \cup H_1$.

Let $f(x;\theta)$ be the density function of a random variable $X$ with unknown parameter $\theta$, and let $X_1, X_2, \ldots, X_n$ be a random sample from this distribution, with observed values $x_1,x_2,\ldots,x_n$. The likelihood function of the sample is

$$L(\theta)=L(\theta;x_1,\ldots,x_n)=\prod_{i=1}^{n}f(x_i;\theta)=f({\bf x};\theta).$$

It is necessary to have a clear idea of what is meant by the parameter space and that subset of it defined by the hypothesis.

Example 9..7

(a)

If $X$ is distributed as bin(n,p) and we are testing $H_0:p=p_0$, then $H_0$ can be written $H_0=\{p:p=p_0\}$ and $H_0 \cup H_1=\{p:\,p \in [0,\,1]\}$.

(b)

If $X$ is distributed as N($\mu$, $\sigma^2$) where both $\mu$ and $\sigma^2$ are unknown and we are testing $H_0:\sigma^2=\sigma_0^2$, then

$$H_0 \cup H_1=\{(\mu,\,\sigma^2):\mu \in(-\infty,\ \infty), \,\sigma^2 \in (0, \, \infty)\}$$

$$H_0=\{(\mu,\,\sigma^2):\,\mu \in (-\infty, \infty), \,\sigma^2=\sigma_0^2\}$$

Now (b) is illustrated in the diagram below.