The concepts introduced above can be extended in several ways. In what follows we provide more rigorous definitions, describe different kinds of hypothesis testing, and introduce terminology.

**Data**

In the most general setup, the observation is
a collection

**Binary Versus M-ary Tests**

When there are two competing hypotheses, we
refer to a binary hypothesis test. When the
number of hypotheses is

#### Example 2

**Phase-Shift Keying**

Suppose we
wish to transmit a binary string of length

In many binary hypothesis tests, one
hypothesis represents the absence of a ceratin
feature. In such cases, the hypothesis is usually
labelled

#### Example 3

**Waveform Detection**

Consider the problem of detecting a known
signal

**Tests and Decision Regions**

Consider the general hypothesis testing
problem where we have

**Simple Versus Composite Hypotheses**

If the distribution of the data under a certain hypothesis is fully known, we call it a simple hypothesis. All of the hypotheses in the examples above are simple. In many cases, however, we only know the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know the variance of the noise. In this case, a hypothesis is said to be composite.

#### Example 4

Consider the problem of detecting the signal

Often a test involving a composite
hypothesis has the form

#### Example 5

Suppose a coin turns up heads with
probability