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Tests and decision regions

Consider the general hypothesis testing problem where we have N d -dimensional observations x 1 , , x N and M hypotheses. If the data are real-valued, for example, then a hypothesistest is a mapping : d N 1 M For every possible realization of the input, the test outputs a hypothesis. The test partitions the input space into a disjoint collection R 1 , , R M , where R k ( x 1 , , x N ) | x 1 x N k The sets R k are called decision regions . The boundary between two decision regions is a decision boundary . depicts these concepts when d 2 , N 1 , and M 3 .

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 onlyknow the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know thevariance of the noise. In this case, a hypothesis is said to be composite .

Consider the problem of detecting the signal s n 2 f 0 n k n n 1 N where k is an unknown delay parameter. Then H 0 : x w H 1 : x s w is a binary test of a simple hypothesis ( H 0 ) versus a composite alternative. Here we are assuming w n 0 2 , with 2 known.

Often a test involving a composite hypothesis has the form H 0 : 0 H 1 : 0 where 0 is fixed. Such problems are called two-sided because the composite alternative "lies on both sides of H 0 ." When is a scalar, the test H 0 : 0 H 1 : 0 is called one-sided . Here, both hypotheses are composite.

Suppose a coin turns up heads with probability p . We want to assess whether the coin is fair( p 1 2 ). We toss the coin N times and record x 1 , , x N ( x n 1 means heads and x n 0 means tails). Then H 0 : p 1 2 H 1 : p 1 2 is a binary test of a simple hypothesis ( H 0 ) versus a composite alternative. This is also a two-sided test.

Errors and probabilities

In binary hypothesis testing, assuming at least one of the two models does indeed correspond to reality, thereare four possible scenarios:

  • Case 1

    H 0 is true, and we declare H 0 to be true
  • Case 2

    H 0 is true, but we declare H 1 to be true
  • Case 3

    H 1 is true, and we declare H 1 to be true
  • Case 4

    H 1 is true, but we declare H 0 to be true
In cases 2 and 4, errors occur. The names given to these errors depend on the area of application. In statistics, theyare called type I and type II errors respectively, while in signal processing they are known as a false alarm or a miss .

Consider the general binary hypothesis testing problem H 0 : x f x , 0 H 1 : x f x , 1 If H 0 is simple, that is, 0 0 , then the size (denoted ), also called the false-alarm probability ( P F ), is defined to be P F 0 declare H 1 When 0 is composite, we define P F sup 0 declare H 1 For 1 , the power (denoted ), or detection probability ( P D ), is defined to be P D declare H 1 The probability of a type II error, also called the miss probability , is P M 1 P D If H 1 is composite, then is viewed as a function of .

Criteria in hypothesis testing

The design of a hypothesis test/detector often involves constructing the solution to an optimizationproblem. The optimality criteria used fall into two classes: Bayesian and frequent.

Representing the former approach is the Bayes Risk Criterion . Representing the latter is the Neyman-Pearson Criterion . These two approaches are developed at length in separate modules.

Statistics versus engineering lingo

The following table, adapted from Kay, p.65 , summarizes the different terminology for hypothesis testing from statistics and signal processing:

Statistics Signal Processing
Hypothesis Test Detector
Null Hypothesis Noise Only Hypothesis
Alternate Hypothesis Signal + Noise Hypothesis
Critical Region Signal Present Decision Region
Type I Error False Alarm
Type II Error Miss
Size of Test ( ) Probability of False Alarm ( P F )
Power of Test ( ) Probability of Detection ( P D )

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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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