What Is Hypothesis Testing?

Hypothesis testing, also known as statistical hypothesis testing, is a statistical inference method used to determine whether the differences between samples and samples and between samples and populations are caused by sampling errors or essential differences. Significance testing is the most commonly used method in hypothesis testing, and it is also the most basic form of statistical inference. The basic principle is to first make some assumptions about the characteristics of the population, and then use statistical reasoning from sampling research to make this assumption. Should be rejected or accepted for inference. Commonly used hypothesis testing methods include Z test, t test, chi-square test, F test, etc. ^[1] .

The basic idea of hypothesis testing is the principle of "small probability events", and its statistical inference method is a contradiction method with some probability properties. The idea of small probability means that a small probability event will not occur in a test. The idea of contradiction is to first test hypotheses, and then use appropriate statistical methods and the principle of small probability to determine whether the hypotheses are true. That is, in order to test whether a hypothesis H ₀ is correct, first assume that the hypothesis H _{0 is} correct, and then make a decision to accept or reject the hypothesis H ₀ according to the sample. If the sample observations cause a "small probability event", the hypothesis H ₀ should be rejected, otherwise the hypothesis H ₀ should be accepted ^[1]

1. The test hypothesis is also called invalid hypothesis, the symbol is H ₀ ; the symbol of the alternative hypothesis is H _{1 ^[2]}

The basic idea of hypothesis testing is to make statistical judgments using the principle of "small probability events", and whether the "small probability events" occur is related to the sample obtained from one sampling and the selected significance level , due to the randomness and selection of the samples The significance level is different, so the test results may not match the real situation, so it is assumed that the test is likely to make mistakes ^[1]

In radar detection, the target is the source that generates the hypothesis. It can use two hypotheses: H ₁ and H ₀ , which represent the presence (H ₁ ) and non-existence (H ₀ ) of the target, respectively. This is a simple binary hypothesis test. The problem of binary digital communications is also a simple hypothesis test. If the hypothesis contains target unknown parameters, it is a composite hypothesis test. The m-ary communication problem is also a composite hypothesis test. If the unknown parameter changes randomly, it is a hypothesis test of a random parameter signal ^[5] .

The best criterion commonly used in communication systems and radar systems is the minimum error probability criterion, that is, the maximum posterior probability criterion. Take radar detection as an example: the target is the source, and two hypotheses that it can use are H ₁ and H ₀ . After receiving the sample X (radar echo), the receiver determines whether H ₁ is true (the target exists) or H ₀ is true (the target non-existence probability can be expressed as p (H ₁ / x) and p (H ₀ / x), called the posterior probability. The decision rule for the maximal posterior probability criterion is, if ^[5]

Then it is judged that H ₁ is true (select H ₁ ); otherwise it is judged that H ₀ is true ^[5] .