Neyman–Pearson lemma
In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman-Pearson lemma is part of the Neyman-Pearson theory of statistical testing, which introduced concepts like errors of the second kind, power function, and inductive behavior.<ref name="org/euclid.ndml">Wald: Chapter II: The Neyman-Pearson Theory of Testing a Statistical Hypothesis: Wald: Chapter II: The Neyman-Pearson Theory of Testing a Statistical Hypothesis</ref> The previous Fisherian theory of significance testing postulated only one hypothesis. By introducing a competing hypothesis, the Neyman-Pearsonian flavor of statistical testing allows  investigating the two types of errors. The trivial cases where one always rejects or accepts the null hypothesis are of little interest but it does prove that one must not relinquish control over one type of error while calibrating the other. Neyman and Pearson accordingly proceeded to restrict their attention to the class of all formula_1 level tests while subsequently minimizing type II error, traditionally denoted by formula_2. Their seminal paper of 1933, including the Neyman-Pearson lemma, comes at the end of this endeavor, not only showing the existence of tests with the most power that retain a prespecified  level of type I error (formula_1), but also providing a way to construct such tests. The Karlin-Rubin theorem extends the Neyman-Pearson lemma to settings involving composite hypotheses with monotone likelihood ratios. Proposition. Consider a test with hypotheses formula_4 and formula_5, where the probability density function (or probability mass function) is formula_6 for formula_7. Denoting the rejection region by formula_8, the Neyman-Pearson lemma states that a most powerful (MP) test satisfies the following: for some formula_9, Also, if there is at least one MP test that satisfies the two conditions, the Neyman-Pearson lemma states that every existing  formula_1-level MP test should obey the likelihood ratio inequalities. Note that the most powerful test may not always be unique as can be inferred from the lemma. In fact, it may not exist at all. In practice, the likelihood ratio is often used directly to construct tests — see likelihood-ratio test. However it can also be used to suggest particular test-statistics that might be of interest or to suggest simplified tests — for this, one considers algebraic manipulation of the ratio to see if there are key statistics in it related to the size of the ratio (i.e. whether  a large statistic corresponds to a small ratio or to a large one). Proof. Define the rejection region of the null hypothesis for the Neyman–Pearson (NP) test as where formula_18 is chosen so that formula_19 Any alternative test will have a different rejection region that we denote by formula_20. The probability of the data falling within either region formula_21 or formula_22 given parameter formula_23 is For the test with critical region formula_20 to have significance level formula_1, it must be true that formula_27, hence It will be useful to break these down into integrals over distinct regions: where formula_30  is the complement of region . Setting formula_31, these two expressions and the above inequality yield that The powers of the two tests are formula_33 and formula_34, and we would like to prove that: However, as shown above this is equivalent to: formula_36 in what follows we show that the above inequality holds: Example. Let formula_38 be a random sample from the formula_39 distribution where the mean formula_40 is known, and suppose that we wish to test for formula_41 against formula_42. The likelihood for this set of normally distributed data is We can compute the likelihood ratio to find the key  statistic in this test and its effect on the test's outcome: This ratio only depends on the data through formula_45. Therefore, by the Neyman–Pearson lemma, the most powerful test of this type of hypothesis for this data will depend only on formula_45. Also, by inspection, we can see that if formula_47, then formula_48 is a decreasing function of formula_45. So we should reject formula_50 if formula_45 is sufficiently large. The rejection threshold depends on the size of the test. In this example, the test statistic can be shown to be a scaled Chi-square distributed random variable and  an exact critical value can be obtained. Application in economics. A variant of the Neyman–Pearson lemma has found an application in the seemingly unrelated domain of the economics of land value. One of the fundamental problems in consumer theory is calculating the demand function of the consumer given the prices. In particular, given a heterogeneous land-estate, a price measure over the land, and a subjective utility measure over the land, the consumer's problem is to calculate the best land parcel that he can buy – i.e. the land parcel with the largest utility, whose price is at most  his budget. It turns out that this problem is very similar to the problem of finding the most powerful statistical test, and so the Neyman–Pearson lemma can be used. Uses in electrical engineering. The Neyman–Pearson lemma is quite useful in electronics engineering, namely in the design and use of radar systems, digital communication systems, and in signal processing systems. In radar systems, the Neyman–Pearson lemma is used in first setting the rate of missed detections to a desired (low) level, and then minimizing the rate of false alarms, or vice versa. Neither false alarms nor missed detections  can be set at arbitrarily low rates, including zero. All of the above goes also for many systems in signal processing. Uses in particle physics. The Neyman–Pearson lemma is applied to the construction of analysis-specific likelihood-ratios, used to e.g. test for signatures of new physics against the nominal Standard Model prediction in proton-proton collision datasets collected at the LHC.