Neyman–Pearson Lemma
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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.The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?: Journal of the American Statistical Association: Vol 88, No 424
The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?: Journal of the American Statistical Association: Vol 88, No 424
/ref>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 Empire of Chance
The Empire of Chance
/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 \alpha level tests while subsequently minimizing type II error, traditionally denoted by \beta. 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 (\alpha), 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.


Statement

Consider a test with hypotheses H_0: \theta = \theta_0 and H_1:\theta=\theta_1, where the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(or probability mass function) is \rho(x\mid \theta_i) for i=0,1. For any hypothesis test with rejection set R, and any \alpha\in , 1/math>, we say that it satisfies condition P_\alpha if * \alpha = Pr_(X\in R) ** That is, the test has
size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume ...
\alpha (that is, the probability of falsely rejecting the null hypothesis is \alpha). * \exists \eta \geq 0 such that \begin x\in& R\setminus A\implies \rho(x\mid \theta_1) > \eta \rho(x\mid \theta_0) \\ x\in& R^c\setminus A \implies \rho(x\mid\theta_1) < \eta \rho(x\mid \theta_0) \end
where A is a set ignorable in both \theta_0 and \theta_1 cases: Pr_(X\in A) = Pr_(X\in A) = 0. ** That is, we have a strict likelihood ratio test, except on an ignorable subset. For any \alpha\in , 1/math>, let the set of level \alpha tests be the set of all hypothesis tests with size at most \alpha. That is, letting its rejection set be R, we have Pr_(X\in R)\leq \alpha. In practice, the likelihood ratio is often used directly to construct tests — see
likelihood-ratio test In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after ...
. 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).


Example

Let X_1,\dots,X_n be a random sample from the \mathcal(\mu,\sigma^2) distribution where the mean \mu is known, and suppose that we wish to test for H_0:\sigma^2=\sigma_0^2 against H_1:\sigma^2=\sigma_1^2. The likelihood for this set of normally distributed data is :\mathcal\left(\sigma^2\mid\mathbf\right)\propto \left(\sigma^2\right)^ \exp\left\. We can compute the likelihood ratio to find the key statistic in this test and its effect on the test's outcome: :\Lambda(\mathbf) = \frac = \left(\frac\right)^ \exp\left\. This ratio only depends on the data through \sum_^n (x_i-\mu)^2. Therefore, by the Neyman–Pearson lemma, the most powerful test of this type of
hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can testable, test it. Scientists generally base scientific hypotheses on prev ...
for this data will depend only on \sum_^n (x_i-\mu)^2. Also, by inspection, we can see that if \sigma_1^2>\sigma_0^2, then \Lambda(\mathbf) is a
decreasing function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
of \sum_^n (x_i-\mu)^2. So we should reject H_0 if \sum_^n (x_i-\mu)^2 is sufficiently large. The rejection threshold depends on the
size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume ...
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 they can buy – i.e. the land parcel with the largest utility, whose price is at most their 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 Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current ...
, namely in the design and use of
radar Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
systems, digital communication systems, and in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
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 The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. I ...
prediction in proton-proton collision datasets collected at the
LHC The Large Hadron Collider (LHC) is the world's largest and highest-energy particle collider. It was built by the European Organization for Nuclear Research (CERN) between 1998 and 2008 in collaboration with over 10,000 scientists and hundr ...
.


See also

* Error exponents in hypothesis testing * ''F''-test *
Lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...
* Wilks' theorem


References

* E. L. Lehmann, Joseph P. Romano, ''Testing statistical hypotheses'', Springer, 2008, p. 60


External links

* Cosma Shalizi gives an intuitive derivation of the Neyman–Pearson Lemm
using ideas from economics

cnx.org: Neyman–Pearson criterion
{{DEFAULTSORT:Neyman-Pearson Lemma Theorems in statistics Statistical tests Articles containing proofs Lemmas