Neyman–Pearson lemma

Neyman–Pearson lemma is a fundamental theorem in the field of statistics that provides a method for choosing between two hypotheses in a way that controls the Type I error rate. Formulated by Jerzy Neyman and Egon Pearson in the 1930s, the lemma is a cornerstone of statistical theory and has widespread applications in fields such as medicine, biology, psychology, and economics.

Overview

The Neyman–Pearson lemma addresses the problem of testing a null hypothesis (H₀) against an alternative hypothesis (H₁), given a set of observations. The lemma states that for a given size of the test (the maximum allowable probability of making a Type I error, i.e., rejecting the null hypothesis when it is true), the test that maximizes the power (the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true) is the likelihood ratio test. This test compares the likelihood of the observed data under the two hypotheses and decides in favor of the hypothesis that makes the observed data more likely.

Mathematical Formulation

The likelihood ratio test is defined by the statistic:

\[\Lambda(x) = \frac{L(\theta_1 | x)}{L(\theta_0 | x)},\]

where \(L(\theta | x)\) is the likelihood of the parameter \(\theta\) given the data \(x\), \(\theta_0\) is the parameter under the null hypothesis, and \(\theta_1\) is the parameter under the alternative hypothesis. The decision rule is to reject the null hypothesis if \(\Lambda(x)\) exceeds a certain threshold, which is determined based on the desired significance level of the test.

Applications

The Neyman–Pearson lemma is widely used in various scientific disciplines to make informed decisions based on experimental data. In medicine, it is employed in clinical trials to determine whether a new treatment is more effective than a standard treatment. In psychology, it helps in making decisions about the validity of experimental results. The lemma also finds applications in quality control and machine learning, where it is used to optimize decision-making processes under uncertainty.

Limitations

While the Neyman–Pearson lemma provides a powerful framework for hypothesis testing, it has limitations. It requires the specification of a single alternative hypothesis, which may not always be practical. Additionally, the lemma does not provide guidance on choosing the significance level or the power of the test, which are subjective decisions that can affect the conclusions drawn from the test.

See Also

• Statistical hypothesis testing
• Type I and type II errors
• Likelihood ratio
• Jerzy Neyman
• Egon Pearson