| rdfs:comment
| - En estadística, el lema fonamental de Neyman-Pearson és un resultat que descriu el criteri òptim per distingir dues hipòtesis simples H0: θ=θ0 i H1: θ=θ1. (ca)
- En estadística, el lema fundamental de Neyman-Pearson es un resultado que describe el criterio óptimo para distinguir dos hipótesis simples y . El lema debe su nombre a sus dos creadores, Jerzy Neyman y Egon Pearson. (es)
- ネイマン・ピアソンの補題（ネイマン・ピアソンのほだい）とは、統計学的仮説検定に関する補題。 2つの仮説H0: θ=θ0 と H1: θ=θ1 の間で仮説検定を行う際に、H1を支持しH0を排除するような、次に示す尤度比による尤度比検定： （ただしここで とする）が、サイズ（危険率、第一種過誤） の仮説検定の中で最もパワー（検出力）が大きい、というものである（は第二種過誤）。、が単純仮説であればとなる。 「αを決めておき、その中で検出力が最も大きい検定法を選択する」という方針をネイマン・ピアソンの基準という。この補題はその方法を具体的に与えるものである。ただしこの尤度比検定法が直接用いられるよりも、近似が用いられることが多い。 (ja)
- In statistica, il lemma fondamentale di Neyman-Pearson asserisce che, quando si opera un test d'ipotesi tra due ipotesi semplici H0: θ=θ0 e H1: θ=θ1, il rapporto delle funzioni di verosomiglianza che rigetta in favore di quando rappresenta il test di verifica più potente a livello di significatività α per una soglia k. Se il test è il più potente per tutti i , si dice che è quello uniformemente più potente (in inglese UMP) tra le alternative del set. Il lemma deve questo nome ai suoi formulatori, Jerzy Neyman e Egon Pearson. (it)
- Lemat Neymana-Pearsona – twierdzenie z obszaru statystyki opublikowane przez Jerzego Neymana i Egona Pearsona w 1933. Stanowi – w amalgamacie z wcześniejszą propozycją Ronalda Fishera – jedną z podstaw procedury weryfikacji hipotez w podejściu częstościowym. (pl)
- Das Neyman-Pearson-Lemma, auch Fundamentallemma von Neyman-Pearson oder Fundamentallemma der mathematischen Statistik genannt, ist ein zentraler Satz der Testtheorie und somit auch der mathematischen Statistik, der eine Optimalitätsaussage über die Konstruktion eines Hypothesentests macht. Gegenstand des Neyman-Pearson-Lemmas ist das denkbar einfachste Szenario eines Hypothesentests, das auch Neyman-Pearson-Test genannt wird: Dabei ist sowohl die Nullhypothese als auch die Alternativhypothese einfach, d. h., sie entsprechen jeweils einer einzelnen Wahrscheinlichkeitsverteilung, deren zugehörige Wahrscheinlichkeitsdichten nachfolgend mit und bezeichnet werden. Dann, so die Aussage des Neyman-Pearson-Lemmas, erhält man den besten Test durch eine Entscheidung, bei der die Nullhypothese ve (de)
- 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 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 (en)
- En statistique, selon le lemme de Neyman-Pearson, lorsque l'on veut effectuer un test d'hypothèse entre deux hypothèses H0 : θ = θ0 et H1 : θ = θ1, pour un échantillon , alors le test du rapport de vraisemblance, qui rejette H0 en faveur de H1 lorsque , où est tel que , est le test le plus puissant de niveau . Ce lemme est nommé d'après Jerzy Neyman et Egon Sharpe Pearson dans un papier de 1931. (fr)
- Em estatística, o método ou lema de Neyman–Pearson foi introduzido pelo matemático polonês Jerzy Neyman e pelo matemático britânico Egon Pearson em um artigo de 1933. Este lema afirma que, quando se realiza um teste de hipóteses entre duas hipóteses simples e , o teste de razão de verossimilhança que rejeita em favor de quando em que é o teste mais potente ao nível de significância para o limiar . Se o teste for o mais potente para todo , pode ser considerado o uniformemente mais potente (UMP) para alternativas no conjunto . (pt)
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| proof
| - Given any hypothesis test with rejection set , define its statistical power function .
Existence:
Given some hypothesis test that satisfies condition, call its rejection region .
For any level hypothesis test with rejection region we have
except on some ignorable set .
Then integrate it over to obtain
.
Since and , we find that .
Thus the rejection test is a UMP test in the set of level tests.
Uniqueness:
For any other UMP level test, with rejection region , we have from Existence part,
.
Since the test is UMP, the left side must be zero. Since the right side gives
, so the test has size .
Since the integrand is nonnegative, and integrates to zero, it must be exactly zero except on some ignorable set .
Since the test satisfies condition, let the ignorable set in the definition of condition be .
is ignorable, since for all , we have .
Similarly, is ignorable.
Define . It is the union of three ignorable sets, thus it is an ignorable set.
Then we have
and . So the rejection test satisfies condition with the same .
Since is ignorable, its subset is also ignorable. Consequently, the two tests agree with probability whether or . (en)
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