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neymen
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There are 6 letters in NEYMEN ( E1M3N1Y4 )
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In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933.Suppose one is performing a hypothesis test between two simple hypotheses H0: θ = θ0 and H1: θ = θ1 using the likelihood-ratio test with threshold * * * * η * * * {\displaystyle \eta } * , which rejects H0 in favour of H1 at a significance level of * * * * α * = * P * ( * Λ * ( * X * ) * ≤ * η * ∣ * * H * * 0 * * * ) * , * * * {\displaystyle \alpha =P(\Lambda (X)\leq \eta \mid H_{0}),} * where * * * * * Λ * ( * x * ) * := * * * * * * L * * * ( * * θ * * 0 * * * ∣ * x * ) * * * * * L * * * ( * * θ * * 1 * * * ∣ * x * ) * * * * * * {\displaystyle \Lambda (x):={\frac {{\mathcal {L}}(\theta _{0}\mid x)}{{\mathcal {L}}(\theta _{1}\mid x)}}} * and * * * * * * L * * * ( * θ * ∣ * x * ) * * * {\displaystyle {\mathcal {L}}(\theta \mid x)} * is the likelihood function. * Then, the lemma states that * * * * Λ * ( * x * ) * * * {\displaystyle \Lambda (x)} * is the most powerful test at significance level α. * If the test is most powerful for all * * * * * θ * * 1 * * * ∈ * * Θ * * 1 * * * * * {\displaystyle \theta _{1}\in \Theta _{1}} * , it is said to be uniformly most powerful (UMP) for alternatives in the set * * * * * Θ * * 1 * * * * * {\displaystyle \Theta _{1}} * . * 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). |