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The ratio of the density functions above is monotone in the parameter , so satisfies the monotone likelihood ratio property. In statistics , the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs).
The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses. [1] Consider a scalar measurement having a probability density function parameterized by a scalar parameter θ, and define the likelihood ratio () = / ().
The connection with the statistical problem of estimating a monotone density is discussed in Groeneboom (1985). [2] Chernoff's distribution is now known to appear in a wide range of monotone problems including isotonic regression .
If X n converges in probability to X, and if P(| X n | ≤ b) = 1 for all n and some b, then X n converges in rth mean to X for all r ≥ 1. In other words, if X n converges in probability to X and all random variables X n are almost surely bounded above and below, then X n converges to X also in any rth mean. [10] Almost sure representation ...
Estimated change in probability: Based on table above, a likelihood ratio of 2.0 corresponds to an approximately +15% increase in probability. Final (post-test) probability: Therefore, bulging flanks increases the probability of ascites from 40% to about 55% (i.e., 40% + 15% = 55%, which is within 2% of the exact probability of 57%).
Neyman–Pearson lemma [5] — Existence:. If a hypothesis test satisfies condition, then it is a uniformly most powerful (UMP) test in the set of level tests.. Uniqueness: If there exists a hypothesis test that satisfies condition, with >, then every UMP test in the set of level tests satisfies condition with the same .
The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has Fatou's lemma and the dominated convergence theorem as direct consequence.
Probability mass function for Fisher's noncentral hypergeometric distribution for different values of the odds ratio ω. m 1 = 80, m 2 = 60, n = 100, ω = 0.01, ..., 1000 Biologist and statistician Ronald Fisher