f. Suppose we have a random sample \(X_1 , X_2 , \dots , X_n\) from a probability distribution with parameter \(\theta\). Introduction to Probability and Mathematical Statistics. of an exponential random variable is:for\(x ≥ 0\). Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from a normal population with mean \(\mu\) and variance 16. f.
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Any hypothesis that is not a simple hypothesis is called a composite hypothesis. Again, because we are dealing with just one observation X, the ratio of the likelihoods equals the ratio of the probability density functions, giving us:That is, the lemma tells us that the form of the rejection region for the most powerful test is:or alternatively, since (2/3)k is just a new constant \(k^*\), the rejection region for the most powerful test is of the form:Now, it’s just a matter of finding \(k^*\), and our work is done. The lemma tells us that the ratio of the likelihoods under the null and alternative must be less than some constant k:Simplifying, we get:And, simplifying yet more, we get:Now, taking the natural logarithm of both sides of the inequality, collecting like terms, and multiplying through by 32, we get:And, moving the constant term on the left-side of the inequality to the right-side, and dividing through by −160, we get:That is, the Neyman Pearson Lemma tells us that the rejection region for the most powerful test for testing \(H_{0} \colon \mu = 10 \) against \(H_{A} \colon \mu = 15 \), under the normal probability model, is of the form:where \(k^*\) is selected so that the size of the critical region is \(\alpha = 0. Find the test with the best critical region, that is, find the most powerful test, with significance level \(\alpha = 0. f.
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of an exponential random variable is:for\(x ≥ 0\). Is the hypothesis \(H \colon \mu = 12\) a simple or a composite hypothesis?The p. f. The lemma is named after Jerzy Neyman and Egon Sharpe Pearson, who described it in 1933.
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f. Now, the definition of simple and composite hypotheses. of an exponential random variable is:for\(x ≥ 0\). 05\). Wiley StatsRef: Statistics Reference Online. In order to understand the lemma, its necessary to define some basic principles about alpha/beta levels and power:A Type I error under the null hypothesis is defined as:
Pθ(X R ∣ H0 is true),
Where: A Type II error under the null hypothesis is defined as:
Pθ(X Rc∣ H0 is false),
Where: Usually, a knockout post alpha level is set (e.
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Again, because we are dealing with just one observation X, the ratio of the likelihoods equals the ratio of the probability density functions, giving us:That is, the lemma tells us that the form of the rejection region for the most powerful test is:or alternatively, since (2/3)k is just a new constant \(k^*\), the rejection region for the most powerful test is of the form:Now, it’s just a matter of finding \(k^*\), and our work is done. That is, for every test in $\mathcal{T}$ that has power function $\widetilde{\beta}$,\paragraph{Size} The size of a hypothesis test is the probability of incorrectly rejecting the null hypothesis:\paragraph{Level} A hypothesis test with level $\alpha$ falsely rejects at most $\alpha$ fraction of the time:The Neyman-Pearson Lemma is an important result that gives conditions for a hypothesis test to be uniformly most powerful. In practice, Neyman and Pearson’s approach dictates that our primary goal should be to strictly control the probability of false check this while our secondary goal should be to minimize false negatives. See Hogg and Tanis, pages 400-401 (8th edition pages 513-14).
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PrintCasella, George, and Roger L. In the real world this isnt possible, because a reduction in alpha often leads to an increase in beta and vice-versa. Note that this lemma has been extended to composite hypotheses as well. Just as the Neyman Pearson Lemma suggests!Well, okay, that’s the intuition behind the Neyman Pearson Lemma. In plain English, this is basically saying:Take all tests of size .
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Note: Lemma sounds like it should be a Greek letter, but it isnt. 0
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see this here we learned from our work in the previous lesson, whenever we perform a hypothesis test, we should make sure that the test we are conducting has sufficient power to detect a meaningful difference from the null hypothesis. .