Hypothesis testing problems basically boil down to the following (informal) question: given some data that you have assumed to have come from some class of data generating processes (d. g. 05, this test has the largest possible value for the power under the alternative hypohthesis, that is, when \(\theta = 2\). Then, we can apply the Nehman Pearson Lemma when testing the simple null hypothesis \(H_0 \colon \mu = 3\) against the simple alternative hypothesis \(H_A \colon \mu = 4\). d. Find the test with the best critical region, that is, find the most powerful test, with significance level \(\alpha = 0.
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Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from an exponential distribution with parameter \(\theta\). , but the variance \(\sigma^2\) is not. In theory, we could try to minimize each type of error equally. That said, how can we be sure that the T-test for a mean \(\mu\) is the “most powerful” test we could use? Is there instead a K-test or a V-test or you-name-the-letter-of-the-alphabet-test that would provide us with more power? A very important result, known as the Neyman Pearson Lemma, will reassure us that each of the tests we learned in Section 7 is the most powerful test for testing statistical hypotheses about the parameter under the assumed probability distribution. The ideal scenario would be for both probabilities to be zero or at least extremely small.
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Suppose X is a single observation (that’s one data point!) from a normal population with unknown mean \(\mu\) and known standard deviation \(\sigma = 1/3\). d. They reasoned that in most cases, false positives (i. f. 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.
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Because both the null and alternative hypotheses are simple hypotheses, we can apply the Neyman Pearson Lemma in an attempt to find the most powerful test. In particular, it says that a likelihood ratio test is the way to go for testing simple hypotheses against each other. Mathematically, it is written as:Level test: sup () ( 0)Casella and Berger (2002) use the above definitions to define a simple hypothesis test that is uniformly most powerful (UMP), which is the essence of the Neyman-Pearson Lemma:
Let C be a class of tests for testing H0: 0 versus H1: c1. Is the hypothesis \(H \colon \theta 2\) a simple or a composite hypothesis?Again, the p. 0.
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f. A Course in Mathematical Statistics. We say that C is the most powerful size \(\alpha\) test. Any hypothesis that is not a simple hypothesis is called a look at this site hypothesis. During the testing procedure, when we receive some data $X$, we compute a test statistic $T(X)$, and check whether $T(X)$ exceeds some critical value. Before we can present the lemma, however, we need to:If \(X_1 , X_2 , \dots , X_n\) is a random sample of size n from a distribution with probability density (or mass) function \f(x; \theta)\), then the joint probability density (or mass) function of \(X_1 , X_2 , \dots , X_n\) is denoted by the likelihood function \(L (\theta)\).
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That said, how can we be sure that the T-test for a mean \(\mu\) is the “most powerful” test we could use? Is there instead a K-test or a V-test or you-name-the-letter-of-the-alphabet-test that would provide us with more power? A very important result, known as the Neyman Pearson Lemma, will reassure us that each of the tests we learned in Section 7 is the most powerful test for useful reference statistical hypotheses about the parameter under the assumed probability distribution. of an exponential random variable is:for\(x ≥ 0\). f. The Best rejection region is one that minimizes the probability of making a Type I or a Type II error:
The probability that X is in the rejection region R is the statistical power of the test.
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