Next, a test is chosen which minimizes Type II errors (). Then, if C is a critical region of size \(\alpha\) and k is a constant such that:and:then C is the best, that is, most powerful, critical region for testing this post simple null hypothesis \(H_0 \colon \theta = \theta_0\) against the simple alternative hypothesis \(H_A \colon \theta = \theta_a\). Hallin, M. A simple hypothesis test is one where the unknown parameters are specified as single values. In order to find the most powerful test at a certain alpha level (with threshold ), you would look for the likelihood-ratio test which rejects the null hypothesis in favor of the alternate hypothesis when
where
.
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In this case, because we are dealing with just one observation X, the ratio of the likelihoods equals the ratio of the normal probability curves:Then, the following drawing summarizes the situation:In short, it makes intuitive sense that we would want to reject \(H_0 \colon \mu = 3\) in favor of \(H_A \colon \mu = 4\) if our observed x is large, that is, if our observed x falls in the critical region C. d. Consider the expressionIf $f(x; \theta_1) \kappa f(x; \theta_0)$, then $\phi(x) = 1$, and the whole expression is nonnegative. of an exponential random variable is:for\(x ≥ 0\). Let C and D be critical regions of size \(\alpha\), that is, let:Then, C is a best critical region of size \(\alpha\) if the power of the test at \(\theta = \theta_a\) is the largest among all possible hypothesis tests.
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The goal would be to maximize this power, so that the null hypothesis is rejected as much as possible when the alternate is true. Therefore, the hypothesis \(H \colon \mu = 12\) is a composite hypothesis. The lemma tells us that the ratio of the likelihoods under the null and alternative must be less than some constant k. And, finally, the definition of a best critical region of size \(\alpha\). of a normal random variable is:for \(−∞ x ∞, −∞ \mu ∞\), and \(\sigma 0\).
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The p. If it does, we reject the null hypothesis. Like all hypothesis tests, the simple hypothesis test requires a rejection region the smallest sample space which defines when the null hypothesis should be rejected. Suppose \(X_1 , X_2 , have a peek at this site , X_n\) is a random sample from an exponential distribution with parameter \(\theta\).
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For example, when deciding whether a new pharmaceutical drug has any positive effect on patients, the FDA’s default belief will be that most medicines have no effect, and they will require overwhelming evidence to believe that some medicine does have an effect. Under the hypothesis \(H \colon \theta 2\), the p. e. Given a measure μ on a measurable space \((\mathcal{X},\mathcal{A})\) and given nonnegative measurable real functions f 0 and f 1 on \((\mathcal{X},\mathcal{A})\) satisfying the condition \({\int \nolimits \nolimits }_{\mathcal{X}}{f}_{i}(x)d\mu (x) = 1\), for i = 1,2, consider the family \(\mathcal{S} = \mathcal{S}(\alpha )\) of all \(\mathcal{A}\)-measurable subsets S of \(\mathcal{X}\)such thatFind all sets in \(\mathcal{S}\!(\!\alpha \!)\) maximizing the integral \({\int \nolimits internet }_{\mathcal{S}}{f}_{1}(x)d\mu (x)\).
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For example:In contrast, the hypothesis 2 7 isnt simple; its a composite hypothesis test that doesnt state a specific value for 2. 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. 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 \(L (\theta)\). This borders on a philosophical argument, although practically it can be seen that this is a reasonable choice in many situations. .