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110年 - 110 國立臺灣大學_碩士班招生考試_農藝研究所生物統計學甲組:統計理論#102885
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7.
(b) (6 points) Please prove the Neyman-Pearson Theorem.
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1. (10 points) Let X~Beta(a,β) and Y~Beta(a + β,γ) be two independent variables of beta distribution. Find the distribution of XY by making the transformations U=XY, V=Y. The pif of beta distribution, Beta(a,b), is given by
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(a) (5 points) Find the MLE (maximum likelihood estimator) of 6. Is it an unbiased estimator?
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(b) (10 points) Use the MLE in (a) to obtain a 100 x (1 - c)% confidence interval for θ. Hint: is distributed as a Chi-square distribution.
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(a) (5 points) Find the method of moment estimator of θ.
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(b) (10 points) Find the limiting distribution of as n→∞ by the Delta method, where Tis the method of moment estimator of θ.
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(a) (5 points) Find the uniform minimum variance unbiased estimator (UMVUE) of g(μ).
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(b) (5 points) Find the Cramer-Rao lower bound (CRLB) for the variance of unbiased estimator of g (μ). Is the CRLB attained by the variance of the UMVUE?
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(a) (2 points) When significant level is set as 0.05, fine the rejection region of X.
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(b) (2 points) When X, reject H0. Find the probability of type I error.
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(c) (2 points) When X , reject H0. Find the probability of typeerror.
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110年 - 110 國立臺灣大學_碩士班招生考試_農藝研究所生物統計學甲組:統計理論#102885
110年 · #102885
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