課程名稱︰高等統計推論二 課程性質︰應數所數統組必修 課程教師︰江金倉 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/4/11 考試時限（分鐘）：11:20~12:10 試題 : 2 1. (10%) Let X ,...,X be a random sample from N(μ,σ ). Find the maximum 1 n likelihood estimator of Φ((x-μ)/σ), where Φ(‧) is the cumulative distribution function of the standard normal distribution and x is a given value. 2. (10%) Suppose that Y ~ χ , i=1,...,k, are mutually independent. Try to find i υ i k the Satterthwaite approximation for Σ a Y , where a 's are known constant. i=1 i i i 3. (15%) Let X ,...,X be a random sample from a p.d.f f(x|θ) with θ∈Θ and 1 n 0 0 _ dim(Φ)=1, θ be an unbiased estimator of θ which attains the Cramer-Rao lower n ^ ^ 2 bound. Suppose that θ is a solution of ∂l(θ|X ,...,X ) = 0 and -∂l(θ|X ,.. n θ n 1 n θ 1 .,X ) > 0 for all θ, where l(θ|X ,...,X ) is the log-likelihood function. n 1 n ^ _ Show that θ=θ. n n 4. (15%) Let (Z ,δ),...,(Z ,δ) be a random sample with Z = min{X ,Y }, and 1 1 n n i i i δ= I(Z = X ), i=1,...,n. Moreover, let X and Y be independent exponential i i i random variables with rates μ and λ. Find the maximum likelihood estimators of λ and μ. 5. (20%) Show the monotonicity of expectation-maximization (EM) sequence. 6. (15%)(15%) Show that the uniformly minimum variance unbiased estimator (UMVUE) is unique and uncorrelated with zero unbiased estimators. --

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