Unformatted text preview: Mathematical Statistics Quiz #3 6/11/2009 1. State the definitions of the following terms and give an interpretation of each of the terms. (1)Consistent estimator. (2)Sufficient statistics. (3)The maximum likelihood estimator (MLE). (4)The minimum variance unbiased estimator (MVUE). (5)Likelihood function of a random sample. 2. State the Central Limit Theorem (CLT) in detail and prove the CLT by moment generating function. 3. If X and S are the sample mean and sample variance of a random sample of size n from a normal population with mean μ and variance σ . (1)What are the distributions of X and S ? Prove them. (2)Show that X is an MVUE of μ. (3)Show that S is a consistent estimator of σ . (4)Find the MLE of σ , denote it by σ . 4. Let Y Y Y denote the order statistics of a random sample of size n from a distribution
θ having p.d.f. f x ,1 θ, zero elsewhere. (1)Find the p.d.f. of Y . (2)If 0 1, show that P c
Y θ Y θ 1 PY 1 θ c.
Y (3)Find c such that P c 1 0.99. The interval Y , Y is called the 99% confidence interval of θ. If n confidence interval of θ. 5 and if the observed value of Y is 1.8, find a 99% 5. Let X , X , ,X be a random sample from f x, µ, σ ∑ √ e µ ,0 ∞, µ R, σ 0. (1)What is this distribution called? Show that T σ. (2)Find the MLE of µ and . σ . (3)Find the MLE of d 6.
S ⁄X √T T lnX is a sufficient statistic for µ for a given , where S , r and T are known constants. (1) Let Θ be an estimator of the parameter θ. Show that the mean square error of θ, EΘ θ VΘ bias Θ . (2) For a sample from normal distribution, show that the MLE of σ , σ , is consistent. ...
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This note was uploaded on 05/09/2011 for the course DIF 1486 taught by Professor Yow-jenjou during the Spring '11 term at National Chiao Tung University.
- Spring '11