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# Mathematical Statistics STAT 710

UW

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This 5 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 710 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/205090/stat-710-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.

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Date Created: 09/17/15

TA Yuan Jiang Email jiangy statwiscedu STAT 710 Discussion 3 February 57 2008 1 Admissibility Example 1 Let X be an observation from the negative binomial distribu tion with a known size 7 and an unknown probability p E 071 Show that X 1r 1 is an admissible estimator of p 1 under the squared error loss Example 2 Let X17 7X be iid from the uniform distribution U097 6 17 19 E R Consider the estimation of 6 under the square error loss Let 7Tt9 be a continuous and positive Lebesgue pdf on R Derive the Bayes estimator wrt the prior 7139 and show that it is a consistent estimator of 0 2 Minimaxity Example 3 Let X be a single sample from the geometric distribution with mean p l7 where p E 071 Show that 1 X is a minimax estimator of p under the loss function Lp7 a a p2101 Of ce 1275A M80 1 Phone 262 1577 TA Yuan Jiang Email jiangy statwiscedu STAT 710 Discussion 1 January 297 2008 1 Bayes actions Example 1 Let X17 7X be iid binary random variables with P X1 1 p E 071 Find the Bayes action wrt the uniform prior on 071 in the 1H2 7 121iz7 problem of estimating p under the loss function L p7a Example 2 Consider the estimation problem in Example 41 with the loss function L 67 a w 6 g 6 7 a2 7 where w 6 2 0 and f8 w 6 g 62dll lt 00 Show that the Bayes action is W few 0909 f9 ltzgtdnt f9 109 f9 96 dH Example 3 Let X1L7 7 X be a random sample of random variables with the Lebesgue density 2 x27Te w 9 ZI9OO7 where 6 E R is unknown Find the generalized Bayes action for estimating 6 under the squared error loss7 when the improper prior of 6 is the Lebesgue measure on R Of ce 1275A M80 1 Phone 262 1577 TA Yuan Jiang Email jiangy statwiscedu STAT 710 Discussion 7 February 197 2008 1 Maximum Likelihood Estimator Example 1 Let X1L7 7X be a random sample from the uniform distri bution on 66 Find the MLE of 6 when i 9 E 0700 ii 6 E 70070 iii 0 e 726 7 0 Example 2 Let X1L7 7X be a random sample from the Weibull distri bution with Lebesgue density 046quotl0quot le maQ omx7 where a gt 0 and 6 gt 0 are unknown Show that the likelihood equation are equivalent to Ma n l 21 log Xi and 6 n l 21 Xi 7 where ha 221 Xf 1 2 X5 log Xii 04 1 and that the likelihood equations have a unique solution Example 3 Let X1L7 7X 7 n 2 2 be a random sample from a distribution having Lebesgue density fg 7 where 6 gt 07j 127 fen is the density of N062 and mm 26 1e lml9 i Obtain an MLE of 6M7 ii Show whether the MLE ofj in part is consistent iii Show that the MLE of 6 is consistent and derive its nondegenerated asymptotic distribution Of ce 1275A M80 1 Phone 262 1577 TA Yuan Jiang Email jiangy statwiscedu STAT 710 Discussion 23 April 227 2008 1 Construction of Con dence Sets Example 1 Let X17 7Xn be a random sample of random variables with Lebesgue density 6a eur lllmmx7 where 6 gt 0 and a gt 0 1 When 6 is known7 derive a con dence interval for a with con dence co ef cient 1 7 a by using the cumulative distribution function of the smallest order statistic X0 ii When both a and 6 are unknown and n 2 27 derive a con dence interval for 6 with con dence coef cient 1 7 a by using the cumulative distribution function of T H1XiX1 iii Show that the con dence intervals in and ii can be obtained using pivotal quantities iv When both a and 6 are unknown7 construct a con dence set for 176 with con dence coef cient 1 7 a by using a pivotal quantity Example 2 Let X be a sample of size 1 from the negative binomial dis tribution with a known size 7 and an unknown probability p E 071 Using the cumulative distribution function of T X 7 73 show that a level 1 7 a con dence interval for p is 1 F272T041 1 T1F2T127a27 1 F272T041 T where 041 042 047 FILM is the 1 7 ath quantile of the F distribution FM Of ce 1275A M80 1 Phone 262 1577 TA Yuan Jiang Email jiangy statwiscedu STAT 710 Discussion 24 April 247 2008 1 Construction of Con dence Sets cont d Example 1 Let T be a statistic having the noncentral F distribution FTIM6 with unknown 6 2 0 and known positive integers T1 and 75 Show that the cumulative distribution function of T7 Fgt7 is nonincreasing in 6 for each xed t gt 0 and use this result to construct a con dence interval for 6 with con dence coef cient 1 7 a Example 2 Let X1L7 7Xn be a random sample from Nu702 with un known 6 M7 02 Consider the prior Lebesgue density 7T6 7T1Ml03927r203927 where 7T1Ml0392 is the density of NWO7 03027 1 1 1 2 W W 5 1 gtIltomgtltazgt7 and 07 037 17 and b are known 1 Find the posterior of M and construct a level 1 7 a HPD credible set for M ii Show that the credible set in converges to the con dence interval X 7 Ag27X tn1a2l as 037 17 and b converge to some limits7 where X is the sample mean7 2 is the sample variance7 and tnim is the 1 7 ath quantile of the t distribution tn1 Of ce 1275A MSC 1 Phone 262 1577

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