# ssrs 2012 barcode font (b) The median of the posterior density, obtained using computer software, is 0.393. in Software Creation QR Code ISO/IEC18004 in Software (b) The median of the posterior density, obtained using computer software, is 0.393.

(b) The median of the posterior density, obtained using computer software, is 0.393.
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11.49. In Problem 11.6, find the Bayes estimate with squared error loss for u in each case and evaluate it assuming n 500 and x 200.
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(a) From Theorem 11-6 we know that the Bayes estimate here is the posterior mean. The mean of the beta density with parameters x 2 and n x 1 is (x 2)>(n 3) 0.4016. (b) Similar to the preceding. The Bayes estimate is (x (c) The Bayes estimate is (x 4)>(n 5) 0.4040. 3)>(n 4) 0.4028.
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11.50. In Problem 11.6, part (a), find the Bayes estimate with squared error loss for the population standard deviation, 2nu(1 u).
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The required estimate is the posterior expectation of !nu(1
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u), which equals
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!n 2, n x
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u 2 (1 1) 3
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11.51. In Problem 11.6, find the Bayes estimate with absolute error loss for u in each case assuming n and x 200.
B x
5 ,n 2 2, n
3 2
By Theorem 11-6, the Bayes estimate of u with absolute error loss is the median of the posterior distribution of u. Since there is no closed form expression for the median of a beta density, we have obtained the following median values using computer software: (a) 0.4015; (b) 0.4027; (c) 0.4038.
11.52. In Problem 11.14, estimate l using a Bayes estimate (a) with squared error loss and (b) with absolute error loss.
The posterior density was obtained in Problem 11.14 as a gamma with parameters 25 and 0.091. (a) The Bayes estimate with squared error loss is the posterior mean, which in this case is ab 2.275.
(b) The Bayes estimate with absolute error loss is the posterior median. Using computer software to calculate the median of the gamma posterior distribution, we get the estimate 2.245.
(x1, x2, c, xn) of size n is taken from a population with density function 11.53. A random sample x f (xu u) 3ux2e ux3, 0 x ` , where u has a prior gamma density with parameters a and b. Find the Bayes estimate for u with squared error loss.
p(uu x) ~ une b . 1 b a x3
uax3
ua 1e
~ u(n
1 R a) 1e uQax3 b ,
which is a gamma density with parameters a
n and
The posterior mean estimate of u is therefore
b(a 1
n) b a x3
CHAPTER 11 Bayesian Methods
11.54. In Problem 11.24, find the Bayes estimate of e
with respect to the squared error loss function.
` ` tu un 1 e ua xt du
The Bayes estimate is E(e
tu u x)
tup(uu x) du
n 1 3u e 0
u(t a xt) du
(n) nx)n #
11.55. The random variable X is normally distributed with mean u and variance s2. The prior distribution of u is standard normal. (a) Find the Bayes estimator of u with squared error loss function based on a random sample of size n. (b) Is the resulting estimator unbiased (see page 195) (c) Compare the Bayes estimate to the maximum likelihood estimate in terms of the squared error loss.
estimate is E[(cx u)2 u x] c2x2 # # # nu nX (b) Since E , the estimator is biased. It is, however, asymptotically unbiased. n s2 n s2 (c) The maximum likelihood estimate of u is x. The squared error loss for this estimate is x2 1. Clearly, since # # c 1, the loss is less for the Bayes estimate. For large values of n, the losses are approximately equal. The Bayes estimate is the posterior mean of u, given by found by maximizing the likelihood L ~ ux(1 dL du xux 1(1 u)n
(a) By Theorem 11-3, the Bayes estimate is
nx # n . With c , the squared error loss for this n s2 n s2 2x2 2cx 0 1 c # 1. #
11.56. In Problem 11.22, show that the Bayes estimate of u is the same as the maximum likelihood estimate.
a x n . The maximum likelihood estimate is a b u)n x with respect to u (see page 198). Solving the equation 0 for u, we get the maximum likelihood estimate x>n.
x)ux(1
11.57. In Problem 11.48, find the Bayes estimate for 1>u with squared error loss function. The Bayes estimate is the expectation of 1>u with respect to the posterior distribution of u: