Statistical Decision Theory And Bayesian Analysis 2nd Edition James O. Berger - Solutions

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In the situation of Example 13, suppose that a > 2 and that 8 is generalizedBayes with respect to (0) =2/[(1+ 02)7]. Construct an estimator with smallerrisk than 8. (It is interesting that n is a proper prior, and yet 8 is inadmissible.)36. In Example 15, verify the claims for b=c p-1.
Suppose that X~P(4); note that the natural parameter is 0 = log 4. It is desired to estimate & under squared-error loss, andㅠ(0) = (1+0) 1(0,0) (0)Find conditions on r such that the generalized Bayes rule with respect to # is admissible.
In Example 12, prove that the conditions of Theorem 14 are satisfied if and only if at most two of the characteristic roots of B equal 1.
If π(0) = 1 in the situation of Theorem 14, show that the generalized Bayes estimator of is "(x) = x.
Assume that X~ N(0, 1), and that it is desired to estimate under loss L(0,a) = 10-a. Show that &0(x) = x is an admissible estimator.
Assume that X ~ B(1, 0) is observed, and that it is desired to estimate 0 € [0, 1]under squared-error loss. Since the loss is convex, attention can be restricted to nonrandomized decision rules. A nonrandomized rule can be written as a vector (y, z), where 0≤y≤1 and 0≤z ≤1 are the
If L(0,a) = exp{(0-a)²}, where= = R', show that Condition (c) of Theorem 11 is violated.
Verify that the following loss functions satisfy Condition (c) of Theorem 11:(a) L(0,a) =0-a, k > 0, 0 = A = R¹;(b) L(0,a) =exp{|0-a|}, O = & = R¹;(c) L(0,a) = (log[a/0])², 0 = & = [1, ∞).
Suppose that X~N, (0, I), and that it is desired to estimate under a quadratic loss. If &(x) = ф(x)x is generalized Bayes, where ф is a real valued function,show that can never be negative.
Assume that X~ , (0, Σ)(p=2), where I is known, and that it is desired to estimate under a quadratic loss. Consider the estimatorδ(x) = (Σ1+(x'Cx)-1A)Σ, (8.37)where C is positive definite. Show that unless A is a constant multiple of C,such an estimator cannot be generalized Bayes, and hence
Verify equation (8.28). (You may assume that all integrals exist, and that differentiating under the integral sign is valid.)
Suppose that X~N (0, I), p≥2, and that it is desired to test Ho: 0=0 versus H: 0≠0, under "0-1" loss. Which of the following rejection regions yield admissible, and which yield inadmissible tests:(a) {x: x> K};(b) {x: x K};(d) {x: at least one coordinate is positive};(e) {x: all coordinates are
Verify that the N (0, I) distribution is in the exponential family, and identify c(0), h(x), and O.
Assume that X~ Be(0, 1), that A = {ao, a₁}, and that the loss function is of the form (8.23). A test is proposed for which Eo[4(X)] = 0.5 and E, [(Х)] = 0.3.Find a test that is R-better than or R-equivalent to ф.
If X~N(0,1) and 0 is to be estimated under squared-error loss, show that 8(x)= x is not a proper prior Bayes rule.
Assume that X~N(0, 1), that ==(-0, ∞), and that L(0,a) =(0-a)².Someone proposes the estimator(0 if x
Assume that X~N(0, 1), that 0 = 4= (-00, 00), and that L(0,a) = (0-a)2.Someone proposes the estimator 80(x) =-cx +b, where c > 0. Find an R-better monotone estimator.
In Definition 9, show that the definition of a nonrandomized monotone decision rule is consistent with the definition of a randomized monotone decision rule.
In the situation of Exercise 15, show that the monotone decision rule , defined byФ1(х) = 1(-0,-0.1)(x), Ф2(x) = 4-0.1,0.1](x), ф3(х) = 1(0.1,0) (x)is inadmissible.
Assume that X~B(5, 0), that =[0, 1], that = {a1, a2, a3}, and that the loss function satisfies (8.13) with 01 = and 02=. Find a monotone decision rule which is R-better than or R-equivalent to the decision rule , defined by 41(x) = 40,3)(x), 42(x) =1(1,4)(x), 43(x) =112,5)(x).
Assume that X~N(0, 1), that =[-1, 1], that A = {a₁, a2, a3}, and that L(0, a₁) = (0+1)², L(0, a2) = 0², and L(0, a3)=(0-1)2. Show that this is a monotone multiple decision problem.
Prove part (ii) of Theorem 5.
Assume that X₁,..., X, is a sample from the U(0, 0+1) distribution, and that it is desired to test Ho: 0≤0 versus H,: 0 > 0. Defining t₁ =min{x;} and t2 = max{x}, show that(112)is a UMP test of size a.[0 if t₁
If X ~C(0, 0), show that the sufficient statistic X has monotone likelihood ratio.
Find a counterexample to part (ii) of Theorem 4 if the size of the test is allowed to be zero.
Verify that the following distributions have monotone likelihood ratio:(a) Be(a, B), B fixed,(b) Be(a, B), a fixed (MLR in -Х),(c) U(0, 0),(d) f(xa, B) =(2B)-1exp{-x-a/B}, B fixed,(e) f(x0) =exp{-(x -0)}I(00)(x).
Let be a most powerful test of size 0< a a, providing the distributions of X under 00 and differ.
If X~ €(0, 1), show that the test if 1
Find the minimal complete class of decision rules for a sample of size n = 2 in the situation of(a) Exercise 45 of Chapter 5,(b) Exercise 46 of Chapter 5.
List violated if S is not(a) closed from below,(b) bounded from below.Use a modification of Exercise 24 of Chapter 5 (namely, a different choice of P) to show that the conclusion of Theorem 1 can be true even when S is not closed from below.
Give an example, in terms of the risk set S, in which the conclusion of Theorem
Prove that if the class of admissible rules is complete, it is minimal complete.
Prove that if is a complete class and contains no proper essentially complete subclass, then is a minimal complete class.
Prove Lemma 2.
Prove Lemma 1.
It is desired to test the null hypothesis that a sequential sample has a U(0, 2)common density versus the alternative hypothesis that the common density is U(1, 3). The decision loss is 0 for a correct decision and 20 for an incorrect decision. It is possible, at any stage, to take observations in
Assume that X₁, X₂,... is a sequential sample from a N(0, o2) density (o known), and that it is desired to estimate under squared-error decision loss.It is possible, at any stage, to take observations in batches of any size. A batch of m observations costs clog(m+1). If the parameter has a
In the situation of Example 21, verify that & is inadmissible if it satisfies (7.90).
In the situation of Example 20, determine the 100(1- a)% HPD credible set for when =0 has prior probability πο >0 and the 0 0 have prior density(1-) times a N(0, A) density. (Note that the set will for sure include the point 0 = 0.)
To see that classical significance is affected by interim analysis, suppose we can observe independent N(0, 1) observations, X, and X₂. We are interested in the significance level for testing Ho: 0 = 0 versus H₁: 0≠0.(a) For the stopping rule "take both observations," show that an a = 0.05
Let X₁, X2,... be a sequential sample from the P(0) density. Suppose the stopping rule is to stop sampling at time n≥ 2 with probability [E x /Σ x], n =2,3,... (define 0/0 = 1). Suppose the first five observations are {3, 1, 2, 5, 7},and sampling then stops. Find the likelihood function for 0.
Prove that (7.88) is true if the indicated densities {fa} exist.
Suppose that X₁, X2,... is a sequential sample from a N(0, 1) distribution, and that it is desired to test Ho: 0 =0 versus H₁: 0≠0. The experimenter reports that he used a proper stopping rule, and obtained the data 3, -1, 2, 1.(a) What could a frequentist conclude?(b) What could a
In the situation of Example 13, show that the following procedure is minimax:stop sampling for the first n for which (b- a*)²≤ (2c)1/3, and estimate by(+)/2, where a maxisisn{x;} - and b = minisisn{x;} +.
In the situation of Exercise 7, show that a minimax sequential procedure is to choose the integer n which minimizes (n+1) 1+nc, take a sample of size n, and estimate by (n+1)-11 x
In the situation of Exercise 6, show that a minimax sequential procedure is to choose the integer n which minimizes (n+1)-¹+nc, take a sample of size n, and estimate 0 by (n+1)- x
In the situation of Exercise 5, show that a minimax sequential procedure is to choose the integer n which minimizes n1+nc, take a sample of size n, and estimate by nΣ x
In the situation of Exercise 4, show that a minimax sequential procedure is to choose the integer n which minimizes n+nc, take a sample of size n, and estimate o by nΣΧρ.
In the situation of Exercise 42, assume that the decision loss is "0-20" loss,that each observation costs c = 0.001, and that the prior probability that equalsis. Find the Bayes sequential procedure. (You may use the small c approximations to the stopping boundaries, given in (7.84).)
Assume that X1, X2,... is a sequential sample from a N(0,1) density, and that it is desired to test Ho: 0 = 0 versus H₁: 0 = 1. The loss in incorrectly deciding Ho is 30, in incorrectly deciding H is 15, and is 0 otherwise. The cost of each observation is c = 0.001, and the prior probability that
For the situation considered in Section 7.5, namely testing a simple nul hypothesis versus a simple alternative hypothesis, show that the asymptoticallypointwise optimal procedure defined in Subsection 7.4.10 (you may assume(7.44) is satisfied) is an SPRT. Find the stopping boundaries of this SPRT,
Consider the situation of Exercise 39.(a) For any a
For 0< ao
Prove that (7.73) is true under the condition that Pa, (Z|
Prove part (ii) of Theorem 11.
It is known that one of two given coins is fair, and that the other coin has probability of coming up heads when flipped. It is desired to determine which of the coins is the fair coin. The coins, tossed simultaneously and independently, produce a sequential sample (X1, Y₁), (X2, Y2),..., where
Consider the situation of Example 15.(a) Show that, for the SPRT with a = -jv and b = kv, B(0) and E,N are given exactly by 1-[0/(1- 0)][(1-0)/0]k-[0/(1-0)1 B(0)=k + j if 0≠, if 0 =, 1 FN=120-1 jk k{1-[0/(1-0)J}-j{[(1-0)/0]k - 1}[(1-0)/0]*-[0/(1-0)] if 0 if 0=(b) If 00= each observation costs c =
Assume that X₁, X2,... is a sequential sample from the P(0) density, and that it is desired to test Ho: 0 = 1 versus H: 0=2.(a) Using the Wald approximations, find the SPRT for which =0.05 andα₁ = 0.1.(b) Calculate B(0) and Ë,N at 0 = 1, 0 =2, and 9 = 1/log 2, for the SPRT in (a).42. Assume
Assume that X₁, X₂,... is a sequential sample from the (0) density, and that it is desired to test Ho: 0= versus H₁: 0 = 1.(a) Using the Wald approximations, find the SPRT for which α = αι = 0.05.(b) Determine B() and B(log 2) for the SPRT in (a).(c) Calculate Ě2N, E, N, Elog2N, and
Assume that X₁, X₂,... is a sequential sample from the density f(x|0) =e(x-0)I(0) (x),
It is desired to test the null hypothesis that a sequential sample has a U(0, 2)common density versus the alternative hypothesis that the common density is U(0, 1). The cost of each observation isc, and the decision loss is "0-K," loss.Let w, denote the prior probability of H;.(a) Show that the
It is desired to test the null hypothesis that a sequential sample has a U (0, 2)common density versus the alternative hypothesis that the common density is U(1, 3). The cost of each observation isc, and the decision loss is "0-K" loss.Let , denote the prior probability of H,.(a) Show that the
In the situation of Example 2, verify that (7.47) is satisfied for B = k/2.
Verify that (7.47) is satisfied and find an asymptotically pointwise optimal procedure for the situation of (a) Example 7, (b) Exercise 26.
Assume that X₁, X2,... is a sequential sample from a B(1, 0) density, that it is desired to test Ho: 0= versus H₁: 0=under "0-1"loss, that each observation costsc, and that 0
Complete the analysis in Example 13, showing that h*(r) (defined by (7.43))is a solution to (7.39) for r>1.
Prove that Theorem 6 is true for a general loss, L(0,a, n), providing the condition in Exercise 31 is satisfied.
Prove that Theorem 5 is true for a general loss, L(0,a, n), providing the followingcondition is satisfied:lim E"E[L(9, 8(X), n)-inf L(0,a, n)]=0.
Prove that the inequalities in (7.24) of Subsection 7.4.8 are valid.
Find the 2-step inner look ahead procedure, diL, for the situation of (a) Example 6, (b) Exercise 15.
Find the 1-step inner look ahead procedure,d, for the situation of (a) Example6, (b) Example (7), (c) Exercise 15, (d) Exercise 16.
Assume that X₁, X2,... is a sequential sample from a P(0) density, that L(0,a, n) = (0-a)²+n, and that 0 has a (1, 1) prior density. Find the Bayes 3-inner truncated procedure, d.
Assume that X₁, X₂,... is a sequential sample from a P(0) density, that L(0,a, n) = (0-a)²+nc, and that has a G(a, B) prior density. Show that the fixed sample size look ahead procedure, dpS, is the same as the 1-step lookahead procedure, di
Show that in Example 10, Case 2, the fixed sample size look ahead procedure,dEs, is as indicated.
For the situation of Exercise 9, describe how the fixed sample size look aheadprocedure, dES, would be implemented. (Note the relevance of (7.11).) Carry out the procedure for the sequential sample 0.5, 2, 1, 1.5, 0, 2.5, 2, 1,...
Find the 2-step look ahead procedure,d, for the situation of (a) Example 6,(b) Exercise 15, (c) Exercise 16.
Find the 1-step look ahead procedure,d, for the situation of (a) Example 7,(b) Exercise 13 (with =3), (c) Exercise 15, (d) Exercise 16.
Use Theorem 4 to show that the procedure found in Exercise 16 is actually the sequential Bayes procedure, dT.
Use Theorem 4 to show that the procedure found in Exercise 14 is actually the sequential Bayes procedure, d".
Prove Theorem 4.
Assume that X₁, X2,... is a sequential sample from a B(1, 0) density, that L(0,a, n) = (0-a²) +Σi [0.01(1+5/i)], and that 0 has a U(0, 1) prior density.Use Theorem 3 to show that the Bayes sequential procedure must be truncated at m = 17.
Prove Theorem 1.
Assume that X₁, X₂,... is a sequential sample from a B(1,0) density, and that a decision between two possible actions, a and ai, must be made. The decision loss is L(0,a) = 0 and L(0,a) = 1 - 0, while each observation costs = . Find the Bayes 2-truncated procedure, d², when has a U(0, 1) prior
Assume that X1, X2,... is a sequential sample from a P(0) density, that L(0,a, n) = (0-a)²+n(2), and that has a G(1, 1) prior density. Find the Bayes 3-truncated procedure, d³.
In the situation of Example 7, find the Bayes 3-truncated procedure, d3, when c =0.01.
Assume that X₁, X2,... is a sequential sample from a B(1, 0) density, and that it is desired to test Ho: 0 = versus H: 0 =. The decision loss is L(00, ao) = L(0, a₁) = 0, L(00, a₁) = 10, and L(01, a0) = 20, while the cost of each observation isc, = 1.(a) Letting o denote the prior probability
Find a Bayes sequential decision procedure for the situation in Exercise(a) 4, (b) 5, (c) 6, (d) 7.
In the situation of Exercise 4 of Chapter 1, assume that a computer consultant can be hired to help in the prediction of the team's winning proportion 0. The consultant will report X~B(1, 0). (This is a rather silly report, but you will appreciate the ease in calculation.) How much money would you
Assume that X1, X2,... is a sequential sample from a N(6, o) density (o known), and that it is desired to test Ho: 00, versus H,: 00, (
Assume that X1, X2,... is a sequential sample from a N(0, 1) density, and that it is desired to test Ho: 00 versus H,: 0>0. The decision loss is zero for a correct decision and 10 for an incorrect decision. The cost of a fixed sample of size n is C(n) n(0.1). If the prior density of 0 is N(0, 4),
Assume that X, X2,... is a sequential sample from a N(0, 1) density, and that it is desired to test Ho: 0-0 versus H: 0-1. The decision loss is L(80,a) = L(0,a) 0, L(0, a,)-1, and L(0,a) -2. The cost of a fixed sample of size nis C(n) = n(0.1). If the prior probabilities of 0, and 8, are and },
Assume that X1, X2,... is a sequential sample from a ((1+0)) density, that it is desired to estimate under decision loss L(8, a)-(8-a)/(1+0), and that the cost of a fixed sample of size n is C(n)=nc. If 0 has the prior density (0|a, B)= T(a+B) (a)(B) 0-1 (0+1)-(+)(0,)(0),
Assume that X1, X2, is a sequential sample from a 4(1, 0) density, that it is desired to estimate & under decision loss L(0, a)=(-a)2/02, and that the cost of a fixed sample of size n is C(n) =nc. If 8 has an 99(a, ) prior density, approximate the optimal fixed sample size.
Assume that X,, X, is a sequential sample from a P(8) density, that it is desired to estimate under decision loss L(0, a)=(-a)/0, and that the cost of a fixed sample of size n is C(n) nc. If 8 has a (a, ) prior density, approximate the optimal fixed sample size.
Assume that X1, X2,... is a sequential sample from a 38(1, 0) density, that it is desired to estimate 8 under decision loss L(e, a)=(-a)2/0(1-0), and that the cost of a fixed sample of size n is C(n)=nc. If 8 has a (0, 1) prior density, approximate the optimal fixed sample size.
Consider the situation of Exercise 1, but assume that the decision loss is L(0,a) 10-a. Find the optimal fixed sample size rule and its Bayes risk.
Do the preceding exercise for (a) C(n) [log(1+n)](0.01). (b) C(n) (0.01).
Assume that X1, X2,... is a sequential sample from a N(0, 1) density, that it is desired to estimate under squared-error decision loss, and that the cost of a fixed sample of size n is C(n) n(0.01). If 8 has a (1,4) prior distribution, find the optimal fixed sample size rule and its Bayes risk.
In Example 26, let & be the group consisting of the transformations g1.c(x) = x+c and g-1.0(x) = -x+c', where c and c' vary over R¹.(a) Show that the decision problem is invariant under , and find and(b) Show that 80(x) = x is the only nonrandomized decision rule which is invariant under G(c)
Consider the situation of Example 22.(a) Show that the given decision problem is actually invariant under the larger group G={8в.с: 8в.с (x, y) = (Bx, cBy), where B is nonsingular and c>0}, and findand(b) Show that there are no invariant decision rules for this problem. (Clearly,too much

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Statistical Decision Theory And Bayesian Analysis 2nd Edition James O. Berger - Solutions

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