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Theory Of Point Estimation 2nd Edition Erich L. Lehmann, George Casella - Solutions

7.14 Let Xj (j = 1,...,n) be independently distributed with densities fj (xj |θ) (θ realvalued), let Ij (θ) be the information Xj contains about θ, and let Tn(θ) = n j=1Ij (θ) be the total information about θ in the sample. Suppose that θˆn is a consistent root of the likelihood equation
7.13 Generalize the preceding problem to the situation in which (a) E(Xi) = α + βti and var(Xi) = 1 and (b) E(Xi) = α + βti and var(Xi) = σ2 where α, β, and σ2 are unknown parameters to be estimated.
7.12 Let Xi (i = 1,...,n) be independent normal with variance 1 and mean βti (with ti known). Discuss the estimation of β along the lines of Example 7.7.
7.11 Find suitable normalizing constants for δn of Example 7.7 when (a) γi = i, (b)γi = i2, and (c) γi = 1/ i.
7.10 Show that the estimator δn of Example 7.7 satisfies (7.14).
7.9 In Example 7.7, show that Yn is less informative than Y .[Hint: Let Zn be distributed as P(λ∞i=n+1γi) independently of Yn. Then, Yn + Zn is a sufficient statistic for λ on the basis of (Yn, Zn) and Yn + Zn has the same distribution as Y .]
7.8 (a) If the cdf F is symmetric and if log F(x) is strictly concave, so is log[1−F(x)].(b) Show that log F(x) is strictly concave when F is strongly unimodal but not when F is Cauchy.
7.7 In Example 7.6, suppose that pi = 1−F(α + βti) and that both log F(x) and log[1−F(x)] are strictly concave. Then, the likelihood equations have at most one solution.
7.6 Show that the likelihood equations (7.11) have at most one solution.
7.5 In the preceding problem, find the efficiency gain (if any)(a) in part (a) resulting from the knowledge that ρ = ρ(b) in part (b) resulting from the knowledge that σ = σ and τ = τ .
7.3 In Example 7.4, determine the joint distribution of (a) (σˆ 2, τˆ2) and (b) (σˆ 2, σˆ 2 A).7.4 Consider samples (X1, Y1),..., (Xm, Ym) and (X1, Y 1),..., (Xn, Y n) from two bivariate normal distributions with means zero and variance-covariances (σ2, τ 2,ρστ )and (σ2, τ 2,
7.2 For the situation of Example 7.3 with m = n:(a) Show that a necessary condition for (7.5) to converge to N(0, 1) is that √n(λˆ −λ) →0, where λˆ = σˆ 2/τˆ2 and λ = σ2/τ 2, for σˆ 2 and τˆ2 of (7.4).(b) Use the fact that λ/λ ˆ has an F-distribution to show that √n(λˆ
7.1 Prove Theorem 7.1.
6.17 In Example 6.10, show that the conditions of Theorem 5.1 are satisfied
6.15 Let X1,...,Xn be iid with E(Xi) = θ, var(Xi) = 1, and E(Xi − θ)4 = µ4, and consider the unbiased estimators δ1n = (1/n)X2 i − 1 and δ2n = X¯ 2 n − 1/n of θ 2.(a) Determine the ARE e2,1 of δ2n with respect to δ1n.(b) Show that e2,1 ≥ 1 if the Xi are symmetric about θ.(c) Find
6.14 Let X1,...,Xn be iid as N(0, σ2).(a) Show that δn = k|Xi|/n is a consistent estimator of σ if and only if k = √π/2.(b) Determine the ARE of δ with k = √π/2 with respect to the MLE *X2 i /n.
6.13 For the situation of Example 6.9, consider as another family of distributions, the contaminated normal mixture family suggested by Tukey (1960) as a model for observations which usually follow a normal distribution but where occasionally something goes wrong with the experiment or its
6.12 Show that the efficiency (6.27) tends to 0 as |a − θ|→∞.
6.11 Let X, ...,Xn be iid according to the Poisson distribution P(λ). Find the ARE ofδ2n = [No. of Xi = 0]/n to δ1n = e−X¯ n as estimators of e−λ.
6.10 Verify the limiting distribution asserted in (6.21).
6.9 Consider the situation leading to (6.20), where (Xi, Yi),i = 1,...,n, are iid according to a bivariate normal distribution with E(Xi) = E(Yi) = 0, var(Xi) = var(Yi) = 1, and unknown correlation coefficient ρ
6.8 Verify the matrices (a) (6.17) and (b) (6.18).
6.7 In Example 6.4, show that the Sjk given by (6.15) are independent of (X1,..., Xp)and have the same joint distribution as the statistics (6.13) with n replaced by n − 1.[Hint: Subject each of the p vectors (Xi1,...,Xin) to the same orthogonal transformation, where the first row of the
6.6 In Example 6.4, verify the MLEs ξˆi and σˆjk when the ξ ’s are unknown.
6.5 Let X1,...,Xn be iid from a H(α, β) distribution with density 1/(H(α)βα) × xα−1 e−x/β .(a) Calculate the information matrix for the usual (α, β) parameterization.(b) Write the density in terms of the parameters (α, µ)=(α, α/β). Calculate the information matrix for the (α,
6.4 If θ = (θ1,...,θr, θr+1,...,θs) and if cov ∂∂θi L(θ ), ∂∂θj L(θ )= 0 for any i ≤ r < j, then the asymptotic distribution of (θˆ1,..., θˆr) under the assumptions of Theorem 5.1 is unaffected by whether or not θr+1,...,θs are known.
6.3 Verify (6.5).
6.2 In Example 6.1, verify Equation (6.4).
6.1 In Example 6.1, show that the likelihood equations are given by (6.2) and (6.3).
5.6 Show that there exists a function f of two variables for which the equations ∂f (x, y)/∂x =0 and ∂f (x, y)/∂y = 0 have a unique solution, and this solution is a local but not a global maximum of f .
5.5 Prove Corollary 5.4.
5.4 Let (X0,...,Xs) have the multinomial distribution M(p0,...,ps; n).(a) Show that the likelihood equations have a unique root.(b) Show directly that the MLEs pˆi are asymptotically efficient.
5.3 Let X1,...,Xn be iid according to N(ξ,σ2).(a) Show that the likelihood equations have a unique root.(b) Show directly (i.e., without recourse to Theorem 5.1) that the MLEs ξˆ and σˆ are asymptotically efficient.
5.2 (a) Show that (5.26) with the remainder term neglected has the same form as (5.15)and identify the Ajkn.(b) Show that the resulting ajk of Lemma 5.2 are the same as those of (5.23).(c) Show that the remainder term in (5.26) can be neglected in the proof of Theorem 5.3.
5.1 (a) If a vector Yn in Es converges in probability to a constant vectora, and if h is a continuous function defined over Es, show that h(Yn) → h(a) in probability.(b) Use (a) to show that the elements of ||Ajkn||−1 tend in probability to the elements of B as claimed in the proof of Lemma
4.19 There is a connection between the EM algorithm and Gibbs sampling, in that both have their basis in Markov chain theory. One way of seeing this is to show that the incomplete-data likelihood is a solution to the integral equation of successive substitution sampling (see Problems 4.5.9-4.5.11),
4.18 The EM algorithm can also be implemented in a Bayesian hierarchical model to find a posterior mode. Recall the model (4.5.5.1), X|θ ∼ f (x|θ),|λ ∼ π(θ|λ), ∼ γ (λ), where interest would be in estimating quantities from π(θ|x). Sinceπ(θ|x) = π(θ,λ|x)d λ, where π(θ,λ|x)
4.17 Verify (4.30).
4.16 Maximum likelihood estimation in the probit model of Section 3.6 can be implemented using the EM algorithm. We observe independent Bernoulli variablesX1,...,Xn, which depend on unobservable variables Zi distributed independently as N(ζi, σ2), where Xi =0 if Zi ≤ u 1 if Zi > u.Assuming that
4.15 For the one-way layout with random effects (Example 3.5.1), the EM algorithm is useful for computing ML estimates. (In fact, it is very useful in many mixed models;see Searle et al. 1992, Chapter 8.) Suppose we have the model Xij = µ + Ai + Uij (j = 1,...,ni, i = 1,...,s)where Ai and Uij are
4.14 In the two-way layout (see Example 3.4.11), the EM algorithm can be very helpful in computing ML estimators in the unbalanced case. Suppose that we observe Yijk : N(ξij , σ2), i = 1,... ,I, j = 1,... ,J, k = 1,...,nij , where ξij = µ+ αi + βj + γij . The data will be augmented so that
4.13 For the situation of Example 4.10:(a) Show that the M-step of the EM algorithm is given byµˆ =4 i=1 ni j=1 yij + z1 + z2/12,αˆi =2 j=1 yij + zi/3 − ˆµ, i = 1, 3=3 j=1 yij/3 − ˆµ, i = 2, 4.(b) Show that the E-step of the EM algorithm is given by zi = E Yi3|µ = µ, α ˆ i =
4.12 For the mixture distribution of Example 4.7, that is, Xi ∼ θg(x) + (1 − θ)h(x), i = 1, . . . , n, independent where g(·) and h(·) are known, an EM algorithm can be used to find the ML estimator of θ. Let Z1, ··· , Zn, where Zi indicates from which distribution Xi has been drawn, so
4.11 In the EM algorithm, calculation of the E-step, the expectation calculation, can be complicated. In such cases, it may be possible to replace the E-step by a Monte Carlo evaluation, creating the MCEM algorithm (Wei and Tanner 1990). Consider the following MCEM evaluation of Q(θ|θˆ(j ),
4.10 Show that if the EM complete-data density f (y, z|θ) of (4.21) is in a curved exponential family, then the hypotheses of Theorem 4.12 are satisfied.
4.9 Consider the following 12 observations from a bivariate normal distribution with parameters µ1 = µ2 = 0, σ2 1 , σ2 2 , ρ:x1 1 1 -1 -1 2 2 -2 -2 * * * *x2 1 -1 1 -1 * * * * 2 2 -2 -2 where “∗” represents a missing value.(a) Show that the likelihood function has global maxima at ρ =
4.8 Without using Theorem 4.8, in Example 4.13 show that the EM sequence converges to the MLE.
4.7 In Theorem 4.8, show that σ11 = σ12.
4.6 In Example 4.7, if η = ξ , show how to obtain a √n-consistent estimator by equating sample and population second moments.
4.5 In Example 4.7, show that l(θ) is concave.
4.4 In Example 4.5, evaluate the estimators (4.8) and (4.14) for the Cauchy case, using for θ˜n the sample median.
4.3 Show that the density (4.4) with = (0,∞) satisfies all conditions of Theorem 3.10.
4.2 Show that the density (4.1) with = (0,∞) satisfies all conditions of Theorem 3.10 with the exception of (d) of Theorem 2.6.
4.1 Let u(t) =c t0 e−1/x(1−x)dx for 0
3.29 To establish the measurability of the sequence of roots θˆ∗n of Theorem 3.7, we can follow the proof of Serfling (1980, Section 4.2.2) where the measurability of a similar sequence is proved.(a) For definiteness, define θˆn(a) as the value that minimizes |θˆ − θ0| subject toθ0 −
3.28 Under the assumptions of Theorem 3.7, suppose that θˆ1n and θˆ2n are two consistent sequences of roots of the likelihood equation. Prove that Pθ0 (θˆ1n = θˆ2n) → 1 as n → ∞.[Hint:(a) Let Sn = {x : x = (x1,...,xn) such that θˆ1n(x) = θˆ2n(x)}. For all x ∈ Sn, there
3.27 Let X1,...,Xn be iid according to θg(x) + (1 − θ)h(x), where (g, h) is a pair of specified probability densities with respect to µ, and where 0
3.26 In Example 3.12, show directly that (1/n)T (Xi) is an asymptotically efficient estimator of θ = Eη[T (X)] by considering its limit distribution.
3.25 For X1,...,Xn iid as DE(θ , 1), show that (a) the sample median is an MLE of θand (b) the sample median is asymptotically normal with variance 1/n, the information inequality bound.
3.24 Check that the assumptions of Theorem 3.10 are satisfied in Example 3.12.
3.23 Let X1,...,Xn be iid according to N(θ, aθ 2),θ > 0, where a is a known positive constant.(a) Find an explicit expression for an ELE of θ.(b) Determine whether there exists an MRE estimator under a suitable group of transformations.[This case was considered by Berk (1972).]
3.22 Under the assumptions of Theorem 3.2, show thatLθ0 +1√n− L(θ0) +1 2I (θ0)/I (θ0)tends in law to N(0, 1).
3.21 Let X1,...,Xn be iid according to a Weibull distribution with density fθ (x) = θ xθ−1 e−xθ, x> 0,θ > 0, which is not a member of the exponential, location, or scale family. Nevertheless, show that there is a unique interior maximum of the likelihood function.
3.20 If X1,...,Xn are iid according to the gamma distribution H(θ , 1), the likelihood equation has a unique root.[Hint: Use Example 3.12. Alternatively, write down the likelihood and use the fact that H(θ)/ H(θ) is an increasing function of θ.]
3.19 If X1,...,Xn are iid as C(θ , 1), then for any fixed n there is positive probability (a)that the likelihood equation has 2n − 1 roots and (b) that the likelihood equation has a unique root.[Hint: (a) If the x’s are sufficiently widely separated, the value of L(θ) in the neighborhood of
3.18 In Problem 3.15(b), with f the Cauchy density C(0, a), the likelihood equation has a unique root aˆ and √n(aˆ −a) L→ N(0, 2a2).
3.17 If X1,...,Xn are iid with density f (xi −θ) or af (axi) and f is the logistic density L(0, 1), the likelihood equation has unique solutions θˆ and aˆ both in the location and the scale case. Determine the limit distribution of √n(θˆ − θ) and √n(aˆ − a).
3.16 For each of the following densities, f (·), determine if (a) it is strongly unimodal and (b) xf (x)/f (x) is strictly decreasing for x > 0. Hence, comment on whether the respective location and scale parameters have unique MLEs:(a) f (x) =1√2πe− 1 2 x2, −∞
3.15 (a) A density function isstrongly unimodal, or equivalently log concave, if log f (x)is a concave function. Show that such a density function has a unique mode.(b) Let X1,...,Xn be iid with density f (x −θ). Show that the likelihood function has a unique root if f (x)/f (x) is monotone,
3.14 Let X have the negative binomial distribution (2.3.3). Find an ELE of p.
3.13 Consider a sample X1,...,Xn from a Poisson distribution conditioned to be positive, so that P(Xi = x) = θ x e−θ /x!(1 − e−θ ) for x = 1, 2,.... Show that the likelihood equation has a unique root for all values of x.
3.12 Let X be distributed as N(θ , 1). Show that conditionally given a
3.11 Verify the nature of the roots in Example 3.9.
3.10 In Example 3.6 with 0
3.9 Prove (3.9).
3.8 Prove the existence of unique 0 < ak < ak−1, k = 1, 2,..., satisfying (3.4).
3.7 Show that Theorem 3.2 remains valid if assumption A1 is relaxed to A1: There is a nonempty set 0 ∈ such that θ0 ∈ 0 and 0 is contained in the support of each Pθ .
3.6 When is finite, show that the MLE is consistent if and only if it satisfies (3.2).
3.5 Let X take on the values 0 and 1 with probabilities p and q, respectively. When it is known that 1/3 ≤ p ≤ 2/3, (a) find the MLE and (b) show that the expected squared error of the MLE is uniformly larger than that of δ(x)=1/2.[A similar estimation problem arises in randomized response
3.4 Suppose X1,...,Xn are iid as N(ξ , 1) with ξ > 0. Show that the MLE is X¯ when X >¯ 0 and does not exist when X¯ ≤ 0.
3.3 Let X1,...,Xn be iid according to N(ξ,σ2). Determine the MLE of (a) ξ when σ is known, (b) σ when ξ is known, and (c) (ξ,σ) when both are unknown.
3.2 In the preceding problem, show that the MLE does not exist when p is restricted to 0
3.1 Let X have the binomial distribution b(p, n), 0 ≤ p ≤ 1. Determine the MLE of p(a) by the usual calculus method determining the maximum of a function;(b) by showing that px qn−x ≤ (x/n)x [(n − x)/n]n−x .[Hint: (b) Apply the fact that the geometric mean is equal to or less than the
2.14 In Example 2.7, show that if θn = c/√n, then Rn(θn) → a2 + c2(1 − a)2.
2.13 Let bn(θ) = Eθ (δn) − θ be the bias of the estimator δn of Example 2.5.(a) Show that bn(θ) = −(1 − a)√n √4 n−√4 n xφ(x − √nθ) dx;(b) Show that bn(θ) → 0 for any θ = 0 and bn(0) → (1 − a).(c) Use (b) to explain how the Hodges estimator δn can violate (2.7)
2.12 In Example 2.7 with Rn(θ) given by (2.11), show that Rn(θ) → 1 for θ = 0 and that Rn(0) → a2.
2.11 Construct a sequence {δn} satisfying (2.2) but for which the bias bn(θ) does not tend to zero.
2.10 In the preceding problem, construct δn such that w(θ) = v(θ) for all θ = θ0 and θ1 and < v(θ) for θ = θ0 and θ1.
2.9 Let δn be any estimator satisfying (2.2) with g(θ) = θ. Construct a sequence δn such that √n(δn − θ) L→ N[0, w2(θ)] with w(θ) = v(θ) for θ = θ0 and w(θ0) = 0.
2.8 Verify the asymptotic distribution claimed for δn in Example 2.5.
2.7 For the situation of Problem 2.6:(a) Calculate the mean squared errors of both δn and X(n) as estimators of θ.(b) Show lim n→∞E(X(n) − θ)2 E(δn − θ)2 = 2.
2.6 Let X1,...,Xn be iid as U(0, θ). From Example 2.1.14, δn = (n + 1)X(n)/n is the UMVU estimator of θ, whereas the MLE is X(n). Determine the limit distribution of (a)n[θ − δn] and (b) n[θ − X(n)]. Comment on the asymptotic bias of these estimators.[Hint: P(X(n) ≤ y) = yn/θ n for any
2.5 If X1,...,Xn are iid n(µ, σ2), show that Sr = [1/(n−1)(xi − ¯x)2]r/2 is an asymptotically unbiased estimator of σr.
2.4 If X1, ..., Xn are a sample from a one-parameter exponential family (1.5.2), thenT (Xi) is minimal sufficient and E[(1/n)T (Xi)] = (∂/∂η)A(η) = τ . Show that for any function g(·) for which Theorem 1.8.12 holds, g((1/n)T (Xi)) is asymptotically unbiased for g(τ ).
2.3 Assume that the distribution of Yn = √n(δn − g(θ)) converges to a distribution with mean 0 and variance v(θ). Use Fatou’s lemma (Lemma 1.2.6) to establish that varθ (δn) → 0 for all θ.
2.2 If kn[δn − g(θ)] L→ H for some sequence kn, show that the same result holds if kn is replaced by kn, where kn/kn → 1.
2.1 Let X1,...,Xn be iid as N(0, 1). Consider the two estimators Tn =X¯ n if Sn ≤ an n if Sn > an, where Sn = (Xi − X¯ )2, P(Sn > an)=1/n, and T n = (X1 + ··· + Xkn )/kn with kn the largest integer ≤ √n.(a) Show that the asymptotic efficiency of T n relative to Tn is
1.39 Let bm,n, m, n = 1, 2,..., be a double sequence of real numbers, which for each fixed m is nondecreasing in n. Show that limn→∞ limm→∞ bm,n = limm,n→∞ inf bm,n and limm→∞ limn→∞ bm,n = limm,n→∞ sup bm,n provided the indicated limits exist (they may be infinite) and
1.38 (a) In Problem 1.37, determine to what values var(Yn)
1.37 Let Yn be distributed as N(0, 1) with probability πn and as N(0, τ 2 n ) with probability 1−πn. If τn → ∞ and πn → π, determine for what values of π the sequence {Yn} does and does not have a limit distribution.

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