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principles of managerial statistics

Questions and Answers of Principles Of Managerial Statistics

=+Here, the approximation means that the neglected term is of lower order in a suitable sense. This leads to the following approximation (whose justification, of course, requires some regularity
=+where θ represents the true parameter vector. Thus, we haveˆθ − θ ≈ − ∂M∂θ−1 M(θ)166 5 Inequalities≈ −E∂M∂θ −1 M(θ).
=+where W(θ) is a matrix depending on θ and u(y, θ) is a vector-valued function of Y and θ satisfying E{u(Y,θ)} = 0 if θ is the true parameter vector (in other words, the estimating equation is
=+5.18. (Estimating equations) A generalization of the WLS (see Example 5.8) is the following. Let Y denote the vector of observations and θ a vector of parameters of interest. Consider an estimator
=+1 ≤ i ≤ s, where Di is the diagonal matrix whose diagonal elements are the eigenvalues of Ai. This is called simultaneous diagonalization (see Appendix A.1). Suppose that A and B are
=+5.17. Many of the “cautionary tales” regarding extensions of results for nonnegative numbers to nonnegative definite matrices are due to the fact that matrices are not necessarily commutative.
=+projection to L(X) is defined as PX⊥ = I−PX , where I is the identity matrix.Show that PX ≥ 0 and PX⊥ ≥ 0.
=+5.16. For any matrix X of full rank, the projection matrix onto L(X), the linear space spanned by the columns of X, is defined as PX = X(XX)−1X(the definition can be generalized even if X is
=+5.7 Exercises 165
=+(iv) Show that the right-side inequality in (iii) is sharper in that exp 1 3(2n − 1)≤ 1 +1 3n, n = 1, 2,....
=+(iii) Prove the following sharper inequality (see below):1 ≤ E(eX¯) ≤ exp 1 3(2n − 1).
=+where b < 2. (Hint: Take the logarithm of both sides of the inequality.)(ii) Suppose that X1,...,Xn are i.i.d. and distributed as Uniform[−1, 1].Let X¯ = n−1 n i=1 Xi. Show that 1 ≤ E(eX¯)
=+5.12. This exercise is related to Example 5.5.(i) Use the monotone function technique to prove the following inequality:ex ≤ 1 + x + x2 2 − b, |x| ≤ b,
=+(ii) Show that the function h(λ) defined by (5.20) attains its maxima on(0, 2B−1) at λ∗ given by (5.21), and the maxima is given by (5.22).(iii) What is the reason for maximizing h(λ)?
=+5.10. This exercise is regarding the latter part of Example 5.6.(i) By using the same arguments, show that I2 ≤ exp−λ − λB2 2 − λB n.
=+5.7. Let xi,...,xn be real numbers. Define a probability on the space X = {x1,...,xn} by 164 5 Inequalities P(A) = # of xi ∈ A nfor any A ⊂ X. Show that P(A ∩ B) ≤ P(A)P(B).[Hint: Note that
=+5.6. Verify the identity (5.13).
=+(iii) the pdf of the Logistic(0, 1) distribution, which is given by f(x) =e−x(1 + e−x)−2, −∞
=+5.5. A pdf f(x) is called log-concave if log{f(x)} is concave. Show that the following pdf’s are log-concave:(i) the pdf of N(0, 1);(ii) the pdf of χ2ν, where the degrees of freedom ν ≥ 2;
=+(iv) Which approximation [(i) or (ii)] do you think is better? Any general comment(s) on the use of the delta method in moment approximation.
=+(iii) How does the sample size n affect the approximation to E(Y )? In other words, does the accuracy of the approximation improve as n increases?[Hint: First use the dominated convergence theorem
=+(i) Let g(x1,...,xn) = n/(n + n i=1 x2 i ). What are the approximations to the mean and variance of Y = g(X1,...,Xn)?(ii) If we let g(t1,...,tn) = n/(n+n i=1 ti), and Ti = X2 i , 1 ≤ i ≤ n,
=+4.30. Let X1,...,Xn be i.i.d. such that E(X1) = 0, E(X2 1 ) = 1, and E(X4 1 ) < ∞. Consider approximation to the mean and variance of Y = n n + n i=1 X2 iusing the delta method of Exercise 4.29.
=+(iii) Obtain the exact mean and variance of T −1 given the suitable range of α and compare the results with the above delta-method approximations.How do the values of α and β affect the
=+(ii) Note that the exact mean and variance of T −1 can be obtained in this case, given a suitable range of α. What is the range of α so that E(T −1)exists? What is the range of α so that E(T
=+(i) Suppose that T ∼ Gamma(α, β) with the pdf given in Exercise 4.24(ii).Use the above delta method to approximate the mean and variance of T −1.126 4 Asymptotic Expansions
=+where μi = E(Ti), 1 ≤ i ≤ k, and ∂g/∂ti is evaluated as (μ1,...,μk). This leads to the following approximations:E{g(T1,...,Tk)} ≈ g(μ1,...,μk), var{g(T1,...,Tk)} ≈ k i=1
=+4.29 [Delta method (continued)]. In Example 4.4 we introduced the delta method for distributional approximations. The method can also be used for moment approximations. Let T1,...,Tk be random
=+(iii) Let η = (ηi)1≤i≤m denote the vector of small-area means and ˆη =(ˆηi)1≤i≤m denote the vector of EBLUPs. Define the overall MSPE of the EBLUP as MSPE(ˆη) = E(|ηˆ−η|2)=E{m
=+a random variable with a χ2 k-distribution. Find the expression.]
=+MSPE(˜ηi) = A A + 1 + xi(XX)−1xi A + 1 , where X = (xi)1≤i≤m.(ii) Show that MSPE(ˆηi) = A A + 1 + xi (XX)−1xi A + 1 +2{1 − xi(XX)−1xi}(A + 1)(m − p)+4{1 −
=+4.9 Exercises 125ηˆi denote the BLUP and EBLUP, respectively, where the MoM estimator of A is used for the EBLUP [see Example 4.18 (continued) or Exercise 4.25].(i) Show that
=+4.28. Consider a special case of the Fay–Herriot model (Example 4.18) in which Di = D, 1 ≤ i ≤ m. This is known as the balanced case. Without loss of generality, let D = 1. Consider the
=+where α > 0 and β > 0, so that θ1 = α and θ2 = β.
=+Γ(α)βα xα−1e−x/β, x> 0, where α > 0 and β > 0 are known as the shape and scale parameters, respectively, so that θ1 = α and θ2 = β.(iii) X1 ∼ Beta(α, β), whose pdf is given by
=+4.24. Let X1,...,Xn be i.i.d. with the following pdf or pmf depending onθ = (θ1, θ2). Obtain the Fisher information matrix (4.78) in each case.(i) X1 ∼ N(μ, σ2), where μ ∈ (−∞, ∞)
=+4.23. Let X1,...,Xn be i.i.d. observations with the pdf or pmf f(x|θ), where θ is a univariate parameter. Here, the pdf is with respect to the Lebesgue measure, whereas the pmf may be regarded as
=+(ii) Show that if q(x)=(x − μ)2/2σ2 for some μ ∈ R and σ2 > 0, the Laplace approximation (4.63) is exact.
=+4.20. This exercise is related to Example 4.16.(i) Show that in this case the exact value of (4.62) is given by√νπΓ( ν2 )Γ( ν+1 2 ) , and the Laplace approximation (4.63) is, 2νπν + 1.
=+ν → ∞? (Hint: Consider the relative error of the approximation defined as|approximate − exact|/exact.)
=+X, including E(X−1), can be obtained, so that one can directly compare the accuracy of the approximation. Does the approximation improve as
=+4.17. Suppose that X has a χ2ν-distribution, where ν > 2. Use the elementary expansion (4.55) with l = 4 and without the remaining term to approximate E(X−1). Note that closed-form
=+4.9 Exercises 123
=+4.7. Obtain the two-term Edgeworth expansion [i.e., (4.27)] for the following distributions of Xi:(i) Xi ∼ the double exponential distribution DE(0, 1), where the pdf of DE(μ, σ) is given by
=+(iv) Make a nice plot that compares the histograms and a nice table that compares the percentiles for the increasing sample size. What do you conclude?
=+(iii) In addition to the histograms, obtain the 5th and 95th percentiles based on the 1000 values of ξn for each case and sample size and compare the percentiles with the corresponding standard
=+(i)]; the second sequence is generated independently from the Beta(α, β) distribution with α = 2 and β = 6 [case (ii)]. Based on each sequence, computeξn = (√n/σ)(X¯ − μ), where n is
=+CLT in regard to Example 4.5.(i) Generate two sequences of random variables. The first sequence is generated independently from the Beta(α, β) distribution with α = β = 2 [case
=+4.6. In this exercise you are asked to study empirically the convergence of
=+4.5. Let X1,...,Xn be i.i.d. observations generated from the following distributions, where n = 30. Construct the histograms of the empirical distribution of X¯ based on 10,000 simulated values.
=+−→b, where b is a positive constant, andξnn ≤ supθ|l(θ)| ≤ cn.
=+(c) Show that n−1/2l(0) d−→ N(0, σ2) as n → ∞, and determine σ2.(d) Show that there is a sequence of positive random variables ξn and a constant c > 0 such that ξn P
=+4.2. This is regarding Example 4.3.(a) Show that l(0) = −n i=1{Xi + 2 log(1 + e−Xi )}, l(0) = n i=1 1 − e−Xi 1 + e−Xi, l(θ) = −2n i=1 eθ−Xi(1 + eθ−Xi )2 .(b) Show that
=+3.22. Suppose that X1,...,Xn are i.i.d. observations whose mgf [defined by (2.9)] exists for some t > 0. Show that X(n) = OP{log(n)}, where X(n) i
=+(ii) Show thatn i=1 X2i n i=1 Xi= 1 nn i=1 X2 i − 1 n2n i=1 X2 i n i=1 Xi − n+1 n3n i=1 X2 i n i=1 Xi − n2+ OP(n−3/2).(iii) What are the orders of the first three terms on
=+3.21. Let X1,...,Xn be independent Exponential(1) random variables.(i) Prove the identity 1n i=1 Xi= 1 n −n i=1 Xi − n n2 + (n i=1 Xi − n)2 n3 − (n i=1 Xi − n)3 n3 n i=1 Xi.
=+3.8 Exercises 79(iii) Show that eX· = oebn0.5+δfor any constants δ, b > 0 (no matter how small).
=+for any constant a > 0 (no matter how how large). In other words, for any a > 0, eX·/ea√n is not bounded in probability.
=+(ii) If you think (i) is straightforward, show that eX· is not OPea√n
=+3.19. Let X1,...,Xn be i.i.d. random variables such that E(Xi) = 0 and var(Xi) = σ2, where σ2 ∈ (0, ∞).(i) Show that X· = n i=1 Xi = OP(√n).
=+(i) Show that log(Fn) = O(n).(ii) Find the limit limn→ Fn/n.(iii) Show that Fn log(Fn) ∼ nFn+1 [the definition of an ∼ bn is in Section 3.2 above (3.6)].
=+How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month in?Not surprisingly,
=+3.17. What sequence is this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...? If you examine the numbers carefully, you will realize that these are the famous Fibonacci numbers, or Fibonacci sequence, defined
=+c1,...,ck and d1,...,dl are constants such that c0d0 = 0. Does the answer depend on the values of k and l?
=+(iii) an = c0 +c1n+···+cknk, bn = d1 +d2n+···+dlnl, where c1,...,ck and d1,...,dl are constants such that ckdl = 0. (Note: The answer depends on the values of k and l.)(iv) an = c0 + c1n−1
=+3.16. Determine the order relation of the following sequences an and bn.(i) an = (n + c)n, bn = nn, where c is any constant.(ii) an = n i=1 i−1, bn = log(n).
=+(ii) an = {log(n)}1−, bn = log(nδ) for any 0 0.(iii) an = exp[{log(n)}], bn = nδ for any 0 0.(iv) an = (n/a)n, bn = n!, where a > 0. (Note: Depending on the value ofa, the conclusion may be
=+3.15. Determine the order relation of the following sequences an and bn:(i) an = c0 +c1n+···+cknk, bn = an for any positive integer k and a > 1, where c1,...,ck are constants.78 3 Big O, Small
=+(vi) Furthermore, let U1,...,Ul be jointly normal, each have N(0, 1) distribution, and the correlations between Uj and Uk be given by (3.28). Show that as δ → 0, P(max1≤j≤l Uj ≤ x) →
=+(v) Show that for any δ > 0 and any l ≥ 1, one can choose θ1,...,θl > 0 such that all pairwise correlations (3.28) are less than δ.
=+(iii) Show that ˆp < 1 with probability tending to 1.(iv) Show by the inequality (3.30) that w.p. → 1, and 2L∗ ≤ M2 − 1 implies that max1≤j≤m Un(θj ) ≤ M.
=+3.14. This problem is associated with Section 3.7.(i) Verify that the LRT statistic is given by (3.21) and (3.22).(ii) For fixed θ, consider the random variable Yi defined therein. Show that for
=+3.8 Exercises 77
=+3.6. Prove Lemma 3.7.
=+3.3. Consider the function f(x) in Example 3.3.(i) Suppose that ap, bq are nonzero. Show that as |x|→∞, f(x) = o(1) if pq.(ii) Suppose that b0 = 0. Show that as x → 0, f(x) = O(1) regardless
=+(iii) The distribution of ξξ is the same as (2.29), where Z1,...,ZM−1 are independent standard normal random variables.
=+2.39. Regarding the distribution of |ξ|2 = ξξ in (2.37), show the following[see the notation below (2.37)]:(i) P B = BP = 0.(ii) P is a projection matrix with rank 1.
=+2.8 Exercises 49
=+i·β for an arbitrary θ = β (not necessarily the true parameter vector), then conditions (i) and (ii) are satisfied.
=+2.36. This exercise refers to Example 2.17.(i) Show that X1,...,Xn are i.i.d. with a distribution whose cf is given by(2.25).(ii) If we define Xi(θ) = Y¯i· − x¯
=+(ii) Show that it is not necessarily true that |ξn|p1(|ξn|≤a)P−→ |ξ|p1(|ξ|≤a)as n → ∞.
=+2.35. Refer to the (iii) ⇒ (i) part of the proof of Theorem 2.17.(i) Show that ηn P−→ η as n → ∞.
=+2.32. Prove that the inequality (2.16) holds for any p > 0. Note that for p ≥ 1, this follows from the convex function inequality, but the inequality holds for 0
=+(ii) Show that ξn, n = 1, 2,..., does not converge to zero in Lp for any p > 0.
=+2.29. Let X ∼ Uniform(0, 1). Define ξn = 2n−11(0
=+where ˆp is the sample proportion which is equal to (X1 + ··· + Xn)/n. This result is also known as normal approximation to binomial distribution. (Of course, the result follows from the CLT,
=+2.27. Let X1,...,Xn be i.i.d Bernoulli(p) observations. Show that np(1 − p)1/2(ˆp − p) d−→ N(0, 1) as n → ∞,
=+2.26. Prove Theorem 2.13.
=+48 2 Modes of Convergence which converges to e−|t| as n → ∞.(iv) Show that the cf of ξ ∼ Cauchy(0, 1) is e−|t|, t ∈ R. Therefore, X¯ d−→ ξas n → ∞.
=+(ii) Show that the cf of Xi is given by max(1 − |t|, 0), t ∈ R.(iii) Show that the cf of X¯ = n−1 n i=1 Xi is given by max 1 − |t|n , 0 n,
=+2.25. Suppose that X1,...,Xn are i.i.d. with the pdf f(x) = 1 − cos(x)πx2 , −∞
=+as n → ∞, where ξ ∼ N(0, e2). (Hint: The result can be established as an application of the CLT; see Chapter 6.)
=+2.17. Let X1, X2,... be i.i.d. Uniform(0, 1] random variables and ξn =(n i=1 Xi)−1/n. Show that√n(ξn −e) d−→ ξ
=+2.16. Suppose that X1,...,Xn are i.i.d. Exponential(1) random variables.Define X(n) as in Exercise 2.15. Show that X(n) − log(n) d−→ ξas n → ∞, where the cdf of ξ is given by F(x) =
=+2.8 Exercises 47 Find a positive number δ such that n−δX(n) converges in distribution to a nondegenerate distribution, where X(n) = max1≤i≤n Xi. What is the limiting distribution?
=+2.15. Suppose that X1, X2,... are i.i.d. with a Cauchy(0, 1) distribution;that is, the pdf of Xi is given by f(x) = 1π(1 + x2), −∞
=+(ii) X1 ∼ Uniform[a, b], where a and b are unknown constants.(iii) X1 ∼ N(μ, σ2), where μ and σ2 are unknown parameters.
=+2.14. Suppose that X1,...,Xn are i.i.d. observations with finite expectation. Show that in the following cases the sample mean X¯ = (X1+···+Xn)/n is a strongly consistent estimator of the
=+where the notations refer to Example 2.7.
=+2.12. Show by similar arguments as in Example 2.7 that I2 ≤ ce−√n,

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