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statistical techniques in business

Questions and Answers of Statistical Techniques in Business

Consider a sequence {Pn, Qn} with likelihood ratio Ln defined in(14.36). Assume L(Ln|Pn) d→ W, where P{W = 0} = 0; show Pn is contiguous to Qn. Also, under (14.41), deduce that Pn is contiguous to
Suppose Qn is contiguous to Pn and let Ln be the likelihood ratio defined by (14.36). Show that EPn (Ln) → 1. Is the converse true?
Fix two probabilities P and Q and let Pn = Pn and Qn = Qn. Show that {Pn} and {Qn} are contiguous iff P = Q.
Fix two probabilities P and Q and let Pn = P and Qn = Q. Show that {Pn} and {Qn} are contiguous iff P and Q are absolutely continuous.
Show the convergence (14.35).
Prove (14.31).
Assume {Pθ, θ ∈ } is L1-differentiable, so that (14.93) holds.For simplicity, assume k = 1 (but the problem generalizes). Let φ(·) be uniformly bounded and set β(θ) = Eθ[φ(X)]. Show β(θ)
Suppose {Pθ, θ ∈ } is a model with an open subset of IRk , and having densities pθ(x) with respect to μ. Define the model to be L1-differentiable atθ0 if there exists a vector of real-valued
Suppose X1,..., Xn are i.i.d. and uniformly distributed on (0, θ).Let pθ(x) = θ−1 I{0 < x < θ} and Ln(θ) = i pθ(Xi). Fix p and θ0. Determine the limiting behavior of Ln(θ0 +
To see what might happen when the parameter space is not open, let f0(x) = x I{0 ≤ x ≤ 1} + (2 − x)I{1 < x ≤ 2} .Consider the family of densities indexed by θ ∈ [0, 1) defined by pθ(x) =
Suppose {Pθ} is q.m.d. at θ0. Show Pθ0+h{x : pθ0 (x) = 0} = o(|h|2)as |h| → 0. Hence, if X1,..., Xn are i.i.d. with likelihood ratio Ln,h defined by(14.12), show that Pnθ0+hn−1/2 {Ln,h =
Suppose {Pθ} is q.m.d. at θ0 with derivative η(·, θ0). Show that, on{x : pθ0 (x) = 0}, we must have η(x, θ0) = 0, except possibly on a μ-null set. Hint:On {pθ0 (x) = 0}, write 0 ≤ n1/2 p
Prove Theorem 14.2.2 using an argument similar to the proof of Theorem 14.2.1.
In Example 14.2.5, show that {[ f (x)]2/ f (x)}dx is finite iff β >1/2.
In Examples 14.2.3 and 14.2.4, find the quadratic mean derivative and I(θ).
Show that the definition of I(θ) in Definition 14.2.2 does not depend on the choice of dominating measure μ.
to construct a family of distributions Pθ with θ ∈ IR2, defined for all small |θ|, such that P0,0 = P, the family is q.m.d. at θ = (0, 0) with score vector at θ = (0, 0) given by(u1(x),
Fix a probability P on S and functions ui(x) such that ui(x)d P(x) = 0 and u2 i (x)d P(x) < ∞, for i = 1, 2. Adapt
To see what might happen when the parameter space is not open, let(iii) Suppose u2(x)d P(x) < ∞. Define pθ(x) = C(θ)2[1 + exp(−2θu(x))]−1 .Show this family is q.m.d. at θ = 0, and
Suppose {Pθ} is q.m.d. at θ0. Show Pθ0+h{x : pθ0 (x) = 0} = o(|h|2)as |h| → 0. Hence, if X1,..., Xn are i.i.d. with likelihood ratio Ln,h defined by(14.12), show that Pnθ0+hn−1/2 {Ln,h =
Suppose {Pθ} is q.m.d. at θ0 with derivative η(·, θ0). Show that, on{x : pθ0 (x) = 0}, we must have η(x, θ0) = 0, except possibly on a μ-null set. Hint:On {pθ0 (x) = 0}, write 0 ≤ n1/2 p
Prove Theorem 14.2.2 using an argument similar to the proof of Theorem 14.2.1.
In Example 14.2.5, show that {[ f (x)]2/ f (x)}dx is finite iff β >1/2.
In Examples 14.2.3 and 14.2.4, find the quadratic mean derivative and I(θ).
Show that the definition of I(θ) in Definition 14.2.2 does not depend on the choice of dominating measure μ.
to construct a family of distributions Pθ with θ ∈ IR2, defined for all small |θ|, such that P0,0 = P, the family is q.m.d. at θ = (0, 0) with score vector at θ = (0, 0) given by(u1(x),
Fix a probability P on S and functions ui(x) such that ui(x)d P(x) = 0 and u2 i (x)d P(x) < ∞, for i = 1, 2. Adapt
Fix a probability P. Let u(x) satisfyu(x)d P(x) = 0 .(i) Assume supx |u(x)| < ∞, so that pθ(x) = [1 + θu(x)]defines a family of densities (with respect to P) for all small |θ|. Show this family
Suppose X and Y are independent, with X distributed as Pθ and Y as P¯θ, as θ varies in a common index set . Assume the families {Pθ} and {P¯θ} are q.m.d. with Fisher Information matrices IX
Suppose gn is a sequence of functions in L2(μ) and, for some function g,(gn − g)2dμ → 0. If h2dμ < ∞, show that hgndμ → hgdμ.
Suppose gn is a sequence of functions in L2(μ); that is, g2 n dμ < ∞.Assume, for some function g,(gn − g)2dμ → 0. Prove that g2dμ < ∞.
Generalize Example 14.2.2 to the case of a multiparameter exponential family. Compare with the result of Problem 14.1.
Generalize Example 14.2.1 to the case where X is multivariate normal with mean vector θ and nonsingular covariance matrix .
In the setting of Section 13.5.3, show that the Bonferroni test that rejects H0 when Mn ≥ z1− αn is equivalent to the test that rejects Hi if min pˆi ≤ α/n, where pˆi = 1 − (Xi).
Prove (13.61).
Prove Lemma 13.5.1 by using Problem 13.32. That is, if 1 < βn =o(n) and Yn,i = exp(δn Xi − δ2 n2 ) , show that E[|Yn,i − 1|I{|Yn,i − 1| > βn}] → 0 . (13.66)Since Yn,i > 0 and βn > 1,
Prove Lemma 13.5.1 as follows. Let η = 1 − √r. Let L˜ n = 1 nn i=1 exp(δn Xi − δ2 n2 )I{Xi ≤ 2 log n} .First, show Ln − L˜ n P→ 0 (using Problem 13.41). Then, show E(L˜ n) = (η2
Under the setting of Lemma 13.5.1 calculate V ar(Ln) and determine which values of r it tends to 0.
Let X1,..., Xn be i.i.d. N(0, 1). Let Mn = max(X1,..., Xn).(i) Show that P{Mn ≥ √2 log n} → 0.(ii) Compute the limit of P{Mn ≥ z1− αn }.
Prove Lemma 13.5.1 by using Problem 13.32. That is, if 1 < βn =o(n) and Yn,i = exp(δn Xi − δ2 n2 ) , show that E[|Yn,i − 1|I{|Yn,i − 1| > βn}] → 0 . (13.66)Since Yn,i > 0 and βn > 1,
Prove Lemma 13.5.1 as follows. Let η = 1 − √r. Let L˜ n = 1 nn i=1 exp(δn Xi − δ2 n2 )I{Xi ≤ 2 log n} .First, show Ln − L˜ n P→ 0 (using Problem 13.41). Then, show E(L˜ n) = (η2
Under the setting of Lemma 13.5.1 calculate V ar(Ln) and determine which values of r it tends to 0.
Let X1,..., Xn be i.i.d. N(0, 1). Let Mn = max(X1,..., Xn).(i) Show that P{Mn ≥ √2 log n} → 0.(ii) Compute the limit of P{Mn ≥ z1− αn }.
(i) If φ(·) denotes the standard normal density and Z ∼ N(0, 1), then for any t > 0,(1 t − 1 t 3 )φ(t) < P{Z ≥ t} ≤ φ(t)t . (Prove the right-hand inequality.(ii) Prove the left inequality
For the Chi-squared test discussed in Section 13.5.1, assume thatδ2 n /√2n → ∞. Show that the limiting power of the Chi-squared test against such an alternative sequence tends to one.
Let Yn,1,..., Yn,n be i.i.d. bernoulli variables with success probability pn, where npn = λ and λ1/2 = δ. Let Un,1,..., Un,n be i.i.d. uniform variables on(−τn, τn), where τ 2 n = 3p2 n.
Prove the second equality in (13.44). In the proof of Lemma 13.4.2, show that κn(n) → 0.
Consider the problem of testing μ(F) = 0 versus μ(F) = 0, for F ∈ F0, the class of distributions supported on [0, 1]. Let φn be Anderson’s test.(i) If|n1/2μ(Fn)| ≥ δ > 2sn,1−α , then
Prove Lemma 13.4.5.
In the proof of Theorem 13.4.2, prove Sn/σ(Fn) → 1 in probability.
Suppose F satisfies the conditions of Theorem 13.4.4. Assume there exists φn such that sup F∈F: μ(F)=0 EF (φn) → α .Show that lim sup n EF (φn) ≤ αfor every F ∈ F.
In Lemma 13.4.2, show that Condition (13.41) can be replaced by the assumption that, for some βn = o(n1/2), lim sup n→∞EGn [|Yn,i − μ(Gn)|I{|Yn,i − μ(Gn)| ≥ βn}] = 0.Moreover, this
Let φn be the classical t-test for testing the mean is zero versus the mean is positive, based on n i.i.d. observations from F. Consider the power of this test against the distribution N(μ, 1).
Assuming F is absolutely continuous with 4 moments, verify(13.39).
In Theorem13.3.2, suppose S2 n is defined with its denominator n − 1 replaced by n. Derive the explicit form for q2(t, F) in the corresponding Edgeworth expansion.
When sampling from a normal distribution, one can derive an Edgeworth expansion for the t-statistic as follows. Suppose X1,..., Xn are i.i.d. N(μ, σ2)and let tn = n1/2(X¯ n − μ)/Sn, where S2 n
Let X1,..., Xn be a sample from N(ξ, σ2), and consider the UMP invariant level-α test of H : ξ/σ ≤ θ0 (Section 6.4). Let αn(F) be the actual significance level of this test when X1,..., Xn
is not robust against nonnormality.
Show that the test derived in
In the preceding problem, investigate the rejection probability when the Fi have different variances. Assume min ni → ∞ and ni /n → ρi .
For i = 1,...,s and j = 1,..., ni , let Xi,j be independent, with Xi,j having distribution Fi , where Fi is an arbitrary distribution with mean μi and finite common variance σ2. Consider testing
The size of each of the following tests is robust against nonnormality:1. the test (7.24) as b → ∞, 2. the test (7.26) as mb → ∞, 3. the test (7.28) as m → ∞.
If i,i are defined as in (13.19), show that n i=1 2 i,i = s.Hint: Since the i,i are independent of A, take A to be orthogonal.
If ξi = α + βti + γui , express Condition (13.20) in terms of the t’s and u’s.
with cn = nk .
Let cn = u0 + u1n +···+ uknk , ui ≥ 0 for all i. Then cn satisfies(13.10). What if cn = 2n? Hint: Apply
Let {cn} and {cn} be two increasing sequences of constants such that cn/cn → 1 as n → ∞. Then {cn} satisfies (13.10) if and only if {cn} does.
Show that (13.10) holds whenever cn tends to a finite nonzero limit, but the condition need not hold if cn → 0.
Suppose (13.20) holds for some particular sequence (n) with fixed s. Then it holds for any sequence (n) ⊆ (n) of dimension s < s.Hint: If is spanned by the s columns of A, let be spanned
In the two-way layout of the preceding problem give examples of submodels (1) and (2) of dimensions s1 and s2, both less than ab, such that in one case Condition (13.20) continues to require ni j
Let Xijk (k = 1,..., ni j; i = 1,,..., a; j = 1,...,b) be independently normally distributed with mean E(Xijk ) = ξi j and variance σ2. Then the test of any linear hypothesis concerning the ξi j
In Example 13.2.3, verify the Huber Condition holds.
Verify (13.15).
Verify the claims made in Example 13.2.1.
Prove Lemma 13.2.3. Hint: For part (ii), use Problem 11.72.
Prove (i) of Lemma 13.2.2.
Determine the maximum asymptotic level of the one-sided t-test when α = .05 and m = 2, 4, 6: (i) in Model A; (ii) in Model B.
Show that the conditions of Lemma 13.2.1 are satisfied and γ has the stated value: (i) in Model B; (ii) in Model C.
In Model A, suppose that the number of observations in group i is ni . if ni ≤ M and s → ∞ show that the assumptions of Lemma 13.2.1 are satisfied and determine γ.
Verify the formula for V ar(X¯) in Model A.
(i) Given ρ, find the smallest and largest value of (13.2) as σ2/τ 2 varies from 0 to ∞.(ii) For nominal level α = 0.05 and ρ = 0.1, 0.2, 0.3, 0.4, determine the smallest and the largest
Under the assumptions of Lemma 13.2.1, compute Cov(X2 i , X2 j) in terms of ρi,j and σ2. Show that V ar(n−1 n i=1 X2 i ) → 0 and hence n−1 n i=1 X2 iP→
Let (Yi, Zi) be i.i.d. bivariate random vectors in the plane, with both Yi and Zi assumed to have finite nonzero variances. Let μY = E(Y1) and μZ =E(Z1), let ρ denote the correlation between Y1
Generalize the previous problem to the two-sample t-test.
(i) Let X1,..., Xn be a sample from N(ξ, σ2). For testingξ = 0 against ξ > 0, show that the power of the one-sided one-sample t-test against a sequence of alternatives N(ξn, σ2) for which
Consider points on a lattice of the form (i, j) where i and j are integers from 0 to n. Each of these (n + 1)2 points can be considered a vertex of a graph. Consider connecting edges adjoining (i, j)
Verify (12.80), (12.81) and (12.82). Based on the bound (12.82), consider an asymptotic regime where p ∼ n−β for some β ≥ 0. For what β does the bound tend to zero, so that a central limit
An alternative characterization of the Wasserstein metric is the following (which you do not have to show): dW (X, Y ) is the infimum of E|X − Y|over all possible joint distributions of (X, Y) such
Use Theorem 12.5.2 to derive a Central Limit Theorem for the sample mean of an m-dependent stationary process. State your assumptions and compare with Theorem 12.4.1.
Complete the details in Example 12.5.1 to get an explicit bound from Theorem 12.5.2 for dW . What conditions are you assuming?
Finish the proof of Theorem 12.5.2 by showing V ar⎛⎝n i=1 j∈Ni Xi X j⎞⎠ ≤ 14D3n i=1 E(|Xi|4) .Hint: Use the arithmetic–geometric mean inequality.
Theorem 12.5.1 provides a bound for dW (W, Z) where W =n−1/2 n i=1 Xi and the Xi are independent with mean 0 and variance one. Extend the result so that V ar(Xi) = σ2 i may depend on i.
Show that, if E(X2) = 1, then E|X| ≤ E(|X|3).
If Z is a real-valued random variable with density bounded by C, then show that, for any random variable W, dK (W, Z) ≤ 2CdW (W, Z) , where dK is the Kolmogorov–Smirnov (or sup or uniform)
Investigate the relationships between dW , dK and dT V , as well as the bounded Lipschitz metric introduced in Problem 11.24. Does convergence of one of them imply convergence of any of the others?
Show that the Wasserstein metric implies weak convergence; that is, if dW (Xn, X) → 0, then Xn d→ X. Give a counterexample to show the converse is false. Prove or disprove the following claim:
Show that, for w > 0, 1 − (w) ≤ min 1 2, 1 w√2πe−w2/2 .Show that this inequality implies fx ≤ √π/2 and f x ≤ 2.
Complete the proof of Lemma 12.5.2 by showing that (12.64) and(12.65) are equivalent, and then showing that (12.66) follows.
Complete the proof of the converse in Lemma 12.5.1. Hint: Use Lemma 12.5.2.
If W ∼ N(0, σ2) with σ = 1, what is the generalization of the characterization (12.61)?

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