Testing Statistical Hypotheses Volume I 4th Edition E.L. Lehmann, Joseph P. Romano - Solutions

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For the location scale model in Problem 15.47, show that, for testingμ ≤ 0 versus μ > 0, argue that the Wald test is LAUMP if β ≥ 1. If σˆn is replaced by any consistent estimator of σ, does the LAUMP property continue to hold? If 1/2
For the location scale model of Example 15.5.2 with f (x) =C(β) exp[−|x|β ], argue that the family is q.m.d. if β > 1/2.
In the location scale model of Example 15.5.2, verify the expressions for the Information matrix. Deduce that the matrix is diagonal if f is an even function.
Let d N(h,C) denote the density of the normal distribution with mean vector h ∈ IRk and positive definite covariance matrix C. Prove that exp( h, x − 1 2 h,Ch)d N(0,C)(x)is the density of N(Ch,C) evaluated at x. Hint:Use characteristic functions.
Assume {Qn,h, h ∈ IRk }is asymptotically normal according to Definition 15.4.1, with Zn and C satisfying (15.62). Show that, under Qn,h, Zn d→N(Ch,C).
Suppose {Qn,h, h ∈ IRk } is asymptotically normal. Show that Qn,h1 and Qn,h2 are mutually contiguous for any h1 and h2.
Suppose {Qn,h , h ∈ IRk }is asymptotically normal according to Definition 15.4.1, with Zn and C satisfying (15.62). Show the matrix C is uniquely determined. Moreover, if Z˜ n is any other sequence also satisfying (15.62), then Zn − Z˜ n → 0 in Qn,h-probability for any h.
Consider the regression model Yi = βxn,i + i , i = 1,..., n , where β ∈ R is unknown, the xn,i are fixed and known, and the i are i.i.d. standard normal. Consider testing the null hypothesis β = 0 versus β > 0 using the UMP level-α test.(i) Let xn,i = i/n. For what constant p is the limiting
Define appropriate extensions of the definitions of LAUMP and AUMP to two-sided testing of a real parameter. Let X1,..., Xn be i.i.d. N(θ , 1).Show that neither LAUMP nor AUMP tests exist for testing θ = 0 against θ= 0.
Suppose X1, ...Xn are i.i.d. N(θ , 1 + θ 2). Consider testing θ = θ0 versus θ>θ0 and let φn be the test that rejects when n1/2[X¯ n − θ0] > z1−α(1 +θ 2 0 )1/2.(i) Compute the limiting power of this test against θ0 + hn−1/2.(ii) Is this test AUMP?
Suppose X1, ...Xn are i.i.d. Poisson(λ). Consider testing the null hypothesis H0 : λ = λ0 versus the alternative, HA : λ>λ0.(i) Consider the test φ1 n with rejection region n1/2[X¯ n − λ0] > z1−αλ1/2 0 , where(zα) = α and is the cdf of a standard normal random variable. Find the
Assume the conditions of Example 15.3.1. Further assume f is strongly unimodal, i.e., − log( f ) is convex. Show the test φ˜n given by (15.43) is AUMP level-α. Hint: Use Problem 15.36.
Assume the conditions of Theorem 15.3.3. Assume φn is LAUMP level-α. Suppose the power function of φn is nondecreasing in θ, for θ ≥ θ0. Showφn is also AUMP level-α.
Let X1,..., Xn be i.i.d. according to a q.m.d. location model f (x −θ ). Let θˆn be any location equivariant estimator satisfying (15.58) (such as an efficient likelihood estimator). For testing θ ≤ 0 against θ > 0, show that the test that rejects when n1/2θˆn > I −1/2(0)z1−α is
For the Cauchy location model of Example 15.3.3, consider the estimator θˆn defined by (15.59). Show that the test that rejects when n1/2θˆn > 21/2z1−αis AUMP. Is the estimator location equivariant? Is the estimator θˆn = θˆn(X1,..., Xn)monotone in the sense it is nondecreasing as any
In the double exponential location model of Example 15.3.2, show that a MLE estimator is a sample median θˆn. The test that rejects the null hypothesis if n1/2θˆn > z1−α is AUMP and is asymptotically equivalent to Rao’s score test in the sense of Problem 15.25.
Suppose Zn is any sequence of random variables such that V arθn (Zn) ≤ 1 while Eθn (Zn) → ∞. Here, θn merely indicates the distribution of Zn at time n. Show that, under θn, Zn → ∞ in probability.
For testing θ0 versus θn, let φ∗n be a test satisfying lim sup n Eθ0 (φ∗n ) = α∗ < αand Eθn (φ∗n ) → β∗.(i) Show there exists a test sequence ψn satisfying lim supn Eθ0 (ψn) = α and a number β such that lim Eθn (ψn) = β ≥ β∗ , and this last inequality is strict
Prove the equivalence of Definition 15.3.2 and the definition in the statement immediately following Definition 15.3.2. What is an equivalent characterization for LAUMP tests?
Prove Theorem 15.3.1.
Prove Lemma 15.3.1 (iii). Hint: Problems 15.12–15.13.
Let X1,..., Xn be i.i.d. N(θ , 1). For testing θ = 0 against θ > 0, let φn be the UMP level-α test. Let φ˜n be the test which rejects if X¯ n ≥ bn/n1/2 or X¯ n ≤ −an/n1/2, where bn = z1−α + n−1/4 and an is then determined to meet the level constraint. Are the tests
for testing θ0 against θ0 + hn−1/2.
Under the q.m.d. assumptions of this section, show that φn,h given by (15.34) and φ˜n given by (15.43) are asymptotically equivalent in the sense of
For testing θ = θ0 versus θ>θ0, define two test sequences φn andψn to be asymptotically equivalent under the null hypothesis if φn − ψn → 0 in probability under θ0. Does this imply that, if θ0 is the true value, the probability the tests reach the same conclusion tends to 1? Show
Suppose X1,..., Xn are i.i.d. N(0, σ2). Let Tn,1 = Y¯n =n−1 n i=1 Yi , where Yi = X2 i . Also, let Tn,2 = (2n)−1 n i=1(Yi − Y¯n)2. For testingσ = 1 versus σ > 1, does the Pitman asymptotic relative efficiency of Tn,1 with respect to Tn,2 exist? If so, find it.Section 15.3
Suppose X1,..., Xn are i.i.d. Poisson with unknown mean θ. The problem is to test θ = θ0 versus θ>θ0. Consider the test that rejects for large X¯ n and the test that rejects for large S2 n = 1 n − 1 ni=1(Xi − X¯ n)2 .Compute the Pitman ARE.
Prove the inequality (15.30). Hint: The quantity (15.29) is invariant with respect to scale. By taking σ2 = 1, the problem reduces to choosing f to minimize f 2 subject to f being a mean 0 density with variance 1. Using the method of undetermined multipliers, it is sufficient to minimize[ f 2(x)
For a double exponential location family, calculate the Pitman AREs among pairwise comparisons of the t-test, the Wilcoxon test, and the Sign test.
Suppose 0 = {θ0}. In order to determine c = c(n, α) in (15.32), define c(n, α) to be c(n, α) = inf{d : Pθ0 {Tn > d} ≤ α} .Argue that this choice of c(n, α) satisfies (15.32). What if Tn > d is replaced by Tn ≥ d?
Under the assumptions of Example 15.2.1 show that the squared efficacy of the Wald test is I(θ0).
Under the assumptions of Theorem 15.2.1, suppose θk → θ0 andβ>α> 0. Show, for any N < ∞, there does not exist a test φk with k ≤ N such that lim inf k Eθk (φk ) ≥ β.
Let f (x) be the triangular density on [−1, 1] defined by f (x) = (1 − |x|)I{x ∈ [−1, 1]} .Let Pθ be the distribution with density f (x − θ ). Find the asymptotic behavior of H(Pθ0 , Pθ0+h) as h → 0, where H is the Hellinger distance. Compare your result with q.m.d. families.Section
Let Pn and Qn be two sequences of probability measures defined on(n, Fn). Assume they are contiguous. Assume further that both of them are product measures, i.e., Pn = n i=1 Pn,i and Qn = n i=1 Qn,i .Let Q − P1 denote the total variation distance between P and Q. Show that sup nn i=1Qn,i
Give an example where Qn − Pn1 → δ > 0 but Pn and Qn are mutually contiguous.
Assume X1,..., Xn are i.i.d. according to a family {Pθ } which is q.m.d. at θ0. Suppose, for some statistic Tn = Tn(X1,..., Xn) and some functionμ(θ ) assumed differentiable at θ0, n1/2(Tn − μ(θn)) d→ N(0, σ2) under θn wheneverθn = θ0 + hn−1/2. Show the same result holds, first
Use problem 15.11 to prove Theorem 14.4.1 when hn → h.
to show that Theorem 14.2.3 (i) remains valid if h is replaced by hn as long as hn falls in a bounded subset of IRk . Also, show part (ii) of Theorem 14.2.3 generalizes if h in the left-hand side of the convergence(14.14) is replaced by hn → h in IRk . Further assume (14.85) and show that, for
Use
For a q.m.d. family, show n H2(Pθ0+hn−1/2 , Pθ0+hn n−1/2 ) → 0 whenever hn → h. Then, show Pnθ0+hn n−1/2 is contiguous to Pnθ0 whenever hn → h.
Suppose Pn − Qn1 → 0. Show that Pn and Qn are mutually contiguous. Furthermore, show that, for any sequence of test functions φn, φnd Pn − φnd Qn → 0.
Let X1,..., Xn be i.i.d. according to a model {Pθ , θ ∈ }, where θis real-valued. Consider testing θ = θ0 versus θ = θn at level-α (α fixed, 0 0.
If I(θ0) is a positive definite Information matrix, show h = 0 if and only if h, I(θ0)h = 0.
Let Pθ be N(θ , 1). Fix h and let θn = hn−1/2. Compute S(Pn 0 , Pnθn)and its limiting value. Compare your result with the upper bound obtained from Theorem 15.1.3.
Consider testing Pnθ0 versus Pnθn and assume n H2(Pθ0 , Pθn ) → 0. Letφn be any test sequence such that lim sup Eθ0 (φn) ≤ α. Show that lim sup Eθn (φn)≤ α.
Prove Lemma 15.1.1.
Let Pθ be uniform on [0, θ]. Let θn = θ0 + h/n. Calculate the limit of n H2(Pθ0 , Pθ0+h/n). If h > 0, let φn be the UMP level-α test which rejects when the maximum order statistic is too large. Evaluate the limit of the power of φn against the alternative θn.
(i) Suppose X is a random variable taking values in a sample space S with probability law P. Let ω0 and ω1 be disjoint families of probability laws.Assume that, for every Q ∈ ω1 and any > 0, there exists a subset A of S (which may depend on ) such that Q(A) ≥ 1 − and such that, if X has
Show that P1 − P01 can also be computed as 2 sup B|P1(B) − P0(B)| , where the supremum is over all measurable sets B. In addition, it may be computed as sup{φ:|φ|≤1}φ(x)d P1(x) −φ(x)d P0(x) , where the supremum is over all measurable functions φ such that supx |φ(x)| ≤ 1.
(i). Let Pi have density pi with respect to a dominating measure μ.Show that P1 − P01 defined by |p1 − p0|dμ is independent of the choice of μand is a metric.(ii). Show the Hellinger distance defined in (15.12) is also independent of μ and is a metri
Let(X j,1, X j,2), j = 1,..., n be independent pairs of independent exponentially distributed random variables with E(X j,1) = θλj and E(X j,2) = λj .Here, θ and the λj are all unknown. The problem is to test θ = 1 against θ > 1.Compare the Rao,Wald, and likelihood ratio tests for this
Suppose X1,..., Xn are i.i.d. according to a quadratic mean differentiable model {Pθ, θ ∈ }, where is an open subset of the real line. Suppose an estimator sequence θˆn is asymptotically linear in the sense that, n1/2(θˆn − θ0) = n−1/2n i=1ψθ0 (Xi) + oPnθ0(1)where Eθ0 [ψθ0 (Xi)]
Suppose X1,..., Xn are i.i.d. random vectors in Rk having the multivariate normal distribution with unknown mean vector μ and identity covariance matrix. Fix a constant c > 0 and consider the shrinkage estimator of μ defined byμˆ n =1 − c nX¯ n2X¯ n , where X¯ n is the sample mean
Suppose X1,..., Xn are i.i.d. N(θ, 1). Consider Hodges’ superefficient estimator of θ (unpublished, but cited in Le Cam (1953)), defined as follows.Let ˆθn be 0 if |X¯ n| ≤ n−1/4; otherwise, let ˆθn = X¯ n. For any fixed θ, determine the limiting distribution of n1/2(θˆn − θ).
Consider the third of the three sampling schemes for a 2 × 2 × K table discussed in Section 4.8, and the two hypotheses H1 : 1 =···= K = 1 and H2 : 1 =···= K .(i) Obtain the likelihood ratio test statistic for testing H1.(ii) Obtain equations that determine the maximum likelihood
Consider testing moment inequalities under the setting of Example 14.4.8. Rather than the likelihood ratio procedure discussed there, consider the following moment selection procedure. Let J = {j : √nX¯ n,j > − log(n)}. Then, reject if the likelihood ratio statistic 2 log(Rn) > c|J |,1−α,
In Example 14.4.8, show (14.91).
In the situation of Problem 14.65, consider the hypothesis of marginal homogeneity H : pi+ = p+i for all i, where pi+ = a j=1 piij , p+i = a j=1 pjii .(i) The maximum-likelihood estimates of the piij under H are given by ˆˆpi j =Yi j /(1 + λi − λj), where the λ’s are the solutions of the
The hypothesis of symmetry in a square two-way contingency table arises when one of the responses A1,..., Aa is observed for each of n subjects on two occasions (e.g., before and after some intervention). If Yi,j is the number of subjects whose responses on the two occasions are (Ai, Aj), the joint
Consider the following model which therefore generalizes model(iii) of Section 4.7. A sample of ni subjects is obtained from class Ai(i = 1,..., a), the samples from different classes being independent. If Yi,j is the number of subjects from the ith sample belonging to Bj(j = 1,..., b), the joint
The problem is to test independence in a contingency table. Specifically, suppose X1,..., Xn are i.i.d., where each Xi is cross-classified, so that Xi = (r,s) with probability pr,s, r = 1,..., R, s = 1,..., S. Under the full model, the pr,s vary freely, except they are nonnegative and sum to 1. Let
Prove (iii) of Theorem 14.4.2. Hint: If θ0 satisfies the null hypothesis g(θ0) = 0, then testing 0 behaves asymptotically like testing the null hypothesis D(θ0)(θ − θ0) = 0, which is a hypothesis of the form considered in part (ii) of the theorem.
Provide the details of the proof to part (ii) of Theorem 14.4.2.
Prove (14.87).
In Example 14.4.6, show that 2 log(Rn) − Qn P→ 0 under the null hypothesis.
In Example 14.4.6, show that Rao’s Score test is exactly Pearson’s Chi-squared test.
(i) In Example 14.4.6, check that the MLE is given by pˆj = Yj /n.(ii) Show (14.83).
Suppose (X1, Y1), . . . , (Xn, Yn) are i.i.d., with Xi also independent of Yi . Further suppose Xi is normal with mean μ1 and variance 1, and Yi is normal with mean μ2 and variance 1. It is known that μi ≥ 0 for i = 1, 2. The problem is to test the null hypothesis that at most one μi is
Suppose X1,..., Xn are i.i.d. with the gamma (g,b) density f (x) = 1(g)bg x g−1 e−x/b x > 0 , with both parameters unknown (and positive). Consider testing the null hypothesis that g = 1, i.e., under the null hypothesis the underlying density is exponential.Determine the likelihood ratio test
Suppose X1,..., Xn are i.i.d. N(μ, σ2) with both parameters unknown. Consider testing the simple null hypothesis (μ, σ2) = (0, 1). Find and compare the Wald test, Rao’s Score test, and the likelihood ratio test.
In Example 14.4.5, determine the distribution of the likelihood ratio statistic against an alternative, both for the simple and composite null hypotheses.
In Example 14.4.5, consider the case of a composite null hypothesis with 0 given by (14.80). Show that the null distribution of the likelihood ratio statistic given by (14.81) is χ2 p. Hint: First consider the case a = 0, so that 0 is a linear subspace of dimension k − p. Let Z = −1/2X, so 2
Prove (14.77). Then, show that[(p)(θ)]−1 ≤ [I(p)(θ)] .What is the statistical interpretation of this inequality?
Suppose X1,..., Xn are i.i.d. Pθ, with θ ∈ , an open subset of IRk .Assume the family is q.m.d. at θ0 and consider testing the simple null hypothesisθ = θ0. Suppose ˆθn is an estimator sequence satisfying (14.62), and consider the Wald test statistic n(θˆn − θ0) I(θ0)(θˆn − θ0).
Suppose a time series X0, X1, X2,... evolves in the following way.The process starts at 0, so X0 = 0. For any i ≥ 1, conditional on X0,..., Xi−1, Xi =ρXi−1 + i , where the i are i.i.d. standard normal. You observe X0, X1, X2,..., Xn.For testing the null hypothesis ρ = 0 versus ρ > 0,
In Example 14.4.7, verify (14.89) and (14.90).
Suppose X1,..., Xn are i.i.d. N(μ, σ2) with both parameters unknown. Consider testing μ = 0 versus μ = 0. Find the likelihood ratio test statistic, and determine its limiting distribution under the null hypothesis. Calculate the limiting power of the test against the sequence of alternatives
Verify that h˜n in (14.61) maximizes L˜ n,h.
Suppose X1,..., Xn are i.i.d., uniformly distributed on [0, θ]. Find the maximum likelihood estimator ˆθn of θ. Determine a sequence τn such that τn(ˆθn −θ) has a limiting distribution, and determine the limit law.
Suppose X1,..., Xn are i.i.d. with common density function pθ(x) = θcθx θ+1 , 0 < c < x, θ > 0 .Here, c is fixed and known and θ is unknown.(i) Show that the maximum likelihood estimator ˆθn is well-defined and determine the limiting distribution of √n(θˆn − θ) under θ.(ii) What is
Let X1, ··· , Xn be i.i.d. with density f (x, θ) = [1 + θ cos(x)]/2π, where the parameter θ satisfies |θ| < 1 and x ranges between 0 and 2π. (The observations Xi may be interpreted as directional data. The case θ = 0 corresponds to the uniform distribution on the circle.) Construct an
Let X1,..., Xn be i.i.d. N(θ, θ2). Compare the asymptotic distribution of X¯ 2 n with that of an efficient likelihood estimator sequence.
Let X1,..., Xn be a sample from a Cauchy location model with density f (x − θ), where f (z) = 1π(1 + z2).Compare the limiting distribution of the sample median with that of an efficient likelihood estimator.
Let(Xi, Yi), i = 1 ... n be i.i.d. such that Xi and Yi are independent and normally distributed, Xi has variance σ2, Yi has variance τ 2 and both have common mean μ.(i) If σ and τ are known, determine an efficient likelihood estimator (ELE) μˆ of μand find the limit distribution of
Prove Corollary 14.4.1. Hint: Simply define θˆn = θ0 +n−1/2 I −1(θ0)Zn and apply Theorem 14.4.1.
Generalize Example 14.4.2 to multiparameter exponential families.
Suppose X1,..., Xn are i.i.d. Pθ according to the lognormal model of Example 14.2.7. Write down the likelihood function and show that it is unbounded.
In Example 14.4.1, show that the likelihood equations have a unique solution which corresponds to a global maximum of the likelihood function.
Assume Xi are independent, normally distributed with E(Xi) = μi .Let Pn be the distribution of (X1,..., Xn) when μi = 0 for all i. Let Qn be the distribution of (X1,..., Xn) when the μi are arbitrary constants. Find a necessary and sufficient condition on μ1, μ2,... so that Pn and Qn are
Suppose X1,..., Xn are i.i.d. according to a model {Pθ : θ ∈ }, where is an open subset of Rk. Assume that the model is q.m.d. Show that there cannot exist an estimator sequence Tn satisfying lim n→∞ sup|θ−θ0|≤n−1/2 Pnθ (n1/2|Tn − θ| > ) = 0 (14.97)for every > 0 and any θ0.
Generalize Corollary 14.3.2 in the following way. Suppose Tn =(Tn,1,..., Tn,k ) ∈ IRk . Assume that, under Pn,(Tn,1,..., Tn,k , log(Ln)) d→ (T1,..., Tk , Z) , where (T1,..., Tk , Z)is multivariate normal with Cov(Ti, Z) = ci . Then, under Qn,(Tn,1,..., Tn,k ) d→ (T1 + c1,..., Tk + ck ) .
Suppose Pθ is the uniform distribution on (0, θ). Fix h and determine whether or not Pn 1 and Pn 1+h/n are mutually contiguous. Consider both h > 0 and h < 0.
Suppose X1,..., Xn are i.i.d. according to a model which is q.m.d.at θ0. For testing θ = θ0 versus θ = θ0 + hn−1/2, consider the test ψn that rejects H if log(Ln,h) exceeds z1−ασh − 1 2σ2 h, where Ln,h is defined by (14.54) and σ2 h =h, I(θ0)h. Find the limiting value of
Reconsider Example 14.3.11. Rather than finding the limiting distribution of Wn under contiguous alternatives, find the limiting distribution of Un(properly normalized) under the same set of alternatives, where Un is the U-statistic introduced in Example 12.3.6. First, find the projection of Un
Verify (14.53) and evaluate it in the case where f (x) = exp(−|x|)/2 is the double exponential density.
Show that σ1,2 in (14.52) reduces to h/√π.
Prove the convergence (14.40).
Suppose Q is absolutely continuous with respect to P. If P{En} →0, then Q{En} → 0.
Suppose Xn has distribution Pn or Qn and Tn = Tn(Xn)is sufficient.Let PT n and QT n denote the distribution of Tn under Pn and Qn, respectively. Prove or disprove: Qn is contiguous to Pn if and only if QT n is contiguous to PT n .
Suppose, under Pn, Xn = Yn + oPn (1); that is, Xn − Yn → 0 in Pnprobability. Suppose Qn is contiguous to Pn. Show that Xn = Yn + oQn (1).

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Testing Statistical Hypotheses Volume I 4th Edition E.L. Lehmann, Joseph P. Romano - Solutions

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