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statistical techniques in business

Questions and Answers of Statistical Techniques in Business

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,
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
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
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 =
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
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
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]
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
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
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
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
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)
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 =
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,
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
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
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
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
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 ,
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
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 −
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
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
(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,
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)
(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
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
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
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
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;
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
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 :
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
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
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
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 =
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
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
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.,
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
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
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
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.
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
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,
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
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
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
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,
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
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
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
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
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
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
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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