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statistical techniques in business

Questions and Answers of Statistical Techniques in Business

Solve the problem corresponding to Example 6.12.1 when(i) X1,..., Xn is a sample from the exponential density E(ξ, σ), and the parameter being estimated is σ;(ii) X1,..., Xn is a sample from the
Generalize the confidence sets of Example 6.11.3 to the case that the Xi are N(ξi, diσ2) where the d’s are known constants.
Let X1,..., Xm; Y1,..., Yn be independently normally distributed as N(ξ, σ2) and N(η, σ2) respectively. Determine the equivariant confidence sets forη − ξ that have smallest Lebesgue measure
In Example 6.12.4, show that(i) both sets (6.60) are intervals;(ii) the sets given by vp(v) > C coincide with the intervals (5.41).
The confidence sets (6.52) are uniformly most accurate equivariant under the group G defined at the end of Example 6.12.3.
Let X1,..., Xr be i.i.d. N(0, 1), and let S2 be independent of the X’s and distributed as χ2ν . Then the distribution of (X1/S√ν,..., Xr /S√ν) is a central multivariate t-distribution, and
Show that in Example 6.12.1,(i) the confidence sets σ2/S2 ∈ A∗∗ with A∗∗ given by (6.45) coincide with the uniformly most accurate unbiased confidence sets for σ2;(ii) if (a,b) is best
In Example 6.12.1, the density p(v) of V = 1/S2 is unimodal.
In Examples 6.12.1 and 6.12.2 there do not exist equivariant sets that uniformly minimize the probability of covering false values.
provides a simple check of the equivariance of confidence sets. In Example 6.12.2, for instance, the confidence sets (6.46) are based on the pivotal vector (X1 − ξ1,..., Xr − ξr), and hence are
Under the assumptions of Problem 6.72, suppose that a family of confidence sets S(x) is equivariant under G∗. Then there exists a set B in the range space of the pivotal V such that (6.75) holds.
Under the assumptions of the preceding problem, the confidence set S(x) is equivariant under G∗.
(i) If G˜ is transitive over X × w and V(X, θ) is maximal invariant under G˜ , then V(X, θ) is pivotal.(ii) By (i), any quantity W(X, θ) which is invariant under G˜ is pivotal; give an example
Let V(X, θ) be any pivotal quantity [i.e., have a fixed probability distribution independent of (θ, ϑ)], and let B be any set in the range space of V with probability P(V ∈ B) = 1 − α. Then
(i) Let (X1, Y1), . . . , (Xn, Yn) be a sample from a bivariate normal distribution, and letρ = C−1⎛⎝(Xi − X¯)(Yi − Y¯)(Xi − X¯)2 (Yi − Y¯)2⎞⎠ , where C(ρ) is determined such
(i) Let X1,..., Xn be independently distributed as N(ξ, σ2), and letθ = ξ/σ. The lower confidence bounds θ for θ, which at confidence level 1 − αare uniformly most accurate invariant under
Counterexample. The following example shows that the equivariance of S(x) assumed in the paragraph following Lemma 6.11.1 does not follow from the other assumptions of this lemma. In Example 6.5.1,
(i) One-sided equivariant confidence limits. Let θ be real-valued, and suppose that, for each θ0, the problem of testing θ ≤ θ0 against θ > θ0 (in the presence of nuisance parameters ϑ)
Let X1,..., Xn; Y1,..., Yn be samples from N(ξ, σ2) and N(η, τ 2), respectively. Then the confidence intervals (5.42) for τ 2/σ2, which can be written as(Yj − Y¯)2 k(Xi − X¯)2 ≤ τ 2σ2
In Example 6.11.1, a family of sets S(x, y) is a class of equivariant confidence sets if and only if there exists a set R of real numbers such that S(x, y) = +r∈R{(ξ, η) : (x − ξ)2 + (y −
The hypothesis of independence. Let (X1, Y1), . . . , (X N , YN ) be a sample from a bivariate distribution, and (X(1), Z1), . . . , (X(N), ZN ) be the same sample arranged according to increasing
In the preceding problem let Ui j = 1 if (j − i)(Z j − Zi) > 0, and= 0 otherwise.(i) The test statistic i Ti , can be expressed in terms of the U’s through the relationN i=1 i Ti = i< j(j −
with C the class of transformations z 1 = z1,z i = fi(zi)for i > 1, where z < f2(z) < ··· < fN (z) and each fi is nondecreasing. If F0 is the class of N-tuples (F1,..., FN ) with F1 =···= FN ,
The hypothesis of randomness.7 Let Z1,..., ZN be independently distributed with distributions F1,..., FN , and let Ti denote the rank of Zi among the Z’s. For testing the hypothesis of randomness
Unbiased tests of symmetry. Let Z1,..., ZN , be a sample, and let φbe any rank test of the hypothesis of symmetry with respect to the origin such that zi ≤ z i for all i implies φ(z1,...,zN ) ≤
An alternative expression for (6.71) is obtained if the distribution of Z is characterized by (ρ, F, G). If then G = h(F) and h is differentiable, the distribution of n and the Sj is given byρm(1
Let Z1,..., ZN be a sample from a distribution with density f (z −θ), where f (z) is positive for all z and f is symmetric about 0, and let m, n, and the Sj be defined as in the preceding
(i) Let m and n be the numbers of negative and positive observations among Z1,..., ZN , and let S1 < ··· < Sn denote the ranks of the positive Z’s among |Z1|,... |ZN |. Consider the N + 1 2 N(N
(i) Let X1,..., Xm; Y1,..., Yn be i.i.d. according to a continuous distribution F, let the ranks of the Y ’s be S1 < ··· < Sn, and let T = h(S1) +···+ h(Sn). Then if either m = n or h(s) +
Continuation.(i) There exists at every significance level α a test of H : G = F which has power> α against all continuous alternatives (F, G) with F = G.(ii) There does not exist a nonrandomized
(i) Let X, X and Y , Y ’ be independent samples of size 2 from continuous distributions F and G, respectively. Then p = P{max(X, X) < min(Y, Y)} + P{max(Y, Y) < min(X, X)}= 1 3 + 2, where = (F
calculated for the observations X1,..., Xm; Y1 − ,... , Yn − .[An alternative measure of the amount by which G exceeds F (without assuming a location model) is p = P{X < Y }. The literature on
and the probability on the right side is calculated for = 0.(ii) Determine the above confidence interval for when m = n = 6, the confidence coefficient is 20 21 , and the observations are x :
(i) If X1,..., Xm and Y1,..., Yn are samples from F(x) and G(y) = F(y − ), respectively, (F continuous), and D(1) < ··· < D(mn)denote the ordered differences Yj − Xi , then P $D(k)
Let X1,..., Xm; Y1,..., Yn be samples from a common continuous distribution F. Then the Wilcoxon statistic U defined in
Let F0 be a family of probability measures over (X , A), and let C be a class of transformations of the space X . Define a class F1 of distributions by F1 ∈ F1 if there exists F0 ∈ F0 and f ∈ C
An alternative proof of the optimum property of the Wilcoxon test for detecting a shift in the logistic distribution is obtained from the preceding problem by equating F(x − θ) with (1 − θ)F(x)
For sufficiently small θ > 0, the Wilcoxon test at levelα = kN n, k a positive integer, maximizes the power (among rank tests) against the alternatives (F, G) with G =(1 − θ)F + θF2.
(i) If X1,..., Xm and Y1,..., Yn are samples with continuous cumulative distribution functions F and G = h(F) respectively, and if h is differentiable, the distribution of the ranks S1
Distribution of order statistics.(i) If Z1,..., ZN is a sample from a cumulative distribution function F with density f , the joint density of Yi = Z(si), i = 1,..., n, is N! f (y1)... f (yn)(s1 −
Under the assumptions of the preceding problem, if Fi = hi(F), the distribution of the ranks T1,..., TN of Z1,..., ZN depends only on the hi , not on F. If the hi are differentiable, the distribution
Let Zi have a continuous cumulative distribution function Fi (i =1,..., N), and let G be the group of all transformations Z i = f (Zi) such that f is continuous and strictly increasing.(i) The
(i) For any continuous cumulative distribution function F, define F−1(0) = −∞, F−1(y) = inf{x : F(x) = y} for 0 < y < 1, F−1(1) = ∞ if F(x) < 1 for all finite x, and otherwise inf{x :
(i) Let Z1,..., ZN be independently distributed with densities f1,..., fN , and let the rank of Zi be denoted by Ti . If f is any probability density which is positive whenever at least one of the fi
Expectation and variance of Wilcoxon statistic. If the X’s and Y ’s are samples from continuous distributions F and G, respectively, the expectation and variance of the Wilcoxon statistic U
Wilcoxon two-sample test. Let Ui j = 1 or 0 as Xi < Yj or Xi > Yj , and let U = Ui j be the number of pairs Xi , Yj with Xi < Yj .(i) Then U = Si − 1 2 n(n + 1), where S1 < ··· < Sn are the
Suppose X = (X1,..., Xk ) is multivariate normal with unknown mean vector (θ1,..., θk ) and known nonsingular covariance matrix . Consider testing the null hypothesis θi = 0 for all i against
For the model of the preceding problem, generalize Example 6.7.3(continued) to show that the two-sided t-test is a Bayes solution for an appropriate prior distribution.
Let X1,..., Xm; Y1,..., Yn be independent N(ξ, σ2) and N(η, σ2)respectively. The one-sided t-test of H : δ = ξ/σ ≤ 0 is admissible against the alternatives (i) 0 < δ < δ1 for any δ1 > 0;
Verify(i) the admissibility of the rejection region (6.27);(ii) the expression for I(z) given in the proof of Lemma 6.7.1.
(i) In Example 6.7.4 show that there exist C0, C1 such that λ0(η)and λ1(η) are probability densities (with respect to Lebesgue measure).(ii) Verify the densities h0 and h1.
(i) The acceptance region T1/√T2 ≤ C of Example 6.7.3 is a convex set in the (T1, T2) plane.(ii) In Example 6.7.3, the conditions of Theorem 6.7.1 are not satisfied for the sets A : T1/√T2 ≤
(i) The following example shows that α-admissibility does not always imply d-admissibility. Let X be distributed as U(0, θ), and consider the tests ϕ1 and ϕ2 which reject when, respectively, X <
The definition of d-admissibility of a test coincides with the admissibility definition given in Section 1.8 when applied to a two-decision procedure with loss 0 or 1 as the decision taken is correct
The following UMP unbiased tests of Chapter 5 are also UMP invariant under change in scale:(i) The test of g ≤ g0 in a gamma distribution (Problem 5.30).(ii) The test of b1 ≤ b2 in Problem
is also UMP similar.[Consider the problem of testing α = 0 versus α > 0 in the two-parameter exponential family with density C(α, τ ) exp − α2τ 2x 2 i − 1 − ατ|xi|, 0 ≤ α <
The UMP invariant test of
Let G be a group of transformations of X , and let A be a σ-field of subsets of X , and μ a measure over (X , A). Then a set A ∈ A is said to be almost invariant if its indicator function is
Inadmissible likelihood ratio test. In many applications in which a UMP invariant test exists, it coincides with the likelihood ratio test. That this is, however, not always the case is seen from the
Invariance of likelihood ratio. Let the family of distributions P ={Pθ, θ ∈ } be dominated by μ, let pθ = d Pθ/dμ, let μg−1 be the measure defined by μg−1(A) = μ[g−1(A)], and suppose
(i) A generalization of equation (6.2) isA f (x) d Pθ(x) =gA f (g−1x) d Pgθ¯ (x).(ii) If Pθ1 is absolutely continuous with respect to Pθ0 , then Pgθ¯ 1 is absolutely continuous with
Envelope power function. Let S(α) be the class of all level-α tests of a hypothesis H, and let β∗α(θ) be the envelope power function, defined byβ∗α(θ) = supφ∈S(α)βφ(θ), where βφ
Consider a testing problem which is invariant under a group G of transformations of the sample space, and let C be a class of tests which is closed under G, so that φ ∈ C implies φg ∈ C, where
Show that(i) G1 of Example 6.6.2 is a group;(ii) the test which rejects when X2 21/X2 11 > C is UMP invariant under G1;(iii) the smallest group containing G1 and G2 is the group G of Example 6.6.2.
The totality of permutations of K distinct numbers a1,..., aK , for varying a1,..., aK can be represented as a subset CK of Euclidean K-space RK , and the group G of Example 6.5.1 as the union of C2,
Almost invariance of a test φ with respect to the group G of either Problem 6.11(i) or Example 6.3.5 implies that φ is equivalent to an invariant test.
For testing the hypothesis that the correlation coefficient ρ of a bivariate normal distribution is ≤ ρ0, determine the power against the alternative ρ = ρ1, when the level of significance α
Let (Xi, Yi) be independent N(μi, σ2) for i = 1,..., n. The parameters μ1,..., μn and σ2 are all unknown. For testing σ = 1 against σ > 1, determine the UMPI level α test. Is the test also
Testing a correlation coefficient. Let (X1, Y1), . . . , (Xn, Yn) be a sample from a bivariate normal distribution.(i) For testing ρ ≤ ρ0 against ρ > ρ0 there exists a UMP invariant test with
Two-sided t-test.(i) Let X1,..., Xn be a sample from N(ξ, σ2). For testing ξ = 0 against ξ = 0, there exists a UMP invariant test with respect to the group X i = cXi , c = 0, given by the
(i) When testing H : p ≤ p0 against K : p > p0 by means of the test corresponding to (6.15), determine the sample size required to obtain powerβ against p = p1, α = 0.05, β = 0.9 for the cases
Let X1,..., Xn be independent and normally distributed. Suppose Xi has mean μi and variance σ2 (which is the same for all i). Consider testing the null hypothesis that μi = 0 for all i. Using
Show that the test of Problem 6.10(i) reduces to(i) [x(n) − x(1)]/S < c for normal versus uniform;(ii) [ ¯x − x(1)]/S < c for normal versus exponential;(iii) [ ¯x − x(1)]/[x(n) − x(1)] < c
Uniform versus triangular.(i) For f0(x) = 1 (0 < x < 1), f1(x) = 2x (0 < x < 1), the test of Problem 6.13 reduces to rejecting when T = x(n)/x¯ < C.(ii) Under f0, the statistic 2n log T is
Normal versus double exponential. For f0(x) = e−x2/2/√2π, f1(x) = e−|x|/2, the test of the preceding problem reduces to rejecting when x 2 i /|xi| < C.(Hogg, 1972.)Note. The corresponding test
Let X1,..., Xn be a sample from a distribution with density 1τ n f x1τ... f xnτ, where f (x) is either zero for x < 0 or symmetric about zero. The most powerful scale-invariant test for
If X1,..., Xn and Y1,..., Yn are samples from N(ξ, σ2) and N(η, τ 2), respectively, the problem of testing τ 2 = σ2 against the two-sided alternatives τ 2 = σ2 remains invariant under the
Let X1,..., Xm; Y1,..., Yn be samples from exponential distributions with densities for σ−1e−(x−ξ)/σ, for x ≥ ξ, and τ −1e−(y−η)/τ for y ≥ η.(i) For testing τ/σ ≤
(i) Let X = (X1,..., Xn) have probability density (1/θn) f [(x1 −ξ)/θ,...,(xn − ξ)/θ], where −∞ < ξ < ∞, 0 < θ are unknown, and where f is even. The problem of testing f = f0 against
Let X, Y have the joint probability density f (x, y). Then the integral h(z) = ∞−∞ f (y − z, y)dy is finite for almost all z, and is the probability density of Z = Y − X.[Since P{Z ≤ b}
In Example 6.1.1, find a maximal invariant and the UMPI level α test.
Consider the situation of Example 6.3.1 with n = 1, and suppose that f is strictly increasing on (0, 1).(i) The likelihood ratio test rejects if X < α/2 or X > 1 − α/2.(ii) The MP invariant test
Prove Theorem 6.3.1(i) by analogy with Example 6.3.1, and(ii) by the method of Example 6.3.2. [Hint: A maximal invariant under G is the set{g1x,..., gN x}.]
(i) A sufficient condition for (6.9) to hold is that D is a normal subgroup of G.(ii) If G is the group of transformations x = ax +b, a = 0, −∞ < b < ∞, then the subgroup of translations x = x
Suppose M is any m × p matrix. Show that MM is positive semidefinite. Also, show the rank of MM equals the rank of M, so that in particular MM is nonsingular if and only if m ≥ p and M is of
(i) Let X be the totality of points x = (x1,..., xn)for which all coordinates are different from zero, and let G be the group of transformations x i =cxi, c > 0. Then a maximal invariant under G is
Let G be a group of measurable transformations of (X , A) leaving P = {Pθ, θ ∈ } invariant, and let T (x) be a measurable transformation to(T , B). Suppose that T (x1) = T (x2) implies T (gx1) =
If X, Y are positively regression dependent, they are positively quadrant dependent.[Positive regression dependence implies that P[Y ≤ y | X ≤ x] ≥ P[Y ≤ y | X ≤ x] for all x < x and y,
(i) The functions (5.78) are bivariate cumulative distributions functions.(ii) A pair of random variables with distribution (5.78) is positively regression dependent. [The distributions (5.78) were
If X and Y have a bivariate normal distribution with correlation coefficient ρ > 0, they are positively regression dependent.[The conditional distribution of Y given x is normal with mean η +
If (X1, Y1), . . . , (Xn, Yn) is a sample from a bivariate normal distribution, the probability density of the sample correlation coefficient R is16 pρ(r) = 2n−3π(n − 3)!(1 − ρ2)1 2
(i) Let (X1, Y1), . . . , (Xn, Yn) be a sample from the bivariate normal distribution (5.73), and let S2 1 = (Xi − X¯)2, S12 = (Xi − X¯)(Yi − Y¯), S2 2 = (Yi − Y¯)2.Then (S2 1 , S12, S2 2
(i) Let (X1, Y1), . . . , (Xn, Yn) be a sample from the bivariate normal distribution (5.69), and let S2 1 = (Xi − X¯)2, S2 2 = (Yi − Y¯)2, S12 =(Xi − X¯)(Yi − Y¯). There exists a UMP
(i) If the joint distribution of X and Y is the bivariate normal distribution (5.69), then the conditional distribution of Y given x is the normal distribution with variance τ 2(1 − ρ2) and mean
Generalize Problems 5.60(i) and 5.61 to the case of two groups of sizes m and n (c = 1).
Determine for each of the following classes of subsets of {1,..., n}whether (together with the empty subset) it forms a group under the group operation of the preceding problem: All subsets
The preceding problem establishes a 1 : 1 correspondence between e − 1 permutations T of G0 which are not the identity and e − 1 nonempty subsets{i1,...,ir} of the set {1,..., n}. If the
to the situation of part (i).[Hartigan (1969).]
(i) Given n pairs (x1, y1), . . . , (xn, yn), let G be the group of 2n permutations of the 2n variables which interchange xi and yi in all, some, or none of the n pairs. Let G0 be any subgroup of G,
Let Z1,..., Zn be i.i.d. according to a continuous distribution symmetric about θ, and let T(1) < ··· < T(M) be the ordered set of M = 2n − 1 subsamples; (Zi1 +···+ Zir )/r, r ≥ 1. If T(0)

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