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statistical techniques in business

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

(i) Generalize the randomization models of Section 14 for paired comparisons (n1 =···= nc = 2) and the case of two groups (c = 1) to an arbitrary number c of groups of sizes n1,..., nc.(ii) Generalize the confidence intervals (5.71) and (5.72) to the randomization model of part (i).
If c = 4, mi = ni = 1, and the pairs (xi, yi) are (1.56,2.01),(1.87,2.22), (2.17,2.73), and (2.31,2.60), determine the points δ(1),..., δ(15) which define the intervals (5.71).
If c = 1, m = n = 3, and if the ordered x’s and y’s are respectively 1.97, 2.19, 2.61 and 3.02, 3.28, 3.41, determine the points δ(1),..., δ(19) defined as the ordered values of (5.72).
Generalization of Corollary 5.11.1. Let H be the class of densities(5.80) with σ > 0 and −∞ < ζi < ∞ (i = 1,..., N). A complete family of tests of H at level of significance α is the class of permutation tests satisfying 1r z∈S(z)φ(z) = α a.e. (5.82)Section 5.12
To generalize Theorem5.11.1 to other designs, let Z = (Z1,..., ZN )and let G = {g1,..., gr} be a group of permutations of N coordinates or more generally a group of orthogonal transformations of N-space. If Pσ,ζ (z) = 1 rr k=1 1(√2πσ)N exp− 1 2σ2 |z − gk ζ|2, (5.80)where |z|2 = z2 i ,
Consider the problem of testing H : η = ξ in the family of densities(5.61) when it is given that σ > c > 0 and that the point (ζ11,..., ζcNc ) of (5.62)lies in a bounded region R containing a rectangle, where c and R are known. Then Theorem 5.11.1 is no longer applicable. However, unbiasedness
Suppose that a critical function φ0 satisfies (5.64) but not (5.66), and let α < 1 2 . Then the following construction provides a measurable critical functionφ satisfying (5.66) and such that φ0(z) ≤ φ(z) for all z. Inductively, sequences of functions φ1, φ2, … and ψ0, ψ1, … are
Continuation. An alternative comparison of the two designs is obtained by considering the expected length of the most accurate unbiased confidence intervals for = η − ξ in each case. Carry this out for varying n and confidence coefficient 1 − α = .95 when σ1 = 1, σ = 2 and when σ1 = 2, σ
Comparison of two designs. Under the assumptions made at the beginning of Section 5.10, one has the following comparison of the methods of complete randomization and matched pairs. The unit effects and experimental effects Ui and Vi are independently normally distributed with variances σ2 1 , σ2
(i) If X1,..., Xn; Y1,..., Yn are independent normal variables with common variance σ2 and means E(Xi) = ξi , E(Yi) = ξi + , the UMP unbiased test of = 0 against > 0 is given by (5.58).(ii) Determine the most accurate unbiased confidence intervals for .[(i): The structure of the problem becomes
In the matched-pairs experiment for testing the effect of a treatment, suppose that only the differences Zi = Yi − Xi are observable. The Z’s are assumed to be a sample from an unknown continuous distribution, which under the hypothesis of no treatment effect is symmetric with respect to the
Confidence intervals for a shift. [Maritz (1979)](i) Let X1,..., Xm; Y1,..., Yn be independently distributed according to continuous distributions F(x) and G(y) = F(y − ) respectively. Without any further assumptions concerning F, confidence intervals for can be obtained from permutation tests of
If c = 1, m = n = 4, α = .1 and the ordered coordinates z(1),...,z(N) of a point z are 1.97, 2.19, 2.61, 2.79, 2.88, 3.02, 3.28, 3.41, determine the points of S(z) belonging to the rejection region (5.53).
Prove Theorem 5.8.1 for arbitrary values of c.Section 5.9
Let T1,..., Ts−1 have the multinomial distribution (2.34), and suppose that (p1,..., ps−1) has the Dirichlet prior density D(a1,..., as) with density proportional to pa1−1 1 ... pas−1 s , where ps = 1 − (p1 +···+ ps−1). Determine the posterior distribution of (p1,..., ps−1) given
Let X1,..., Xm and Y1,..., Yn be independently distributed as N(ξ, σ2) and N(η, τ 2), respectively and let (ξ, η, σ, τ ) have the joint improper prior density π(ξ, η, σ, τ ) dξ dη dσ dτ = dξ dη(1/σ) dσ(1/τ ) dτ . Extend the result of Example 5.7.4 to inferences concerning τ
Let X1,..., Xm and Y1,..., Yn be independently distributed as N(ξ, σ2) and N(η, σ2), respectively, and let (ξ, η, σ) have the joint improper prior density given byπ(ξ, η, σ) dξ dη dσ = dξ dη ·1σ dσ for all − ∞ < ξ, η < ∞, 0 < σ.Under these assumptions, extend the results
Let θ = (θ1,..., θs) with θi real-valued, X have density pθ(x), and a prior density π(θ). Then the 100γ% HPD region is the 100γ% credible region R that has minimum volume.[Apply the Neyman–Pearson fundamental lemma to the problem of minimizing the volume of R.]
If X is normal N(θ, 1) and θ has a Cauchy density b/{π[b2 + (θ −μ)2]}, determine the possible shapes of the HPD regions for varying μ and b.
In Example 5.7.4, show that(i) the posterior density π(σ | x) is of type (c) of Example 5.7.2;(ii) for sufficiently large r, the posterior density of σr given x is no longer of type (c).
In Example 5.7.3, verify the marginal posterior distribution of ξ given x.
Verify the posterior distribution of p given x in Example 5.7.2.
If X1,..., Xn, are independent N(θ, 1) and θ has the improper priorπ(θ) ≡ 1, determine the posterior distribution of θ given the X’s.
Verify the posterior distribution of given x in Example 5.7.1.
(i) Under the assumptions made at the beginning of Section 5.6, the UMP unbiased test of H : ρ = ρ0 is given by (5.44).(ii) Let (ρ, ρ¯) be the associated most accurate unbiased confidence intervals forρ = aγ + bδ, where ρ = ρ(a, b), ρ¯ = ¯ρ(a, b). Then if f1 and f2 are increasing
In Example 5.5.1, consider a confidence interval for σ2 of the form I = [d−1 n S2 n , c−1 n S2 n ], where S2 n =i(Xi − X¯)2 and cn < dn are constants. Subject to the level constraint, choose cn and dn to minimize the length of I. Argue that the solution has shorter length that the uniformly
Scale parameter of a gamma distribution. Under the assumptions of the preceding problem, there exists(i) A UMP unbiased test of H : b ≤ b0 against b > b0 which rejects when Xi >C(/, Xi).(ii) Most accurate unbiased confidence intervals for b.[The conditional distribution of Xi given / Xi , which
Shape parameter of a gamma distribution. Let X1,..., Xn be a sample from the gamma distribution (g,b) defined in Problem 3.36.(i) There exist UMP unbiased tests of H : g ≤ g0 against g > g0 and of H : g = g0 against g = g0, and their rejection regions are based on W = /(Xi/X¯).(ii) There exist
Most accurate unbiased confidence intervals exist in the following situations:(i) If X, Y are independent with binomial distributions b(p1, m) and b(p2, m), for the parameter p1q2/p2q1.(ii) In a 2 × 2 table, for the parameter of Section 4.6.
Let X1,..., Xn be distributed as in Problem 5.15. Then the most accurate unbiased confidence intervals for the scale parameter a are 2C2[xi − min(x1,..., xn)] ≤ a ≤2 C1[xi − min(x1,..., xn)].
Use the preceding problem to show that uniformly most accurate confidence sets also uniformly minimize the expected Lebesgue measure (length in the case of intervals) of the confidence sets.15 Section 5.5
Let S(x) be a family of confidence sets for a real-valued parameterθ, and let μ[S(x)] denote its Lebesgue measure. Then for every fixed distribution Q of X (and hence in particular for Q = Pθ0 where θ0 is the true value of θ)EQ{μ[S(X)]} = θ=θ0 Q{θ ∈ S(X)} dθprovided the necessary
Two-stage t -tests with power independent of σ.(i) For the procedure /1 with any givenc, let C be defined by ∞C tn0−1(y) dy = α.Then the rejection region (n i=1 ai Xi − ξ0)/√c > C defines a level-α test of H : ξ ≤ ξ0 with strictly increasing power function βc(ξ) depending only on
Confidence intervals of fixed length for a normal mean.(i) In the two-stage procedure /1, defined in part (iii) of the preceding problem, let the number c be determined for any given L > 0 and 0 < γ < 1 by L/2 √c −L/2 √c tn0−1(y) dy = γ, where tn0−1 denotes the density of the
Stein’s two-stage procedure.(i) If m S2/σ2 has a χ2-distribution with m degrees of freedom, and if the conditional distribution of Y given S = s is N(0, σ2/S2), then Y has Student’s t-distribution with m degrees of freedom.(ii) Let X1, X2,... be independently distributed as N(ξ, σ2). Let
On the basis of a sample X = (X1,..., Xn) of fixed size from N(ξ, σ2) there do not exist confidence intervals for ξ with positive confidence coefficient and of bounded length.14[Consider any family of confidence intervals δ(X) ± L/2 of constant length L. Letξ1,... ξ2n be such that |ξi −
Suppose X and Y are independent, normally distributed with variance 1, and means ξ and η, respectively. Consider testing the simple null hypothesisξ = η = 0 against the composite alternative hypothesis ξ > 0, η > 0. Show that a UMPU test does not exist.Section 5.4
Let X1,..., Xm and Y1,..., Yn be independent samples from I(μ, σ)and I(ν, τ ), respectively.(i) There exist UMP unbiased tests of τ2/τ1 against one- and two-sided alternatives.(ii) If τ = σ, there exist UMP unbiased tests of ν/μ against one- and two-sided alternatives.[Chhikara (1975).]
Inverse Gaussian distribution.13 Let X1,..., Xn be a sample from the inverse Gaussian distribution I(μ, τ ), both parameters unknown.(i) There exists a UMP unbiased test of μ ≤ μ0 against μ > μ0, which rejects when X¯ > C[(Xi + 1/Xi)], and a corresponding UMP unbiased test of μ = μ0
Gamma two-sample problem. Let X1,... Xm; Y1,..., Yn be independent samples from gamma distributions (g1, b1), (g2, b2), respectively.(i) If g1, g2 are known, there exists a UMP unbiased test of H : b2 = b1 against one- and two-sided alternatives, which can be based on a beta distribution.[Some
Extend the results of the preceding problem to the case, considered in Problem 3.30, that observation is continued only until X(1),..., X(r) have been observed.
Let X1,..., Xn be a sample from the Pareto distribution P(c, τ ), both parameters unknown. Obtain UMP unbiased tests for the parameters c and τ .[Problems 5.15 and 3.8.]
Exponential densities. Let X1,..., Xn be a sample from a distribution with exponential density a−1e−(x−b)/a for x ≥ b.(i) For testing a = 1 there exists a UMP unbiased test given by the acceptance region C1 ≤ 2[xi − min(x1,..., xn)] ≤ C2, where the test statistic has a
Let X1,..., Xm and Y1,..., Yn be samples from N(ξ, σ2) and N(η, σ2). The UMP unbiased test for testing η − ξ = 0 can be obtained through Problems 5.5 and 5.6 by making an orthogonal transformation from (X1,... Xm, Y1,... Yn) to (Z1,..., Zm+n) such that Z1 = (Y¯ − X¯)/√1/m + (1/n), Z2
If m = n, the acceptance region (5.23) can be written as max S2 Y0 S2 X, 0 S2 XS2 Y≤1 − C C , where S2 X = (Xi − X¯)2, S2 Y = (Yi − Y¯)2 and where C is determined by C 0Bn−1,n−1(w) dw = α2.
Let X1,..., Xn and Y1,..., Yn be independent samples from N(ξ, σ2) and N(η, τ 2), respectively. Determine the sample size necessary to obtain power ≥ β against the alternatives τ/σ > when α = .05, β = .9, = 1.5, 2, 3, and the hypothesis being tested is H : τ/σ ≤ 1.
Let Xi = ξ + Ui , and suppose that the joint density f of the U’s is spherically symmetric, that is, a function of U2 i only, f (u1,..., un) = q(u2 i ) .Show that the null distribution of the one-sample t-statistic is independent of q and hence is the same as in the normal case, namely
As in Example 3.9.2, suppose X is multivariate normal with unknown mean ξ = (ξ1,..., ξk ) and known positive definite covariance matrix . Assume a = (a1,..., ak ) is a fixed vector. The problem is to test H :k i=1 ai ξi = δ vs. K :k i=1 ak ξi = δ .Find a UMPU level α test. Hint: First
Let N have the binomial distribution based on 10 trials with success probability p. Given N = n, let X1, ··· , Xn be i.i.d. normal with mean θ and variance one. The data consists of (N, X1, ··· , X N ).(i). If p has a known value p0, show there does not exist a UMP test of θ = 0 versusθ >
Let X1, X2,... be a sequence of independent variables distributed as N(ξ, σ2), and let Yn = [nXn+1 − (X1 +···+ Xn)]/√n(n + 1) . Then the variables Y1, Y2,... are independently distributed as N(0, σ2).
If X1,..., Xn is a sample from N(ξ, σ2), the UMP unbiased tests ofξ ≤ 0 and ξ = 0 can be obtained from Problems 5.5 and 5.6 by making an orthogonal transformation to variables Z1,..., Zn such that Z1 = √nX¯ .[Then ni=2 Z2 i = n i=1 Z2 i − Z2 1 = n i=1 X2 i − nX¯ 2 = n i=1(Xi −
Let X1,..., Xn be independently normally distributed with common variance σ2 and means ζ1,..., ζn, and let Zi = n j=1 ai j X j be an orthogonal transformation (that is, n i=1 ai jaik = 1 or 0 as j = k or j = k). The Z’s are normally distributed with common variance σ2 and means ζi = ai j ξ
Let Z1,..., Zn be independently normally distributed with common variance σ2 and means E(Zi) = ζi(i = 1,...,s), E(Zi) = 0 (i = s + 1,..., n).There exist UMP unbiased tests for testing ζ1 ≤ ζ0 1 and ζ1 = ζ0 1 given by the rejection regions Z1 − ζ0 1 n i=s+1 Z2 i /(n − s)> C0 and |Z1
Let X1,..., Xn be a sample from N(ξ, σ2). Denote the power of the one-sided t-test of H : ξ ≤ 0 against the alternative ξ/σ by β(ξ/σ), and by β∗(ξ/σ)the power of the test appropriate when σ is known. Determine β(ξ/σ) for n = 5, 10, 15, α = .05, ξ/σ = .07, 0.8, 0.9, 1.0, 1.1,
(i) Let Z and V be independently distributed as N(δ, 1) and χ2 with f degrees of freedom respectively. Then the ratio Z ÷ √V/ f has the noncentral t-distribution with f degrees of freedom and noncentrality parameter δ, the probability density of which is 12 pδ(t) = 1 21 2 ( f −1)( 1 2 f
In the situation of the previous problem there exists no test for testing H : ξ = 0 at level α, which for all σ has power ≥ β > α against the alternatives(ξ, σ) with ξ = ξ1 > 0.[Let β(ξ1, σ) be the power of any level α test of H, and let β(σ) denote the power of the most powerful
Let X1,..., Xn be a sample from N(ξ, σ2). The power of Student’s t-test is an increasing function of ξ/σ in the one-sided case H : ξ ≤ 0, K : ξ > 0, and of |ξ|/σ in the two-sided case H : ξ = 0, K : ξ = 0.[If S = 1 n − 1(Xi − X¯)2, the power in the two-sided case is given by 1
soon becomes more powerful than Fisher’s test under (a). For detailed numerical comparisons see Wacholder and Weinberg (1982)and the references given there.
In the 2 × 2 table for matched pairs, in the notation of Section 4.9, the correlation between the responses of the two members of a pair isρ = p11 − π1π2√π1(1 − π1)π2(1 − π2).For any given values of π1 < π2, the power of the one-sided McNemar test of H :π1 = π2 is an increasing
Consider the comparison of two success probabilities in (a) the two binomial situation of Section 4.5 with m = n, and (b) the matched-pairs situation of Section 4.9. Suppose the matching is completely at random, that is, a random sample of 2n subjects, obtained from a population of size N(2n ≤
In the 2 × 2 table for matched pairs, show by formal computation that the conditional distribution of Y given X + Y = d and X = x is binomial with the indicated p.
Let Xijkl (i, j, k = 0, 1, l = 1,..., L) denote the entries in a 2 ×2 × 2 × L table with factors A, B, C, and D, and let l = PABcC Dl PA˜ BCDl PAB˜C D˜ l PA˜ BC D˜ l PABCDl PA˜ B˜C Dl PABC D˜ l PA˜ B˜C˜ Dl.Then(i) under the assumptionl = there exists a UMP unbiased test of the
The UMP unbiased test of H : = 1 derived in Section 4.8 for the case that the B- and C-margins are fixed (where the conditioning now extends to all random margins) is also UMP unbiased when(i) only one of the margins is fixed;(ii) the entries in the 4K cells are independent Poisson variables with
In a 2 × 2 × K table with k = , the test derived in the text as UMP unbiased for the case that the B and C margins are fixed has the same property when any two, one, or no margins are fixed.
In a 2 × 2 × 2 table with m1 = 3, n1 = 4; m2 = 4, n2 = 4; and t1 =3, t1 = 4, t2 = t2 = 4, determine the probabilities that P(Y1 + Y2 ≤ K|Xi + Yi =ti,i = 1, 2) for k = 0, 1, 2, 3.
Rank-sum test. Let Y1,..., YN be independently distributed according to the binomial distributions b(pi, ni),i = 1,..., N where pi = 1 1 + e−(α+βxi) .This is the model frequently assumed in bioassay, where xi denotes the dose, or some function of the dose such as its logarithm, of a drug given
(i) Based on the conditional distribution of X2,..., Xn given X1 = x1 in the model of Problem 4.29, there exists a UMP unbiased test of H : p0 = p1 against p0 > p1 for every α.(ii) For the same testing problem, without conditioning on X1 there exists a UMP unbiased test if the initial probability
Continuation. For testing the hypothesis of independence of the X’s, H : p0 = p1, against the alternatives K : p0 < p1, consider the run test, which rejects H when the total number of runs R = U + V is less than a constant C(m)depending on the number m of zeros in the sequence. When R = C(m), the
Runs. Consider a sequence of N dependent trials, and let Xi be 1 or 0 as the ith trial is a success or failure. Suppose that the sequence has the Markov property15 P{Xi = 1|xi,..., xi−1} = P{Xi = 1|xi−1}and the property of stationarity according to which P{Xi = 1} and P{Xi = 1|xi−1}are
Positive dependence. Two random variables (X, Y ) with c.d.f.F(x, y) are said to be positively quadrant dependent if F(x, y) ≥ F(x,∞)F(∞, y)for all x, y.14 For the case that (X, Y ) takes on the four pairs of values (0, 0), (0, 1),(1, 0), (1, 1) with probabilities p00, p01, p10, p11, (X, Y )
Sequential comparison of two binomials. Consider two sequences of binomial trials with probabilities of success p1 and p2, respectively, and let ρ =(p2/q2) ÷ (p1/q1).(i) If α < β, no test with fixed numbers of trials m and n for testing H : ρ = ρ0 can have power ≥ β against all
Let X and Y be independently distributed with Poisson distributions P(λ) and P(μ). Find the power of the UMP unbiased test of H : μ ≤ λ, against the alternatives λ = .1, μ = .2; λ = 1, μ = 2; λ = 10, μ = 20; λ = .1, μ = 0.4; at level of significance α = 0.1.[Since T = X + Y has the
Negative binomial. Let X, Y be independently distributed according to negative binomial distributions N b(p1, m) and N b(p2, n) respectively, and let qi = 1 − pi .(i) There exists a UMP unbiased test for testing H : θ = q2/q1 ≤ θ0 and hence in particular H : p1 ≤ p2.(ii) Determine the
The singly truncated normal (STN) distribution, indexed by parameters ν and λ has support the positive real line with density p(x; ν, λ) = C(ν, λ) exp(−νx − λx 2) , where C(ν, λ) is a normalizing constant. Based on an i.i.d. sample, show there exists a UMPU test of the null hypothesis
The UMP unbiased tests of the hypotheses H1,..., H4 of Theorem 4.4.1 are unique if attention is restricted to tests depending on U and the T ’s.
Continuation. The function φ4 defined by (4.16), (4.18), and (4.19)is jointly measurable in u and t.[The proof, which otherwise is essentially like that outlined in the preceding problem, requires the measurability in z and t of the integral g(z, t) =z−−∞udFt(u).This integral is absolutely
Measurability of tests of Theorem 4.4.1. The function φ3 defined by(4.16) and (4.17) is jointly measurable in u and t.[With C1 = v and C2 = w, the determining equations for v, w, γ1, γ2 are Ft(v−) + [1 − Ft(w)] + γ1[Ft(v) − Ft(v−)] (4.26)+γ2[Ft(w) − Ft(w−)] = αand Gt(v−) + [1
Suppose P{I = 1} = p = 1 − P{I = 2}. Given I = i, X ∼N(θ, σ2 i ), where σ2 1 < σ2 2 are known. If p = 1/2, show that, based on the data (X, I), there does not exist a UMP test of θ = 0 vs θ > 0. However, if p is also unknown, show a UMPU test exists. [See Examples 10.20-21 in Romano and
Random sample size. Let N be a random variable with a power series distribution P(N = n) = a(n)λn C(λ) , n = 0, 1,... (λ > 0, unknown).When N = n, a sample X1,..., Xn from the exponential family (3.19) is observed.On the basis of (N, X1,..., X N ) there exists a UMP unbiased test of H : Q(θ)
Let Xi(i = 1, 2) be independently distributed according to distributions from the exponential families (3.19) with C, Q, T , and h replaced by Ci , Qi , Ti , and hi . Then there exists a UMP unbiased test of(i) H : Q2(θ2) − Q1(θ1) ≤ c and hence in particular of Q2(θ2) ≤ Q1(θ1);(ii) H :
Let X1,..., Xn be a sample from the uniform distribution over the integers 1,..., θ and let a be a positive integer.1. The sufficient statistic X(n) is complete when the parameter space is = {θ :θ ≤ a}.2. Show that X(n) is not complete when = {θ : θ ≥ a}, a ≥ 2, and find a complete
Let X, Y be independent binomial b(p, m) and b(p2, n), respectively.Determine whether (X, Y ) is complete when(i) m = n = 1,(ii) m = 2, n = 1.
Determine whether T is complete for each of the following situations:(i) X1,..., Xn are independently distributed according to the uniform distribution over the integers 1, 2,..., θ and T = max(X1,..., Xn).(ii) X takes on the values 1,2,3,4 with probabilities pq, p2q, pq2, 1 − 2pq, respectively,
The completeness of the order statistics in Example 4.3.4 remains true if the family F is replaced by the family F1 of all continuous distributions.[Due to Fraser (1956). To show that for any integrable symmetric function φ,φ(x1,..., xn) d F(x1)...d F(xn) = 0 for all continuous F implies φ = 0
Counterexample. Let X be a random variable taking on the values−1, 0, 1, 2, … with probabilities Pθ{X = −1} = θ; Pθ{X = x} = (1 − θ)2θx , x = 0, 1,....Then P = {Pθ, 0 < θ < 1} is boundedly complete but not complete. [Girschick et al. (1946)]
Let X1,..., Xm and Y1,..., Yn be samples from N(ξ, σ2) and N(ξ, τ 2). Then T = ( Xi,Yj, X2 i ,Y 2 j ), which in Example 4.3.3 was seen not to be complete, is also not boundedly complete.[Let f (t) be 1 or −1 as y¯ − ¯x is positive or not.]
Let X1,..., Xn be a sample from (i) the normal distribution N(aσ, σ2), with a fixed and 0 < σ < ∞; (ii) the uniform distribution U(θ − 1 2 , θ +1 2 ), −∞ < θ < ∞; (iii) the uniform distribution U(θ1, θ2),∞ < θ1 < θ2 < ∞. For these three families of distributions the
For testing the hypothesis H : θ = θ0, (θ0 an interior point of ) in the one-parameter exponential family of Section 4.2, let C be the totality of tests satisfying (4.3) and (4.5) for some −∞ ≤ C1 ≤ C2 ≤ ∞ and 0 ≤ γ1, γ2 ≤ 1.(i) C is complete in the sense that given any
Let (X, Y ) be distributed according to the exponential family d Pθ1,θ2 (x, y) = C(θ1, θ2)eθ1 x+θ2 y dμ(x, y) .The only unbiased test for testing H : θ1 ≤a, θ2 ≤ b against K : θ1 > a or θ2 > b or both is φ(x, y) ≡ α.[Take a = b = 0, and let β(θ1, θ2) be the power function of
Let X and Y be independently distributed according to one-parameter exponential families, so that their joint distribution is given by d Pθ1,θ2 (x, y) = C(θ1)eθ1T (x) dμ(x)K(θ2)eθ2U(y) dν(y).Suppose that with probability 1 the statistics T and U each take on at least three values and that
Suppose X has density (with respect to some measure μ)pθ(x) = C(θ) exp[θT (x)]h(x) , for some real-valued θ. Assume the distribution of T (X) is continuous under θ (for any θ). Consider the problem of testing θ = θ0 versus θ = θ0. If the null hypothesis is rejected, then a decision is
Let Tn/θ have a χ2-distribution with n degrees of freedom. For testing H : θ = 1 at level of significance α = .05, find n so large that the power of the UMP unbiased test is ≥ .9 against both θ ≥ 2 and θ ≤ 1 2 . How large does n have to be if the test is not required to be unbiased?
Let X have the Poisson distribution P(τ ), and consider the hypothesis H : τ = τ0. Then Condition (4.6) reduces to C2−1 x=C1+1τ x−1 0(x − 1)!e−τ0 +2 i=1(1 − γi) τCi−1 0(Ci − 1)!e−τ0 = 1 − α, provided C1 > 1.
Let X have the binomial distribution b(p, n), and consider the hypothesis H : p = p0 at level of significanceα. Determine the boundary values of the UMP unbiased test for n = 10 with α = 0.1, p0 = 0.2 and with α = 0.05, p0 = 0.4, and in each case graph the power functions of both the unbiased
p-values. Consider a family of tests of H : θ = θ0 (or θ ≤ θ0), with level-α rejection regions Sα, such that (a) Pθ0 {X ∈ Sα} = α for all 0 < α < 1, and(b) Sα ⊆ Sα for α < α. If the tests Sα are unbiased, the distribution of pˆ under any alternative θ satisfies Pθ{ ˆp ≤
Admissibility. Any UMP unbiased test φ0 is admissible in the sense that there cannot exist another test φ1 which is at least as powerful as φ0 against all alternatives and more powerful against some.[If φ is unbiased and φ is uniformly at least as powerful as φ, then φ is also unbiased.]
In Example 3.9.3, provide the details for Cases 3 and 4.
In Example 3.9.2, Case 2, verify the claim for the least favorable distribution.
Suppose (X1,..., Xk ) has the multivariate normal distribution with unknown mean vector ξ = (ξ1,..., ξk ) and known covariance matrix . Suppose X1 is independent of (X2,..., Xk ). Show that X1 is partially sufficient for ξ1 in the sense of Problem 3.65. Provide an alternative argument for Case
Suppose X is a k × 1 random vector with E(|X|2) < ∞ and covariance matrix . Let A be an m × k (nonrandom) matrix and let Y = AX. Show Y has mean vector AE(X) and covariance matrix AA.

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