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statistical techniques in business

Testing Statistical Hypotheses 3rd Edition Erich L. Lehmann, Joseph P. Romano - Solutions

Uniform vs. triangular.(i) For f0(x) = 1 (0
Normal vs. double exponential. For f0(x) = e−x2/2/√2π, f1(x) = e−|x|/2, the test of the preceding problem reduces to rejecting when x2 i /|xi| < C.(Hogg, 1972.)Note. The corresponding test when both location and scale are unknown is obtained in Uthoff (1973). Testing normality against
Let X1,...,Xn be a sample from a distribution with density 1τ n fx1τ...f xnτ, where f(x) is either zero for x < 0 or symmetric about zero. The most powerful scale-invariant test for testing H : f = f0 against K : f = f1 rejects when∞0 vn−1f1(vx1) ...f1(vxn) dv∞0 vn−1f0(vx1)
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 group G generated by the transformations Xi = aXi +b, Y i = aYi +c, (a = 0), and Xi = Yi, Y i =
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−n)/τ for y ≥ η.(i) For testing τ /σ ≤ ∆ against τ /σ > ∆, there exists a UMP invariant test with respect to the group G : Xi = aXi +b, Y j =
(i) Let X = (X1,...,Xn) have probability density (1/θn)f[(x1−ξ)/θ, . . . , (xn − ξ)/θ], where −∞
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} = b−∞ h(z)dz, it is finite and hence h is finite almost everywhere.]
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 agrees with the likelihood ratio test when f is convex.(iii) When f is concave, the MP invariant
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}.
Prove statements (i)-(iii) of Example 6.3.1.
(i) A sufficient condition for (6.8) to hold is that D is a normal subgroup of G.(ii) If G is the group of transformations x = ax +b, a = 0, −∞
Suppose M is any m × p matrix. Show that MTM is positive semidefinite. Also, show the rank of MTM equals the rank of M, so that in particular MTM is nonsingular if and only if m ≥ p and M is of rank p.
(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 xi = cxi,c > 0. Then a maximal invariant under G is(sgn xn, x1/xn,...,xn−1/xn) where sgn x is 1 or −1 as x is positive or negative.(ii) Let X be the
Let G be a group of measurable transformations of (X , A) leaving P = {Pθ, θ ∈ Ω} invariant, and let T(x) be a measurable transformation to (T , B).Suppose that T(x1) = T(x2) implies T(gx1) = T(gx2) for all g ∈ G, so that G induces a group G∗ on T through g∗T(x) = T(gx), and suppose
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
(i) The functions (5.78) are bivariate cumulative distributions functions.(ii) A pair of random variables with distribution (5.78) is positively regressiondependent. [The distributions (5.78) were introduced by Morgenstem(1956).]
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 + ρτσ−1(x − ξ)and variance τ 2(l − ρ2). Through addition to such a variable of the positive
If (X1, Y1),..., (Xn, Yn) is a sample from a bivariate normal distribution, the probability density of the sample correlation coefficient R is15 pρ(r) = 2n−3π(n − 3)! (1 − ρ2)1 2 (n−1)(1 − r2)1 2 (n−4) (5.85)×∞k=0Γ2#1 2 (n + k − 1)$ (2ρr)k k!or alternatively pρ(r) = n −
(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 ) are independently distributed of (X, ¯ Y¯ ), and their joint distribution is the same as that
(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 unbiased test for testing the hypothesis τ /σ = ∆. Its acceptance region is|∆2S2 1 − S2 2
by making an orthogonal transformation from(Y1,,...,Yn) to (Z1,...,Zn) such that Z1 = √nY¯ , Z2 = viYi.]
(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 η + (ρτ /σ)(x − ξ).(ii) Let (X1, Y1),..., (Xn, Yn) be a sample from a bivariate normal
Generalize Problems 5.60(i) and 5.61 to the case of two groups of sizes m and n (c = 1).Section 5.13
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 {i1,...,ir} with(i) r = 2;(ii) r = even;(iii) r divisible by 3.(iv) Give two other examples of subgroups G0 of
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 permutations T and T correspond respectively to the subsets R = {i1,...,ir} and R = {j1,...,js}, then 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, and let e be the number of elements in G0. Any element g ∈ G0 (except the identity)is characterized by
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) = −∞, T(M+1) = ∞, then Pθ[T(i)
(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 m, n are positive integers with m ≤ n, thenm K=1 m K n K=m + n m− 1
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 1rz∈S(z)φ(z) = α a.e.
To generalize Theorem 5.11.1 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 − gkζ|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 of
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 function φ 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 =
Comparison of two designs. Under the assumptions made at the beginning of Section 12, 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 and
(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
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
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.
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 the
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 Θ 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 ξgiven 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 prior π(θ) ≡ 1, determine the posterior distribution of θ given the X’s.
Verify the posterior distribution of Θ 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 ofH : 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 is
Shape parameter of a gamma distribution. Let X1,...,Xn be a sample from the gamma distribution Γ(g,b) defined in Problem 3.34.(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.14 Section 5.5
Let S(x) be a family of confidence sets for a real-valued parameter θ, 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 aiXi − 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
Stein’s two-stage procedure.(i) If mS2/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.13[Consider any family of confidence intervals δ(X) ± L/2 of constant length L.Let ξ1,...ξ2n be such that |ξi − ξj
Suppose X and Y are independent, normally distributed with variance 1, and means ξ and η, respectively. Consider testing the simple null hypothesis ξ = 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.12 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 ofµ = µ0 against
Gamma two-sample problem. Let X1,...Xm; Y1,...,Yn be independent samples from gamma distributions Γ(g1, b1), Γ(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 3.29, 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 χ2
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 = (Xi +
If m = n, the acceptance region (5.23) can be written as max S2 Y∆0S2 X,∆0S2 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)T and known positive definite covariance matrixΣ. Assume a = (a1,...,ak)T is a fixed vector. The problem is to test H : k i=1 aiξi = δ vs. K : k i=1 akξi = 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, ··· , XN ).(i). If p has a known value p0, show there does not exist a UMP test of θ = 0 versus θ >
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 of ξ ≤ 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¯.[Thenn i=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 aijXj , be an orthogonal transformation (that is, n i=1 aijaik = 1 or 0 as j = k or j = k). The Z’s are normally distributed with common variance σ2 and means ζi = aij ξj
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 6 1 n i=s+1 Z2 i /(n − s)> C0 and |Z1 − ζ0
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 byβ∗(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) andχ2 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 11 pδ(t) = 1 21 2 (f−1)Γ( 1 2
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 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 letΓl = PABcCDlPABCD ˜ l PAB˜CD˜ l PAB˜ CD˜ l PABCDlPA˜BCD ˜ l PABCD˜ l PA˜B˜CD˜ l.Then(i) under the assumption Γl = Γ 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
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 i th 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 letρ = (p2/q2) ÷ (p1/q1).(i) If αλ0.

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