Testing Statistical Hypotheses 2nd Edition E. L. Lehmann - Solutions

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Two clinics using different methods of determining the amounts of a certain hormone in the blood have agreed to conduct a calibration experiment in which, for each of 24 patients, a blood specimen is divided into four aliquots, two being analysed by clinic A and the other two by clinic B. The
Review the aldosterone assay problem of Section 7.2 considering the possibility of a weighted analysis and/or a transformation. Compare your analysis with that of Section 7.2 and decide which you would recommend to the clinic.
Review and prepare a report on Problem 1.5.,
Review Section 5.4.2 and investigate the possibility that gender interacts with the covariates.
In a study of obesity in a certain group of 40 men a clinic has been assigning indices of obesity, breathlessness and unfitness, all measured on the scale of 0 to 1, and is now wondering to what extent these indices relate to height (cm), girth (cm), weight (kg) and a diet index, also measured on a
Consider the use of the symptom data of Problem 1.3 for purposes of differential diagnosis of forms A and B. Do you consider the multivariate kernel approach of Section 5.6 adequate for this purpose?
A study has been carried out to see if the composition of four constituentsa, b,c, d in tissue obtained at biopsy is in any way dependent on the size (mg) of the specimen analysed. The study so far has investigated 16 patients with the results recorded below. You have been asked to investigate the
Refer to Problem 1.3. It has been suggested that the blood compositions of Form A patients are dependent on the three symptoms, whereas the blood compositions of form B patients are not dependent on the three symptoms. Investigate this conjecture.
In a study of lethargic patients the responses of ten men and ten women to different doses of a stimulus are given in the following table.You have been asked to investigate fully the extent to which response depends on the stimulus and gender.
Re-read Section 5.3.3 and apply the analyses described there to obtain an appropriate description of the experience in the hyperplasia group of Cushing’s syndrome, restricting your analysis to the first seven concentrations.In your analysis consider how you can compare this experience with that
A problem in defining normal experience has arisen in a new blood screening situation which isolates a certain constituent of blood plasma.It is now possible to determine the proportions of the three partsa, b, c of this constituent. For twenty-five normal individuals this composition has been
There is a need to describe experience in the levels of two hormones, A and B, in normal individuals so that it may be possible to screen persons suspected of being susceptible to a hormone deficiency condition. The hormone levels (in standard units) of 30 normal persons have been determined and
Two indicators of a certain blood disease have been recorded in 24 patients and the clinic has asked you to provide a means of detecting whether new patients suspected of having the disease are within this previous experience. The clinic is particularly interested in having some easy graphical
A clinic consults you on the possible use of the concentrations(meq/l) of two hormones a and b in the detection of patients who may have an overactive excreting gland. The clinic has determined the hormone levels in 45 individuals known to have normal excretion and the hormone levels are recorded
A clinic is attempting to set up a means of describing normal experience of a newly discovered enzyme and has made determinations of the enzyme concentration in 150 normal individuals. The results in standard units are given below.As consultant you are asked to investigate this possibility and to
Consider the special case of model (4.17), y = XφT + e, where the ei mutually independent N(0, σ2) random variables.Show that the REML estimate of σ2 is given by rss/(n − p − 1) and that the maximised profile restricted loglikelihood is equal to−0.5(n − p − 1){log 2π + log[rss/(n −
Suppose that the use of a kernel density method is being sought for the data set of Problem 4.3. Can you suggest any suitable kernels and how you might proceed to use these in the analysis of the three-part compositional data set?
The compositional biplot for the three part-compositional data set of Problem 4.2 should give an exact representation of the covariance structure and the relationship of the markers to the parts of the composition. Construct the appropriate biplot and verify the exact correspondence.Plot the
Review the software available to you and try to find a program for the singular value decomposition of multivariate data or even a biplot program. Apply the software so that you may construct a biplot for the female data of Problem 1.5. Construct an estimate of the covariance matrix and compare
For readers with access to WinBUGS. For the data set of Problem 1.4 and the regression of final level on initial level and treatment follow the procedure detailed in Section 4.6 to estimate relevant parameters and for this simple situation compare your answers with a standard regression approach.
The data set below, recording the results of three diagnostic tests in standard units, has been recorded with the intention of developing a diagnostic system to differentiate between two types A and B of a recently identified syndrome. Application of standard binary logistic software has given a
Three-part compositions (a,b, c) of skin tissue have been collected from 25 normal individuals and are recorded below. Display the data graphically in a triangular (ternary) diagram in S2. If you are unfamiliar with triangular diagrams refer to the description in Chapter 11.Transform the data to
On the assumption that the variability of D-part compositions in a selected set of cases follows a LD(µ, Σ) distribution and that maximum likelihood estimates ˆµ, Σ for ˆ µ, Σ have been computed show that the atypicality index of a referred patient R with composition zR is given by J qR{qR
The following two data sets consist of two different test results(in standard units) on each of 20 normal patients. Draw the Q-Q plots for each of these data sets and comment on the possible nature of the underlying distributions.
In a study of the value of a diagnostic test result x for the differential diagnosis of two types t = 1, 2 a clinic has arrived at a logistic regression model Pr(t = 1|x, α, β)=1 − Pr(t = 2|x, α, β) = exp(α + βx)1 + exp(α + βx), with information about α, β contained in a posterior
A study of the effects of a treatment for patients with a certain history of blood pressure problems has been carried out. Recorded below are the initial systolic and diastolic blood pressures in mm Hg of 15 patients in the study and their corresponding blood pressures at the end of
A clinic with large experience of the relationship of how a measurement y varies with age x has decided that a reasonable predictive conditional density function for the dependence of y on x is of lognormal formΛ(α + βx, σ2). On the assumption that only large measurements are regarded as
In assessing the infectivity of a patient a clinic has decided that the variability of the total count of bacteria in 100 blood cells is well described by a Poisson distribution with probability function p(y|θ) = exp(−θ)θy/y! (y = 0, 1, 2,...).On the assumption that a Bayesian posterior
Identify some of the hypotheses of interest for problems 1.1, 1.3, 1.4, 1.5 and construct appropriate lattices for the investigation of these hypotheses.
Suppose that a clinic has devised a differential diagnostic system for two types t = 1, 2 and based on a 1 × c feature vector x on the basis of a fitted binary normal model Pr(t = 1|x, β)=1 − Pr(t = 2|x, β) = Φ(xβT ).Suppose further that the referred patients have feature vectors which
Using the general form of the results in Property 3.2 establish for the simple univariate model of linear regression yi = α + βxi + ei (i = 1,...,n), where ei (i = 1,...,n) are independent N(0, σ2) errors and the xi are real numbers, thatβˆ = n i=1(xi − x¯)(yi − y¯)n i=1(xi − x¯)2
Is the form of argument you have used in Problem 2. in any way altered if further information w (for example sex) is available on each individual in the study? In particular does such information alter in any way the assessment of an odds ratio built on the further conditioning pR(u|w, t)?
In a simple case-control study to investigate the effect of exposure v or non-exposure v∗ to a chemical on the possibility of developing (u)or not developing (u∗) a certain disease, a random sample of patients suffering from the disease and another random sample of individuals not suffering
A clinic has already carried out a trial to see how a typical treatment t may be able to control a hormone level v to a lower level y and so from the selected patients has produced an assessed conditional probability pS(y|v, t). Unfortunately it has been necessary to switch to a more efficient and
A clinic has conducted a trial in which a typical patient S referred to it on the basis of information v, and who has subsequently been diagnosed to have disease type u, is allocated treatment t (from among a set of trial treatments) and the outcome y of the treatment recorded. The clinic has
A certain genetic disease is carried only by females and affects only males. There is no direct test which can ascertain whether a female is a carrier, but affected males can be immediately recognised. The daughter of a carrier has probability 0.5 of herself becoming a carrier; the son of a carrier
In a large sample survey of a population, each person was classified into one of two types A and B and for each person two simple binary tests, each with a positive or negative result, were carried out. The table below gives the estimated probability p(u, v) of a person being of type u and giving
SLGD is a sex-linked genetic disease carried only by females and affecting only males. The son of a carrier has probability 0.25 of being SGLD; the daughter of a carrier has a probability 0.75 of being a carrier. Any offspring of a non-carrying mother is normal with respect to the disease. There is
A clinic has developed a new technique of measuring the concentration of five recently discovered enzymes, labelled e1,..., e5, and hopes to devise a means of determining whether new patients referred to the clinic are within normal experience of these enzymes. Forty normal individuals, 20 female
A dispute about the relative efficacy of rival treatments for the enhancement of low hormone level has given rise to a study of thirty patients, each suffering from low hormone levels. The patients have been allocated to one of the two treatments 1 and 2 at random and the enhanced level of hormone
A clinic with a differential diagnostic problem in distinguishing between two forms A and B of a newly identified blood disorder seeks your advice. For appropriate treatment it is important for the clinic to distinguish between these forms and this can be eventually done but often much later than
The assay department of a clinic has consulted you about problems concerning a possible assay procedure. The objective is to find a reasonably reliable means of determining the dosage of drug in a patient’s blood.The idea is to compare two responses of the patient against the two responses
In an early assay technique the immune responsiveness of an infected patient with a bacterial infection was measured by an index computed in the following way. On examination of the blood of a patient R under the microscope the total number nR of bacteria captured by 100 leucocytes was counted.
29. Instead of conditioning the confidence sets 8 E S( X) on a set C, consider a randomized procedure which assigns to each point x a probability IJ!(x) and makes the confidence statement 8 E S( x) with probability IJ!( x) when x is observed.*(i) The randomized procedure can be represented by a
28. (i) Under the assumptions of the preceding problem, the uniformly most accurate unbiased (or invariant) confidence intervals for 8 at confidence level 1 - a are fl = max( X(1) +d, X(II)) - 1 < 8 < min( X(1)' X(II) -d) = B, where d is the solution of the equation 2d" = a 2d" - (2d - 1)II = a if
27. Let Xl" ' " x" be independently distributed according to the uniform distribution U(8, 8 + 1). (i) Uniformly most accurate lower confidence bounds fl for 8 at confidence level 1 - a exist and are given by fl = max(X(I) - k, X(II) - 1), where X(I) = min(XI , · · · , XII)' X(II) = max(XI, ..
26. Let X have probability density f(x - 8), and suppose that EIXI < 00. For the confidence intervals X - c < 8 there exist semirelevant but no relevant subsets. [Buehler (1959).]
25. Let X be a random variable with cumulative distribution function F. If EIXI < 00, then J~ ooF(x) dx and JO'[1 - F(x)] dx are both finite. [Apply integration by parts to the two integrals.]
24. (i) Verify the posterior distribution of 8 given x claimed in Example 9. (ii) Complete the proof of (32).
23. In Example 9, check directly that the set C = {x: x S - k or x k} is not a negatively biased semirelevant subset for the confidence intervals (X -c, X+ c).
22. In Example 8, (i) the problem remains invariant under G' but not under G3 ; (ii) the statistic D is ancillary
21. Let VI" ' " v" be independently distributed as N(O, 1), and given VI = VI " ' " v" = Vn , let X; (i = 1, .. . , n) be independently distributed as N(8vj ,1). (i) There does not exist a UMP test of H : 8 = 0 against K : 8 > O. (ii) There does exist a UMP conditional test of H against K given the
20. Let XI"'" Xm and YI , . . . , Y,. be positive, independent random variables distributed with densities f(x/o) and g(y/'r) respectively. If f and g have monotone likelihood ratios in (x, 0) and ( y,1') respectively, there exists a UMP conditional test of H : 1'/0 sAo against 1'/0> Ao given the
19. Let the real-valued function f be defined on an open interval. (i) If f is logconvex, it is convex. (ii) If f is strongly unimodal, it is unimodal.
18. Verify the density (16) of Example 7.
17. Suppose X = (U, Z), the density of X factors into pu( x) = c(8 ,,'})go(u; z)h,,( z) k( u, z ), and the parameters 8, ,'} are unrelated. To see that these assumptions are not enough to insure that Z is S-ancillary for 8, consider the joint density C( 8, ,'}) e" }where I( u, z) is the indicator
16. In the situation of Example 2, the statistic Z remains S-ancillary when the parameter space is g = {(X, IL): IL X}.
15. In the situation of Example 3, X + Y is binomial if and only if Ii = 1.
14. Assuming the distribution (22) of Chapter 4, Section 9, show that Z is S-ancillary for p = P+/(P+ + p_).
13. A sample of size n is drawn with replacement from a population consisting of N distinct unknown values {al, . . . , aN }. The number of distinct values in the sample is ancillary.
12. Let X, Y have joint density p( x ,y) = 2f(x)f(y)F(8xy), where f is a known probability density symmetric about 0, and Fits cumulative distribution function. Then (i) p(x, y) is a probability density. (ii) X and Yeach have marginal density f and are therefore ancillary, but (X, Y) is not. (iii)
11. Let X be uniformly distributed on (8,8 + 1),0 < 8 < 00, let [X] denote the largest integer X, and let V = X - [X]. (i) The statistic V( X) is uniformly distributed on (0,1) and is therefore ancillary. (ii) The marginal distribution of [X] is given by { [8] [Xl = [8] + 1 with probability 1 -
10. In the preceding problem, suppose the probabilities are given by 1 1 - 8 6 2 1 - 28 6 3 1 - 38 6 4 1 + 8 6 5 1 + 28 6 6 1 + 38 6 Exhibit two different maximal ancillaries.
9. Consider n tosses with a biased die, for which the probabilities of 1, . . . , 6 points are given by 1 1 - 8 12 2 2 -8 12 3 3 - 8 12 4 1 + 8 12 5 2+8 12 6 3+8 12 and let X; be the number of tosses showing i points. (i) Show that the triple ZI = XI + Xs, Z2 = X2 + X4 , Z3 = X3 + X6 is a maximal
8. An experiment with n observations XI" '" Xn is planned, with each X; distributed as N(8,1). However, some of the observations do not materialize (for example, some of the subjects die, move away, or tum out to be unsuitable). Let Ij = 1 or 0 as is observed or not, and suppose the Ij are
7. Let X, Y be independently normally distributed as N(8 ,1), and let V= Y- X and W = { Y - X if X + Y > 0, X - Y if X + Ys O. (i) Both V and W are ancillary, but neither is a function of the other. (ii) (V, W) is not ancillary. [Basu (1959).) [10.5
6. In the preceding problem, suppose that the densities of X under cf and 3&" are ee:" and (1/8)e- x / orespectively. Compare the UMP conditional and unconditional tests of H : 8 = 1 against K : 8 > 1
5. With known probabilities p and q perform either or 3&", with X distributed as N(8, 1) under or N( -8,1) under 3&". For testing H : 8 = 0 against 8> 0 there exist a UMP unconditional and a UMP conditional level-a test. These coincide and do not depend on the value of p .
4. Let Xl" ' " X; be independently distributed, each with probability p or q as N(t aJ) or Na, af). (i) If p is unknown, determine the UMP unbiased test of H : = 0 against K : ~> O. (ii) Determine the most powerful test of H against the alternative ~l when it is known that p = t, and show that a
3. The test given by (3), (8), and (9) is most powerful under the stated assumptions.
2. Under the assumptions of Problem 1, determine the most accurate invariant (under the transformation X' = - X) confidence sets S( X) with p(~ E S( X)I~ + p(~ E S( X) IS') = 2y. Find examples in which the conditional confidence coefficients Yo given and YI given satisfy (i) Yo < YI; (ii) Yo > YI '
1. Let the experiments and 3&" consist in observing X : N(t aJ) and X : Na, at) respectively (ao < al ) , and let one of the two experiments be performed, with P(~) = P( 3&") = t. For testing H : = 0 against = ~l determine values ao, aI' ~l' and a such that (i) a o < al ; (ii) ao > aI' where the ai
32. The following two examples show that the assumption of a finite sample space is needed in Problem 31. (i) Let Xl' " '' Xn be i.i.d. according to a normal distribution N( 0,02 ) and test H : 0 = 00 against K : 0 > 00 ' (ii) Let X and Y be independent Poisson variables with E( X) =.\ and E( Y) =
31. Locally uniformly most powerful tests. If the sample space is finite and independent of 0, the test 'Po of Problem 2(i) is not only LMP but also locally uniformly most powerful (LUMP) in the sense that there exists a value A > 0 such that 'Po maximizes /3", (0) for all ° with 0 < °-00 < A.
30. Suppose in Problem 29(i) the variance 0 2 is unknown and that the data consist of XI"'" Xn together with an independent random variable S2 for which S2/02 has a X2-distribution.1f K is replaced by EO?/02 = r2, then (i) the confidence sets E(O; - XY/S2 .::;; C are uniformly most accurate
29. Let Xl"' " Xn be independent normal with means 01 " " , On and variance 1. (i) Apply the results of the preceding problem to the testing of H : 01 = . . . = On = 0 against K :EO? = r2 , for any fixed r > O. (ii) Show that the results of (i) remain valid if H and K are replaced by H' :E O? .::;;
28. To generalize the results of the preceding problem to the testing of H: 1 vs. K : {1o, 8 E w}, assume: (i) There exists a group G that leaves H and K invariant. (ii) Gis transitive over w. (iii) There exists a probability distribution Q over G which is right-invariant in the sense of Section 4
27. For testing H :10 against K :{II" ' " Is}, suppose there exists a finite group G = {gl" '" gN } which leaves H and K invariant and which is transitive in the sense that givenh,fj' (1 S, j, j') there exists g E G such that gh = h" In generalization of Problems 24, 25, determine a UMP invariant
26. Show that the UMP invariant test of Problem 24 is most stringent.
25. The UMP invariant test 4>0 of Problem 24 (i) maximizes the minimum power over K; (ii) is admissible. (iii) For testing the hypothesis H of Problem 24 against the alternatives K' = {KI , ... , Kn , K{,.. . , K~}, where under Ks : = 0 for all j'" i, = -~ determine the UMP test under a suitable
24. Let XI"'" X; be independent normal variables with variance 1 and means ~I' • • • , ~n' and consider the problem of testing H : ~I = . .. = ~n = 0 against the alternatives K = {KI , • •• , Kn } , where K;: ~j = 0 for j '" i, ~; = (known and positive). Show that the problem remains
23. Let (ZI ' . '. ' ZN) "" (XI' .. . ' Xm , Y1, · · · , Yn ) be distributed according to the joint density (56) of Chapter 5, and consider the problem of testing H: 71 "" against the alternatives that the X's and Y's are independently normally distributed with common variance 0 2 and means 71
22. Let {0t.} be a class of mutually exclusive sets of alternatives such that the envelope power function is constant over each 0t. and that Unto = °-0H' and let % maximize the minimum power over 0t.. If CPt. = cp is independent of d , then cp is most stringent for testing 8 EON.
21. Existence of most stringent tests. Under the assumptions of Problem 1 there exists a most stringent test for testing 8 E 0H against 8 E °-0H.
20. Suppose that the problem of testing 0 E 0H against 8 E OK remains invariant under G, that there exists a UMP almost invariant test CPo with respect to G, and that the assumptions of Theorem 3 hold. Then CPo maximizes info K [w(8)Eecp(X) _ + u(8)] for any weight functions w(8) 0, u(8) that are
19. Let X = (XI" . . , Xp ) and Y = (YI , .. . , ~) be independently distributed according to p-variate normal distributions with zero means and covariance matrices E(X;X.;) = aij and E(Y;lj) = !::J.a;j ' (i) The problem of testing H: !::J. s !::J.o remains invariant under the group G of
18.· (i) In the preceding problem determine the maximin test if to' is replaced by r.a i~d, where the a's are given positive constants. (ii) Solve part (i) with Var(X;) = 1 replaced by Var(X;) = al (known). [(i): Determine the point (~r ,...,~:) in w' for which the MP test of H against K: (~r, ...
17. Let XI" ' " Xn be independently normally distributed with means E(X;) = and variance 1. The test of H : ~I = . . . = ~n = 0 that maximizes the minimum power over w' :r.~ d rejects when r.X;c. [If the least favorable distribution assigns probability 1 to a single point, invariance under
16. Write out a formal proof of the maximin property outlined in the last paragraph of Section 3.
15. Determine whether (21) remains the maximin test if in the model (20) G, is replaced by Gij'
14. Evaluate the test (21) explicitly for the case that Pi is the normal distribution with mean ~i and known variance a2 , and when ( 0 = (I'
13. Show that if 9'0 '" 9'1 and (0' (I are sufficiently small, then Qo '" QI'
12. Prove the formula (15).
11. Show that there exists a unique constant b for which qo defined by (11) is a probability density with respect to ~, that the resulting qo belongs to 9'0 ' and that b --+ 00 as (0 --+ O.
10. If (13) holds, show that ql defined by (11) belongs to 9'1'
9. Double-exponential distribution. Let Xl" ' " X; be a sample from the double-exponential distribution with density }e- 1x - O/. The LMP test for testing 8 S 0 against 8 > 0 is the sign test, provided the level is of the form 1 m a=-2:(n) 2n k k' -0 so that the level-a sign test is nonrandomized.
8. (i) Let X have binomial distribution b( p, n), and consider testing H : p = Po at level a against the alternatives OK :r/« S }Po/qo or 2po/qo . For a = .05 determine the smallest sample size for which there exists a test with power ~.8 against OK if Po = .1, .2, .3, .4, .5. (ii) Let Xl' " '' Xn
7. Let x = (XI""'Xn ) , and let ge(x,~) be a family of probability densities depending on 8 = (81" • • , 8r ) and the real parameter ~, and jointly measurable in x and For each 8, let he(~) be a probability density with respect to a a-finite measure II such that Pe(x) = !ge(x, ~)hea) dll(~)
6. Let le(x) = 8g(x) + (1 - 8)h(x) with 0 s 8 1. Then le(x) satisfies the assumptions of Lemma 1 provided g(x)/h(x) is a nondecreasing function of x.
5. Let ZI"'" Z; be identically independently distributed according to a continuous distribution D, of which it is assumed only that it is symmetric about some (unknown) point. For testing the hypothesis H : D(O) = t,the sign test maximizes the minimum power against the alternatives K: D(O) s q (q

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Testing Statistical Hypotheses 2nd Edition E. L. Lehmann - Solutions

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