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statistical sampling to auditing

Questions and Answers of Statistical Sampling To Auditing

60. In Example 23. show that (i) both sets (55) are intervals; (ii) the sets given by vp( v) > C coincide with the intervals (42) of Chapter 5.
59. The confidence sets (47) are uniformly most accurate equivariant under the group G defined at the end of Example 22.
58. Let Xl •. . .• X, be independent N(O,l), and let S2 be inds>endent of the X's and distributed as X;. Then the distribution of (XI/Sill, . . . • Xr/S..;;) is a central multivariate
57. Show that in Example 20, (i) the confidence sets a2 / S2 E AU with A** given by (40) coincide with the uniformly most accurate unbiased confidence sets for a2 ; (ii) if (a,b) is best with respect
56. In Example 20, the density p(v) of V = 1/S2 is unimodal.
55. In Examples 20 and 21 there do not exist equivariant sets that uniformly minimize the probability of covering false values.
54. (i) Let (Xl ' Yl ) , ... , (Xn , y") be a sample from a bivariate normal distribution, and let -I ( E(X; - x)( Y; - Y) ) p=C , - VE(X; - X)2E(Y; _ y)2 where C(p) is determined such that (
53. (i) Let XI' .. " X; be independently distributed as N(t 0 2), and let 8 = ~jo The lower confidence bounds for 8, which at confidence level 1 - a are uniformly most accurate invariant under the
52. Counterexample. The following example shows that the equivariance of S(x) assumed in the paragraph following Lemma 5 does not follow from the other assumptions of this lemma. In Example 8, let n
51. (i) One-sided equivariant confidence limits. Let () be real-valued, and suppose that for each ()o, the problem of testing () 5 ()o against () > ()o (in the presence of nuisance parameters {;)
50. Let Xl" ' " XII ; Yl , .. . ,y" be samples from N(~, 11 2) and N( 71 , T 2) respectively. Then the confidence intervals (43) of Chapter 5 for T 2 /11 2 , which can be written as [(}j - yf k[( X;
49. In Example 16, a family of sets S(x, y) is a class of equivariant confidence sets if and only if there exists a set 9t of real numbers such that S(x,y) = U {(~,7I) :(x-n2+(Y-7I)2=r2} re9t
48. The hypothesis of independence. Let (XI' YI), . . . ,(XN , YN ) be a sample from a bivariate distribution, and (~1)' ZI)' ... , ( N)' ZN) be the same sample arranged according to increasing
47. In the preceding problem let U;j = 1 if (j - i)(Zj - Zi) > 0, and = 0 otherwise. (i) The test statistic LiT; can be expressed in terms of the U's through the relation N LiT;= LU-i)U +
46. The hypothesis of randomness. Let ZI" ' " ZN be independently distributed with distributions FI , . .. , FN , and let T; denote the rank of Z, among the Z 's. For testing the hypothesis of
45. Unbiased tests of symmetry. Let Z\ , . . . , ZN be a sample, and let cf> be any rank test of the hypothesis of symmetry with respect to the origin such that Zi z; for all i implies cf>(Zt , . . .
44. An alternative expression for (66) is obtained if the distribution of Z is characterized by (p, F, G). If then G = h(F) and h is differentiable, the distribution of n and the S, is given by ( 67)
43. Let Z Z be a sample from a distribution with density f(z - ), where (z) is positive for all z and f is symmetric about 0, and let m, n, and the S, be defined as in the preceding problem. (i) The
42. (i) Let m and n be the numbers of negative and positive observations among Z,..., Z, and let S < < S, denote the ranks of the positive Z's among 1ZZ. Consider the N + N(N-1) distinct sums Z + Z,
41. Continuation. (i) There exists at every significance level a a test of H: G = F which has power >a against all continuous alternatives (F,G) with F + G. (ii) There does not exist a nonrandomized
40. (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')} = + 24, where A (FG)
39. - (i) If X X and Y,..., Y, are samples from F(x) and G(y) = F(y A) respectively (F continuous), and D(1)
38. Let X X Y,..., Y be samples from a common continuous distribution F. Then the Wilcoxon statistic U defined in Problem 27 is distributed symmet- rically about mn even when m n.
37. Let be a family of probability measures over (, ), and let be a class of transformations of the space . Define a class of distributions by Fif there exists F, and fe such that the distribution of
36. 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-6) with (10) F(x) +
35. For sufficiently small > 0, the Wilcoxon test at level a = k(~). k a positive integer, maximizes the power (among rank tests) against the alternatives (F,G) with G (1-0)F+OF.
34. (i) If X, X and Y, Y, are samples with continuous cumulative distribution functions F and G h(F) respectively, and if h is differen- tiable, the distribution of the ranks S < ... < .
33. Distribution of order statistics. (i) If Z., Z is a sample from a cumulative distribution function F with densityf, the joint density of Y, Z,), i = 1,..., n, is (62) N!f(y)... f(y) (1
32. Under the assumptions of the preceding problem, if F, h,(F), the distribu- tion of the ranks T... Ty of Z. Zy depends only on the h,, not on F. If the h, are differentiable, the distribution of
31. Let Z, have a continuous cumulative distribution function F, (i = 1,..., N), and let G be the group of all transformations Z-f(Z) such that fis continuous and strictly increasing. (i) The
30. (i) For any continuous cumulative distribution function F, define F-(0) = -oo, F(y) inf(x: F(x)=y) for 0 < y < 1, F(1) = co if F(x)
29. (i) Let Z. Zy be independently distributed with densities f.....N. and let the rank of Z, be denoted by T. If f is any probability density which is positive whenever at least one of thef, is
28. Expectation and variance of Wilcoxonstatistic. If the X 's and Y's are samples from continuous distributions F and G respectively, the expectation and variance of the Wilcoxon statistic V defined
27. Wilcoxon two-sample test. Let Vi) = 1 or 0 as X; < lj or X; > lj, and let V = EEU,1 be the number of pairs X;. lj with X; < lj. (i) Then V = ES, - tn(n + 1), where SI < . . . < S; are the ranks
26. For the model of the preceding problem, generalize Example 13 (continued) to show that the two-sided t-test is a Bayes solution for an appropriate prior distribution.
25. Let X X Y..... Y be independent N(E, 02) and N(n, 2) respec- tively. The one-sided t-test of H: 8/0 0 is admissible against the alternatives (i) 0 < 8 0; (ii) 8 > 8 for any 82 > 0.
24. Verify (i) the admissibility of the rejection region (22); (ii) the expression for I(Z) given in the proof of Lemma 3.
23. (i) In Example 13 (continued) show that there exist C, C, such that A,(7) and A(n) are probability densities (with respect to Lebesgue measure). (ii) Verify the densities ho and h.
22. (i) The acceptance region T/T C of Example 13 is a convex set in the (T, T) plane. (ii) In Example 13, the conditions of Theorem 8 are not satisfied for the sets A: T/TC and ': > k.
21. (i) The following example shows that a-admissibility does not always imply d-admissibility. Let X be distributed as U(O, 8), and consider the tests !PI and !P2 which reject when respectively X <
20. The definition of d-admissibility of a test coincides with the admissibility definition given in Chapter 1, Section 8 when applied to a two-decision procedure with loss 0 or 1 as the decision
19. Let G be a group of transformations of !!E, and let SiI be a a-field of subsets of !!E, and !J. a measure over (!!E, SiI). Then a set A E SiI is said to be almost invariant if its indicator
18. Inadmissible likelihood-ratio test. In many applications in which a UMP invariant test exists, it coincides with the likelihood-ratio test. That this is,
17. Invariance of likelihoodratio. Let the family of distributions 9' = {Po, 8 E Q} be dominated by p., let Po = dPo/dp., let p.g-I be the measure defined by p.g-l(A) = p.[g-I(A)]. and suppose that
16. (i) A generalization of equation (1) is !f(x) dPo(x) = f f(g-Ix} dPgo(X) . A gA (ii) If Po, is absolutely continuous with respect to POo' then PgOI is absolutely continuous with respect to
15. Envelope power function. Let S(a) be the class of all level-a tests of a hypothesis H, and let /3:(8) be the envelopepower function, defined by /3:(8) = sup /3(8), ES(a) where fJ denotes the
14. Consider a testing problem which is invariant under a group G of transformations of the sample space, and let 't' be a class of tests which is closed under G, so that E 't' implies g E 't',
13. Show that (i) GI of Example 11 is a group ; (ii) the test which rejects when xiiiXfl > C is UMP invariant under G1; (iii) the smallest group containing G, and G2 is the group G of Example 11.
12. Almost invariance of a test If> with respect to the group G of either Problem 6(i) or Example 6 implies that If> is equivalent to an invariant test.
11. For testing the hypothesis that the correlation coefficient p of a bivariate normal distribution is s Po. determine the power against the alternative p = PI when the level of significance a is
10. Testing a correlation coefficient. Let (Xl' yl)•... '(Xn , y") be a sample from a bivariate normal distribution. (i) For testing P S Po against P > Po there exists a UMP invariant test with
9. Two-sided t-test. (i) Let Xl" ' " X" be a sample from N( t 0 2). For testing = 0 against =1' 0, there exists a UMP invariant test with respect to the group XI = eX" c =1' O. given by the two-sided
8. (i) When testing H : P 5 Po against K: P > Po by means of the test corresponding to (11), determine the sample size required to obtain power fJ against P = PI' a = .05, fJ = .9 for the cases Po =
7. If X. ,... • x" and YI,. .. ,y" are samples from Na.02) and N(1/,-r 2 ) respectively, the problem of testing -r 2 = 0 2 against the two-sided alternatives -r 2 =1' 0 2 remains invariant under
6. Let Xl"' " Xm; Yl , . .. , Y" be samples from exponential distributions with densities a-Ie-(. there exists a UMP invariant test with respect to the group G: X: = aX;+b, J}' = aJ} +c, a> 0, - 0
5. (i) Let X = (Xl' . . . , Xn) have probability density (1/0 n)f[(XI - ~)/O" ",(xn - ~)/O] where -00 < < 00 , 0
4. Let X, Y have the joint probability density [i x, y). Then the integral h(z) = J~ "J (y - z, y) dy is finite for almost all z, and is the probability density of Z= Y- X. [Since
3. (i) A sufficient condition for (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, - 00 < b < 00, then the subgroup of translations x' =
2. (i) Let be the totality of points x = (XI" . . , x,,) for which all coordinates are different from zero, and let G be the group of transformations x; = cx i, e > O. Then a maximal invariant under
1. Let G be a group of measurable transformations of (~, JJI) leaving 9 = {Po, (J En} invariant, and let T( x) be a measurable transformation to (ff, !fI). Suppose that T(xl) = T(X2) implies T(gxl) =
79. Consider a one-sided, one-sample, level-a r-test with rejection region t( X) en ' where X = (X\, . .. , Xn ) and t( X) is given by (16). Let an (F) be the rejection probability when X\, ... , Xn
78. Let X\, ,, ,,Xm and Y\,... , y" be independent samples from I(J.L, 0) and I ('" T) respectively. (i) There exist UMP unbiased tests of T2/ T\ against one- and two-sided alternatives. (ii) If T=
hypotheses based on the statistic V = L(I/X; - I/X). (iii) When T = TO ' the distribution of TOV is X~-\ [Tweedie (1957).
77. Inverse Gaussian distribution:" Let Xl"'" X; be a sample from the inverse Gaussian distribution I(p., T), both parameters unknown. (i) There exists a UMP unbiased test of p. P.o against p. > P.o,
76. Let XI" '" X; be a sample from the Pareto distribution P(c, T), both parameters unknown. Obtain UMP unbiased tests for the parameters c and T. [Problem 12, and Problem 44 of Chapter 3.]
75. Gamma two-sample problem . Let Xl"'" Xm ; YI , ... , y" be independent samples from gamma distributions I'(g, bl)' r(g2' b2 ) respectively. (i) If gl' g2 are known, there exists a UMP unbiased
74. Scale parameter of a gamma distribution . Under the assumptions of the preceding problem, there exists (i) A UMP unbiased test of H : b bo against b > bo which rejects when LX; > C(I1X;). (ii)
73. Shape parameter of a gamma distribution . Let Xl' . . .' X; be a sample from the gamma distribution I'(g,b) defined in Problem 43 of Chapter 3. (i) There exist UMP unbiased tests of H : g s go
72. Let (X" Y;), i = 1,... , n, be i.i.d. according to a bivariate distribution F with E(Xh E(y;2) < 00. (i) If R is the sample correlation coefficient, then r;;R is asymptotically normal with mean 0
71. There exist bivariate distributions F of (X, y) for which p = 0 and Var(XY)/[Var(X)Var(Y)] takes on any given positive value.
70. If X, Y are positively regression dependent, they are positively quadrant dependent. [Positive regression dependence implies that (91) P[Y~YIX~xl ~P[Y~YIX~x'l for all x < x' and Y, and (91)
69. (i) The functions (79) are bivariate cumulative distributions functions. (ii) A pair of random variables with distribution (79) is positively regressiondependent.
68. If X and Y have a bivariate normal distribution with correlation coefficient p > 0, they are positively regression-dependent. [The conditional distribution of Y given x is normal with mean n +
67. If (XI' Yd, ...,(Xn , y,,) is a sample from a bivariate normal distribution, the probability density of the sample correlation coefficient R is* (86) 2n - 3 pp(r) = (1 _l)~(n-1)(1 _ r2)~(n -4)
66. (i) Let (XI' YI ) , ... ,(Xn , y,.) be a sample from the bivaria~ normal_distribution (74), and let Sf = '£(X; - X)2, Sl2 = '£(X; - X)(Y; - Y), sf = '£(Y, - Y)2. Then (s], Sl2' Sf) are
65. (i) Let (XI' YI ) , .. . ,(Xn , Yn ) be a sample from the bivariate normal distribution (70), and let Sr = '£( X; - X)2, Sf = '£( Y; - y)2, Sl2 = '£( X; - X)( Y; - Y). There exists a UMP
64. (i) If the joint distribution of X and Y is the bivariate normal distribution (70), then the conditional distribution of Y given X is the normal distribution with variance T2 (1 - p2) and mean
63. Generalize Problems 60(i) and 61 to the case of two groups of sizes m and n (c = 1).
62. Determine for each of the following classes of subsets of {I, ... , n} whether (together with the empty subset) it forms a group under the group operation of the preceding problem: All subsets
61. The preceding problem establishes a 1 : 1 correspondence between e - 1 permutations T of Go which are not the identity and e - 1 nonempty subsets {il , ... , if} of the set {I, ... , n}. If the
60. (i) Given n pairs (XI' YI)' .. . ,(X" , Y,,), let G be the group of 2" permutations of the 2" variables which interchange Xi and Yi in all, some, or no of the n pairs. Let Go be any subgroup of
59. Let Z., .. . , Z" be i.i.d. according to a continuous distribution symmetric about 0, and let 1(1) < ' " < 1( M) be the ordered set of M = 2" - 1 subsarnples (Z;t + ... +Z;)/r, r 1. If 1(0) =
58. (i) Generalize the randomization models of Section 14 for paired comparisons (nl = . . . = n(. = 2) and the case of two groups (c = 1) to an arbitrary number c of groups of sizes nl , .. . ,
57. If m , n are positive integers with m :s; n, then f (~)(i) = (m;:; n)_1. K-l
56. If c = 4, m, = n; = 1, and the pairs (x;, y;) are (1.56,2.01), (1.87,2.22), (2.17,2.73), and (2.31,2.60), determine the points 8(1» . .. , 8(15) which define the intervals (72).
55. 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 8(1» .. . ,8(19) defined as the ordered values of (73).
54. Generalization of Corollary 3. Let H be the class of densities (81) with C1 > 0 and - 00 < r; < 00 (i = 1, ... , N). A complete family of tests of H at level of significance a is the class of
53. To generalize Theorem 7 to other designs, let Z = (ZI" . . , ZN) and let G = {gl ' .. . ' g,} be a group of permutations of N coordinates or more generally a group of orthogonal transformations
52. Consider the problem of testing H :'II = in the family of densities (62) when it is given that a > c > 0 and that the point (ru , .. . , i.. ) of (63) lies in a bounded region R containing a
51. Suppose that a critical function 1/10 satisfies (65) but not (67), and let a < t. Then the following construction provides a measurable critical function 1/1 satisfying (67) and such that I/Io(z)
50. Continuation. An alternative comparison of the two designs is obtained by considering the expected length of the most accurate unbiased confidence intervals for t:. = 'II - in each case. Carry
49. 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
48. (i) If Xl' . . . ' Xn ; Y1, ••. , y" are independent normal variables with common variance a2 and means E(X;) = Ci' E(Y;) = t + l1, the UMP unbiased test of l1 = 0 against l1 > 0 is given by
47. In the matched-pairs experiment for testing the effect of a treatment, suppose that only the differences Z, = Y; - X; are observable. The Z's are assumed to be a sample from an unknown continuous
46. Confidence intervalsfor a shift. (i) Let XI" ' " Xm ; YI , . . . , y" be independently distributed according to continuous distributions F(x) and G(y) = F(y - 6) respectively. Without any further
45. If c = 1, m = n = 4, a = .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
44. Prove Theorem 6 for arbitrary values of c.
43. Let TI , • • • , 'F.- I have the multinomial distribution (34) of Chapter 2, and suppose that (pI , .. . , p,_I) has the Dirichlet prior density D( aI ' , as) with density proportional to
42. Let X... . . , Xm and Y" . .. , y" be independently distributed as Na , ( 2 ) and N(T/, 'T 2 ) , respectively and let (t T/ ,a, T) have the joint improper prior density 7Ta ,1J,a, 'T) d~ d1J do
41. Let XI' "'' Xm and Y1, . .. , y" be independently distributed as N(t a2 ) and N(T/, a2 ) respectively, and let (€, T/ ,a) have the joint improper prior density given by 7T(~ T/,a) d~ dT/ do =

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