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statistical sampling to auditing

Questions and Answers of Statistical Sampling To Auditing

40. Let 8 = (81 " " , 8s ) with 8; real-valued, X have density Po (x), and e a prior density 71'(8). Then the l00y% HPD region is the l00y% credible region R that has minimum volume. [Apply the
39. If X is normal N(8,1) and 8 has a Cauchy density b/{ 7T[b2 + (8 - JL)2]), determine the possible shapes of the HPD regions for varying JL and b.
38. In Example IS, show that (i) the posterior density 7T(alx)is of type (c) of Example 13; (ii) for sufficiently large r, the posterior density of o' given x is no longer of type (c).
37. In Example 14, verify the marginal posterior distribution of given x.
36. Verify the posterior distribution of P given x in Example 13. "For the corresponding result concerning one-sided confidence bounds. see Madansky (1962).
35. If XI" '" x" are independent N(8,1) and 8 has the improper prior 7T(8) == 1, determine the posterior distribution of 8 given the X's.
34. Verify the posterior distribution of e given x in Example 12.
33. (i) Under the assumptions made at the beginning of Section 8, the UMP unbiased test of H: P = Po is given by (45). (ii) Let (p, p) be the associated most accurate unbiased confidence intervals
32. Most accurate unbiased confidence intervals exist in the following situations: (i) If X, Y are independent with binomial distributions b( PI' m) and b(P2 ' n), for the parameter PIQ2/P2QI' (ii)
31. Let XI"' " Xn be distributed as in Problem 12. Then the most accurate unbiased confidence intervals for the scale parameter a are 2 2 - E[x; - min(xp . .. , xn)] a - E[x; - min( xl , · .. ,
30. 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."
29. Let S(x) be a family of confidence sets for a real-valued parameter (J, and let p,fS(x)j denote its Lebesgue measure. Then for every fixed distribution Qof X (and hence in particular for Q = P80
28. Two-stage t-tests with power independent ofa. (i) For the procedure TIl with any givenc, let C be defined by trxltllo_1(Y) dy =a. Then the rejection region O:::'-la;X; - ~o)/IC > C defines a
27. Confidence intervals of fixed length for a normal mean. (i) In the two-stage procedure ill defined in part (iii) of the preceding problem, let the number c be determined for any given L > 0 and
26. Stein's two-stage procedure. (i) If mS2/(12 has a X2 = distribution with m degrees of freedom, and if the conditional distribution of Y given S = s is N(O, (12/S2), then Y has Student's
25. On the basis of a sample X = (XI' " . , XII) of fixed size from N(~, (12) there do not exist confidence intervals for with positive confidence coefficient and of bounded length. [Consider any
24. Let X; = + U;, and suppose that the joint density of the U's is spherically symmetric, that is, a function of EU;2 only, /(u\, o.. , un) = q([ul). Then the null distribution of the one-sample
23. Determine the maximum asymptotic level of the one-sided r-test when a = .05 and m = 2,4,6: (i) in Model A; (ii) in Model B.
22. Show that the conditions of Lemma 1 are satisfied and y has the stated value: (i) in Model B; (ii) in Model C.
21. In Model A, suppose that the number of observations in group i is n., If ni :::; M and s -+ 00, show that the assumptions of Lemma 1 are satisfied and determine y.
20. Verify the formula for Var(X) in Model A.
19. If Y" is a sequence of random variables and c a constant such that E(y" - C)2 -+ 0, then for any a > 0, P(IY" - c]a) -+ 0, that is, Y" tends to c in probability.
18. The Chebyshev inequality. For any random variable Y and constants a> ° andc, E(Y - C)2 a2 P( IY - c]a) .
17. (i) Given p, find the smallest and largest value of (31) as (12/T2 varies from ° to 00 . (ii) For nominal level a = .05 and p = 1,.2,.3, .4, determine the smallest and the largest asymptotic
16. Generalize Problem 15(i) and (ii) to the two-sample r-test.
15. (i) Let X\, . .. , x" be a sample from N(t a2 ). The power of the one-sided one-sample r-test against a sequence of alternatives (tn'a) for which Intil/a -- 8 tends to ~(8 - uo )(ii) The result
14. Corollary 2 remains valid if c; is replaced by a sequence of random variables c" tending to c in probability.
13. Extend the results of the preceding problem to the case, considered in Problem 10, Chapter 3, that observation is continued only until X(11" ' " X(r) have been observed.
12. Exponential densities. Let XI" ' " x" be a sample from a distribution with exponential density a -I e -(x -hl/u for x b.(i) For testing a = 1 there exists a UMP unbiased test given by the
11. Let Xl . . .. ' X", and YI•. . .• y" be samples from Na. ( 2 ) and N(1j, ( 2 ). The UMP unbiased test for testing 71 - = 0 can be obtained through Problems 5 and 6 by making an orthogonal
10. If m = n, the acceptance region (23) can be written as ( s~ t:..oSi ) 1 - C max /loS; ' S~ :::; -C-' where si = E( X, - X)2, S~ = E( Yi - y)2 and where C is determined by l c a o B" _I.,,_I(W) dw
9. Let XI" '" x" and Y...... y" be independent samples from Na, ( 2 ) and N( 71, T2) respectively. Determine the sample size necessary to obtain power f3 against the alternatives Tla > t:.. when a =
8. Let XI' X2 , . •. be a sequence of independent variables distributed as N( t ( 2 ) . and let y" = [nX,, +1 - (XI + .. . +X,,)l! vn(n + 1) . Then the variables Y" Y2 , . • • are independently
7. If X••. .. , X" is a sample from N(~. a2). the UMP unbiased tests of :::; 0 and = 0 can be obtained from Problems 5 and 6 by making an orthogonal transformation to variables Z., ... , Z" such
6. Let XI"' " x" be independently normally distributed with common variance a 2 and means ~., ... , ~". and let Z, = E'j=. aij be an orthogonal transformation (that is, E;'=. aijaik = 1 or 0 as j =
5. Let Zl' . .. ' Z" be independently normally distributed with common variance 0 2 and means E(Z;) = ;(i = 1, . . . , s) , E(Z;) = 0 (i = s + 1, ... , n) . There exist UMP unbiased tests for testing
4. Let Xl' .. . ' X" be a sample from N (~, (12). Denote the power of the one-sided r-test of H: 0 against the alternative ~/o by fJa/o), and by fJ*(~/o) the power of the test appropriate when 0 is
3. (i) Let Z and V be independently distributed as N( 8,1) and X2 with I degrees of freedom respectively. Then the ratio Z IVII has the noncentral t-distribution with I degrees of freedom and
2. In the situation of the previous problem there exists no test for testing H: = 0 at level a , which for all 0 has power P> a against the alternativesa, 0) with = ~l > O. [Let P( ~l' 0) be the
1. Let Xl" ' " Xn be a sample from N(t 0 2). The power of Student's r-test is an increasing function of ~/o in the one-sided case H : s 0, K: > 0, and of I~II0 in the two-sided case H : = 0, K: ':F
36. The UMP unbiased test of H: A = 1 derived in Section 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)
35. Random sample size. Let N be a random variable with a power-series distribution P(N = n) = a(n)An C(A) , n = 0,1, . . . (A > 0, unknown). When N = n, a sample XI"'" Xn from the exponential family
34. Let X, Y, Z be independent Poisson variables with means A,p., JI. Then there exists a UMP unbiased test of H : AP. s Jl2.
33. Let X; (i = 1,2) be independently distributed according to distributions from the exponential families (12) of Chapter 3 with C, Q, T, and h replaced by Ci , Qi' 1;, and hi' Then there exists a
32. Negative binomial. Let X, Y be independently distributed according to negative binomial distributions Nbt p-; m) and Nb(P2' n) respectively, and let qi = 1 - Pi' (i) There exists a UMP unbiased
31. Let X\, .. . , Xn be a sample from the uniform distribution over the integers 1, . .. , 8, and let a be a positive integer. (i) The sufficient statistic X(n) is complete when the parameter space
30. 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.
29. In the 2 X 2 table for matched pairs, in the notation of Section 9, the correlation between the responses of the two members of a pair is For any given values of 'lT1 < 'lT2, the power of the
28. Consider the comparison of two success probabilities in (a) the two-binomial situation of Section 5 with m = n, and (b) the matched-pairs situation of Section 9. Suppose the matching is
27. In the 2 X 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
26. Let X;jk' (i, j , k = 0,1, / = 1, . . . , L) denote the entries in a 2 X 2 X 2 X L table with factors A, B, C, and D, and let r _ PABcD,P1BcD,PABcD,P1BcD, i : PA BCD,P1BCD,PABCD,P1BCD, Then (i)
25. In a 2 X 2 X K table with li k = Ii, 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.
24. In a 2 X 2 X 2 table with ml = 3, nl = 4; m2 = 4, n2 = 4; and II = 3, 11 = 4, 12 = 12= 4, determine the probabilities that P(YI + Y2 klX; + Y; = l j , i = 1,2) for k = 0,1,2,3.
23. Rank-sum test. Let YI , . . . , YN be independently distributed according to the binomial distributions b(p;, n;), i = 1, . . . , N, where 1 P;= 1 + e-ca+!JXj) . This is the model frequently
22. (i) Based on the conditional distribution of X 2 , . •. , X n given Xl = Xl in the model of Problem 20, there exists a UMP unbiased test of H : Po = PI against PI > Po for every a . (ii) For
21. Continuation . For testing the hypothesis of independence of the X's, H : Po = PI' against the alternatives K : Po < PI' consider the run test, which rejects H when the total number of runs R = U
20. Runs. Consider a sequence of N dependent trials, and let X; be 1 or 0 as the ith trial is a success or failure. Suppose that the sequence has the Markov property! P{X; = 1IxI , .. ·, xi- d =
19. Positive dependence. Two random variables (X, Y) with c.d.I, F(x, y) are said to be positively quadrant dependent if F(x, y) F(x, oo)F(oo , y) for all x, y .• For the case that (X, Y) takes on
18. Sequential comparison of two binomials. Consider two sequences of binomial trials with probabilities of success PI and P2 respectively, and let P = (P2/q2) -:- (PI/ql)' (i) If a < fJ , no test
16. The UMP unbiased tests of the hypotheses HI' " '' H4 of Theorem 3 are unique if attention is restricted to tests depending on U and the T 's.#!#17. Let X and Y be independently distributed with
15. Continuation. The function 1/14 defined by (16), (18), and (19) is jointly measurable in u and t.[The proof, which otherwise is essentially like that outlined in the preceding problem, requires
14. Measurability of testsof Theorer: 3. The function l/J3 defined by (16) and (17) is jointly measurable in u and t. [With C1 = v and C2 = w, the determining equations for v. W , 'YI ' 'Y2 are (25)
13. Determine whether T is complete for each of the following situations: (i) XI' . .. , X" are independently distributed according to the uniform distribution over the integers 1,2•. . . , 8 and
12. The completeness of the order statistics in Example 6 remains true if the family $' is replaced by the family of all continuous distributions. [To show that for any integrable symmetric function
11. Counterexample. Let X be a random variable taking on the values -1,0.1,2, . . . with probabilities Po{ X = -I} = 8; P8{X= x} = (1- 8)28x, x = 0.1 •. . . . Then gJ = {P8 , 0 < 8 < I} is
10. Let Xi' ''' ' Xm and Y\,... , y" be samples from Na, ( 2 ) and »«. or 2 ) . Then T = (EXi,Dj,[Xi2,D?), which in Example 5 was seen not to be complete, is also not boundedly complete. [Let f(t)
9. Let X\ , .. . , X; be a sample from (i) the normal distribution N(aa, ( 2 ), with a fixed and 0 < a < 00; (ii) the uniform distribution U( °-!,° + !), - 00 < ° < 00 ; (iii) the uniform
8. For testing the hypothesis H: °= 00 (00 an interior point of 0) in the one-parameter exponential family of Section 2, let rc be the totality of tests satisfying (3) and (5) for some - 00 s C\ s
7. Let (X, Y) be distributed according to the exponential family dP8r-82( X, y) = C( °1 , 02)e8.x+82Y d",(x, y). The only unbiased test for testing H : 01 sa, 02 s b against K : 01 > a or 02 > b or
6. Let X and Y be independently distributed according to one-parameter exponential families, so that their joint distribution is given by dP8! . 82 ( x, y) = C( °1) e 8•T(x) d",( x) K( 02) e 82U(
5. Let T,,/O have a X2-distribution with n degrees of freedom. For testing H : °= 1 at level of significance a = .05, find n so large that the power of the UMP unbiased test is .9 against both ° 2
4. Let X have the Poisson distribution P( 'T), and consider the hypothesis H : 'T = 'To . Then condition (6) reduces to To-1 x-C+1 (x-1)! 1-1 -To + (1 - Y) (C - 1)! C-1 eo=1 1-a, provided C >
3. Let X have the binomial distribution b(p, n), and consider the hypothesis H :p = Po at level of significancea. Determine the boundary values of the UMP unbiased test for n = 10 with a = .1, Po =
2. p-values. Consider a family of tests of H : 8 = 80 (or 8 :5; 80 ) , with level-a rejection regions Sa such that (a) PSo{ X E Sa} = a for all 0 < a < 1, and (b) Sao = (Ia>aoSa for all 0 < ao < 1,
1. Admissibility. Any UMP unbiased test 4>0 is admissible in the sense that there cannot exist another test 4>1 which is at least as powerful as 4>0 against all alternatives and more powerful against
53. Let [, g be two probability densities with respect to JL. For testing the hypothesis H: 0 :5 00 or 0 01 (0 < 00 < 0\ < 1) against the alternatives 00 < 0 < 01 in the family 9= (()f( x) + (1 -
52. Let XI" '" X" be independent N( (J, y), 0 < y < 1 known, and Y1, · · · , Y" independent N((J,I). Then X is more informative than Y according to the definition at the end of Section 4.[If V, is
51. Let Xl " ' " X, be i.i.d. with density Po or PI' so that the MP level-a test of H : Po rejects when Il;'= I r( Xi) c,,, where r( X,) = PI ( X,)/Po( Xi)' or equivalently when (34) 1 In O)ogr(x,)
50. (i) For testing Ho : (J = 0 against HI : (J = (JI when X is N((J, 1), given any o < a < 1 and any 0 < 'IT < 1 (in the notation of the preceding problem), there exists (JI and x such that (a) Ho
49. In the notation of Section 2, consider the problem of testing Ho : P = Po against HI : P = PI' and suppose that known probabilities 'ITo = 'IT and 'lT1 = 1 - 'iT can be assigned to Ho and HI
48. Let X be distributed according to P9 ' (J E n, and let T be sufficient for (J. If q>(X) is any test of a hypothesis concerning (J, then "'(T) given by "'(I) = E(q>(X)It] is a test depending on T
47. Let Xl" ' " Xn be a sample from the inverse Gaussian distribution I(p., 1') with density / l' exp( _. _1'(X _ p./), 2'1Tx 3 2xp.2 X> 0, 1', P. > O. Show that there exists a UMP test for testing
46. Consider a single observation X from W(l, c). (i) The family of distributions does not have monotone likelihood ratio in x. (ii) The most powerful test of H : C = 1 against C = 2 rejects when X <
45. A random variable X has the Weibull distribution W(b,c) if its density is C(X) "-l b t e -(x/b l ', X> 0,b, C > O. (i) Show that this defines a probability density. (ii) If Xl"'" Xn is a sample
44. A random variable X has the Pareto distribution P( C, T) if its density is CT'/X" +I,O < T < X, 0
43. Let Xl' . . . ' X" be a sample from the gamma distribution I'(g,b) with density 1 f( g) bKxg -1e - ' Ih, O bo when g is known; (ii) H: g s go against g> go when b is known. In each case give the
42. Let X, be independently distributed as N(itJ., 1), i = 1, .. . , n. Show that there exists a UMP test of H : tJ. .s 0 against K : tJ. > 0, and determine it as explicitly as possible. Note. The
41. Let Xl' . . .' x" be independently distributed, each uniformly over the integers 1,2, . . . , 8. Determine whether there exists a UMP test for testing H : 8 = 80 at level 1/86' against the
40. Let the distribution of X be given by 'xH 0 1 2 3 where 0 < 8 < .1. For testing H : 8 = .05 against 8 > .05 at level a = .05, determine which of the following tests (if any) is UMP : (i) q,(0) =
39. Let Po, PI' P2 be the probability distributions assigning to the integers 1, . . . ,6 the following probabilities: 1 2 3 4 5 6 Po .03 .02 .02 .01 0 .92 PI .06 .05 .08 .02 .01 .78 P2 .09 .05 .12 0
38. Let Xl" ' " Xm ; Yl , ... , y" be independently, normally distributed with means E and 1/, and variances a2 and 'T 2 respectively, and consider the hypothesis H:'T S a against K: a < 'T. (i) If E
37. Let XI, ,, ,,Xm and lj, ... , y" be independent samples from N(tl) and N( 1/,1), and consider the hypothesis H : 1/ S Eagainst K : 1/ > E. There exists a UMP test, and it rejects the hypothesis
36. Sufficient statistics with nuisance parameters. (i) A statistic T is said to be partially sufficient for 8 in the presence of a nuisance parameter 1/ if the parameter space is the direct product
35. Let X and Y be the number of successes in two sets of n binomial trials with probabilities PI and P2 of success. (i) The most powerful test of the hypothesis H: P2 :S PI against an alternative
34. A counterexample. Typically, as a varies the most powerful level-a tests for testing a hypothesis H against a simple alternative are nested in the sense that the associated rejection regions, say
33. Confidence bounds for a median. Let XI" ' " Xn be a sample from a continuous cumulative distribution function F. Let be the unique median of F if it exists, or more generally let = inf{~' : F( 0
32. Let the variables X; (i = 1, . . . , s) be independently distributed with Poisson distribution P( i)' For testing the hypothesis H :'L~j .s a (for example, that the combined radioactivity of a
31. For testing the hypothesis H': 81 s 8 s 82 (81 s 82 ) against the alternatives 8 < 81 or 8 > 82 , or the hypothesis 8 = 80 against the alternatives 8 oF 80 , in an exponential family or more
30. Extension of Theorem 6. The conclusions of Theorem 6 remain valid if the density of a sufficientstatistic T (which without loss of generality will be taken to be X), say PB( X), is STP3 and is
29. STP3 • Let 8 and x be real-valued, and suppose that the probability densities Pe ( x) are such that Pe :( x )jPe ( x) is strictly increasing in x for 8 < 8' . Then the following two conditions

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