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

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' = x + b is normal but the subgroup x' = ax is not. [The defining property of a normal subgroup is
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 G is (sgn x"' XI/X" , . . . , X,, _I/X,,) where sgn X is 1 or -1 as x is positive or negative. (ii)
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) = T(gx2) for all g E G, so that G induces a group G* on ff through g*T(x) = T(gx), and suppose
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 are i.i.d. according to a distribution FE§', with §' the class of all distributions with mean zero
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= 0, there exist UMP unbiased tests of 11/J.L against one- and two-sided alternatives. [Chhikara
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, which rejects when X> C[L(X; + 1/X;»), and a corresponding UMP unbiased "For additional
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 test of H : b2 = bl against one- and two-sided alternatives, which can be based on a beta
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) Most accurate unbiased confidence intervals forb. [The conditional distribution of LX; givennX;, which
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 against g > go and of H' : g = go against s» go, and their rejection regions are based on W =
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 and variance Var( Xi Y; )IVar X; Var K. (ii) The variance of part (i) can take on any value between
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) implies positive quadrant dependence.]
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 + p'TO-1(x - 0 and variance 'T 2(1 - p2). Through addition to such a variable of the positive quantity
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) '/T(n - 3)! k X L r 2 [Hn+ k - 1)] (2pr) k-O k! or alternatively (87) n - 2 I pp(r) = -'/T-(1 -
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 independently distributed of (X, Y), and their joint distribution is the same as that of (f.7:lx:2,
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 unbiased test for testing the hypothesis T/ a = t:J.. Its acceptance region is 1t:J.2 S2 - S21 I 2 C < ,
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 'II -I- (pT/a)(x - n (ii) Let (XI' YI ) , . . . , (X", y,,) be a sample from a bivariate normal
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 {il , ... , if} with (i) r = 2; (ii) r = even; (iii) r divisible by 3. (iv) Give two other examples of
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 permutations T and T' correspond respectively to the subsets R = {i1, . . . , if} and R' = {il, .. .
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 G, and let e be the number of elements in Go . Any element g E Go (except the identity) is
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) = -00, 1(M+I) = 00, then 1 P8[1(i)
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 , .. . , n(.. (ii) Generalize the confidence intervals (72) and (73) to the randomization model of part (i).
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 permutation tests satisfying (83) 1 - L If>(z')=a r z'ES(z) a.e. Section 14
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 of N-space. If i .: 1 (1 ) (81) P" .r(z) = - [ (~ ) NexP -2"2lz - gkrl 2 ,where Izl2 = Ezl, then
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 rectangle, where c and Ii are known. Then Theorem 7 is no longer applicable . However, unbiasedness of 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) I/I( z) for all z. Inductively, sequences of functions 1/11 '1/12' ... and "'0 ' "'I' ... are
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 this out for varying n and confidence coefficient 1 - a = .95 when al = I, a = 2 and when al = 2, a =
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 and experimental effects 0; and V; are independently normally distributed with variances al, a2 and
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 (59). (ii) Determine the most accurate unbiased confidence intervals for l1. [(i): The structure of
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 distribution, which under the hypothesis of no treatment effect is symmetric with respect to the
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 assumptions concerning F, confidence intervals for 6 can be obtained from permutation tests of the
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 (54).
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 p'tl- I .. . p,u,-I, where Ps = 1- (PI + +Ps-I)' Determine the posterior distribution of (Pi' ... '
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 ds = d~ d1J (l/a) do (1/'T) dr. Extend the result of Example 15 to inferences concerning 'T 2/ a2
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 = d~ dT/ . - de a for all - 00 < ~, T/ < 00 , 0
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 Neyman-Pearson fundamental lemma to the problem of minimizing the volume of R.]
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 for p-= ay + b8, where p = p(a, b), p = p(a, b). Then if /1 and /2 are increasing functions,
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) In a 2 X 2 table, for the parameter t:. of Chapter 4, Section 6.
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 , · .. , xn)]' C2 C1
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 where (Jo is the true value of (J) EQ{p,[S(X)]} = 1 Q{(J E S(X)} d(J 9"'90 provided the necessary
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 level-a test of H : ~o with strictly increasing power function {J,.a) depending only on t (ii) Given any
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 0< y < 1 by (no-I(y) dy = y, -L/2~ where ( " 0- 1 denotes the density of the r-distribution with no -
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 r-distribution with m degrees of freedom . (ii) Let XI' X2 , . .. be independently distributed as N(t (12).
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 family of confidence intervals 6(X) ± LI2 of constant length L. Let ~I"' ~2N be such that It - ~jl > L
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 t-statistic is independent of q and hence the same as in the normal case, namely Student's t with n - 1
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 level of the r-test as (12/T2 varies from ° to 00 .
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 of (i) remains valid if X\, .. . , Xn are a sample from any distribution with mean and finite
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 acceptance region CI s 2E[x; - min(xI""'Xn ) ] s C2 , where the test statistic has a X2-distribution with
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 transformation from (Xl •. . .• X"" YI, · · . •Y,,) to (ZI" '" Z", +,,) such that ZI =
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 = 2'
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 = .05. f3 = .9, t:.. = 1.5. 2, 3, and the hypothesis being tested is H : TI a s 1.
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 distributed as N(0. a2). Section 3
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 that ZI = Iii X. [Then n n n n L zl = L zl - zf = L xl - nX2 = L (x, - x(] i - 2 i - I i-I i=1
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 = k or j ,;, k) . The Z's are normally distributed with common variance a2 and means tj = Ea ii~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 ~l ~~? and ~l =~? given by the rejection regions Zl - ~:) n L Z//(n - s) ; - s+1 IZI - r?1 n L
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 known. Determine fJa/o) for n = 5,10,15, a = .05, ~/o = 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and in each
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 noncentrality parameter 6, the probability density of which is* 1 100 I h(t) = 21
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 power of any level a test of H, and let P( 0) denote the power of the most powerful test for testing =
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 O. [If I 1 -2 S = - i..(X; - X) , n - 1 the power in the two-sided case is given by 1- p{_ CS _ In~ s
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) only one of the margins is fixed; (ii) the entries in the 4K cells are independent Poisson variables
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 (12) of Chapter 3 is observed. On the basis of (N, XI' " . , XN ) there exists a UMP unbiased test
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 UMP unbiased test of (i) H: Q2(02) - QI(OI) s c and hence in particular of Q2(02) :::;; Ql(OI); (ii)
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 test for testing H : °= q2/ql :::;; 00 and hence in particular H' :PI s P2 . (ii) Determine the
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 is O={8:8~a} (ii) Show that X(II) is not complete when g = {6: 6 a}, a 2, and find a complete
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 one-sided McNemar test of H :'IT\ = 'lT2 is an increasing function of p. [The conditional 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 completely at random, that is, a random sample of 2n subjects, obtained from a population of size N (2n N),
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) under the assumption r, = r there exists a UMP unbiased test of the hypothesis r s ro for any fixed
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 assumed in bioassay, where X; denotes the dose, or some function of the dose such as its logarithm, of a
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 the same testing problem, without conditioning on Xl there exists a UMP unbiased test if the initial
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 + V is less than a constant C( m) depending on the number m of zeros in the sequence. When R = C(
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 = P{X; = 1lxi-d and the property of stationarity according to which P{X; = I} and P{X; = 1Ixi_l} are

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