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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

8.33. (Sec. 8.10.1) Show that if A cR~ is convex and monotone in majorization, then A* is convex. [Hint: Show where (px + qy) px +qy. 2 = (ZZ)' = Rm.]
8.32. (Sec. 8.10.1) Show that the boundary of A has m-measure O. [Hint: Show that(closure of A) CA U C, where C = {VI U - IT' is singular}.]
8.31. (Sec. 8.10.1) Show that if A and B are positive definite and A - B is positive semidefmite, then B- 1 - A -I is positive semidefinite.
8.30. (Sec. 8.10.1) Admissibility of the Bartlett-Nanda-Pillai trace test. Show that the acceptance region tr X'(ZZ' +XX,)-IX 5; c satisfies the conditions of Theorem 8.10.1.
8.29. (Sec. 8.10.1) Admissibility of the Lawley-Hotelling test. Show that the accep·tance region tr XX'(ZZ,)-I 5; c satisfies the conditions of Theorem 8.10.1.
8.28. (Sec. 8.10.1) Admissibility of the likelihood ratio test. Show that tile acceptance region I ZZ' 1/ I ZZ' + XX' I ~ c satisfies the conditions of Theorem 8.10.1. [Hint:The acceptance region can
8.27. (Sec. 8.10.1) Likelihood ratio test as a Bayes procedure. Let "'I'"'' wm+n be independently normally distributed with covariance matrix I and means coEw;= 'Y;, i = 1, ... , m, ocE",; = 0, i = m
8.26. (Sec. 8.10.1) Bartlett-Nanda-Pillai V-test as a Bayes procedure. Let"'1."'2"'" "'m+n be independently normally distributed with covariance matrix! and means xE",; = 'Y;, i = 1, ... , m, ooE",;
8.25. (Sec. 8.6) Find the cdf of tr HG - I for p = 2. [Hint: Use the distribution of the roots given in Chapter 13.]
8.24. (Sec. 8.6) Interpret the transformations referred to in Theorem 8.6.1 in the original terms; that is, H: PI = P'r and z~l).
8.23. (Sec. 8.9) Let XI be the yield of a process and .1'2 a quality measure. Let ZI = 1, Zz = ± 10° (temperature relative to average) :, = ±O.75 lrelatiw measure of flow of one agent), and Z4 =
8.22. (Sec. 8.9) The Latin square. Let Yij' i, j = 1, ... , r, be distributed according to N(ll-ij' I), where ooEY,'j = Il-ij ='1 + Ai + Vj + Il-k and k = j - i + 1 (mod r) with L:Aj = L:vj = L:ll-k
8.21. (Sec. 8.9) Let Yij (a p-component vector) be distributed according to N(ll-i!, l:)'where ooEYij = Il-ij = Il- + Ai + Vj + "Yij' L:iAi = 0 = L:jVj = L:i"Yij = '"5.. j "Yij; the "Yij are the
8.20. (Sec. 8.6) The multivariate beta density. Let Hand G be independently distributed according to WO:, m) and WC~:, n), respectively. Let C be a matrL\such that CC' = H + G, and letfor Land [- L
8.19. (Sec. 8.6.5) Prove for Ii ~ 0, i = 1, ... , p, P I· P P L 1 ~ I :;:; log n (I + Ii):;:; L Ii'i= 1 ' 1= I ;= I Comment: The inequalities imply an ordering of the value, of the
8.18. (Sec. 8.5) In casep = 3, ql = 4, and n = N - q = 20, find the 50'7
8.17. (Sec. 8.5) Usc the asymptotic expansion of the distribution to compute Prf-k logU3•3•n :$;M*} for(a) n = 8, M* = 14.7,(b) n = 8, M* = 21.7,(c) n = 16, M* = 14.7,(d) n = 16, M* =
8.16. (Sec. 8.4)(a) Show that when p is even, the characteristic function of Y= log Up•m•n , say
8.15. (Sec. 8.4) For p :$; m find ooEUh from the density of G and H. [Hint: Use the fact that the density of K+L:~lViVi' is W(I,s+t} if the density of K is WeI,s) and V], ... ,V; are independently
8.14. (Sec. 8.4) Find PrfU4•4•n ~ u}.
8.13. (Sec. 8.4) Find Pr{U4.3.n ~ u}.
8.12. (Sec. 8.4) Show that the cdf of U3•3•n is[Hint: Use Theorem 8.4.4. The region {O:$; ZI :$; 1,0 :$; Z2 :$; 1, ZfZ2 :$; u} is the union of fO :$;ZI:$; 1,0 :$;Z2:$; u} and fO :$;Zl:$; u/z2, u
8.11. (Sec. 8.4) Prove Lemma 8.4.1 by showing that the density of PIll and P2w is - - K exp[-tr(B2 -B2) A22 (B20 B)']. -
8.10. (Sec. 8.4) By comparing Theorem 8.2.2 and Problem 8.9, prove Lemma 8.4.1.
8.9. (Sec. 8.3) Prove 10 = (124) (z(!). a - = (C-C2AA21)(A1-A2422421):
8.8. (Sec. 8.3) Let ql = qz· How do you test the hypothesis PI = P2?
8.7. (Sec. 8.3) Let Zq" = 1, let qz = 1, and let A* i,j=1,..., 9-9-1. Prove that (in - B)(4u - AzA2'42)(Bin - B) = (Bin - Bi)" (Bin - B).
8.6. (Sec. 8.3) Let q = 2, ZI" = w" (scalar), zz" = 1. Show that the U-statistic for testing the hypothesis PI = 0 is a monotonic function of a T2-statistic, and give the TZ-statistic in a simple
8.5. (Sec. 8.3) In the following data [Woltz, Reid, and Colwell (1948), used by R. L. Anderson and Bancroft (1952») the variables are Xl' rate of cigarette bum;xc' the percentage of nicotine; ZI'
8.~. (Sec. 8.2) Show that ~ minimizes the generalized variance I r. (X"'''~Z'')(X''-~Z,,)'I·(1'=1
8.3. (Sec. 8.2) Prove Theorem 8.2.3.
8.2. (Sec. 8.2) Show that Theorem 3.2.1 is a special case of Theorem 8.2.1.[Hint: Let q = 1, Zo = 1, ~ = JL.)
8.1. (Sec. 8.2.2) Consider the following sample (for N = 8):Let Z20 = 1, and let Zlo be the amount of fertilizer on the ath plot. Estimate P for this sample. Test the hypothesis ~I = 0 at ('Ie 0.01
7.37. (Sec. 7.5) Show that if X~-I and X~-z are independently distributed, then x~· I X~ z is distrihuted ~s (:d'l ,l" /4. [Hint: In the joint density of x = X~-I and y = x.~ _ z substitute z =
7.36. (Sec. 7.2) Dirichlet distribution. Let YI , •.• , Y,n be independently distributed as,\'~-variables with Pi' ... ' Pm degrees of freedom, respectively. Define Z; =YjL7~1}j, i = [, ... ,m.
7.35. Let the density of Y he I(y) = K for y' y 5, P + 2 and 0 elsewhere. Prove that K = r
7.34. (Sec. 7.8) Verify (II).
7.33. (Sec. 7.R) Prove L,/I.,G) is a multiple of (g_(J"Yct>-I(g-U). Hint: Transform so ~ = I. Then show= .~(2I 0) tI 0 I'
7.32. (Sec. 7.8) Prove LiI., G) and LI(I., G) are invariant with respect to transformations G* = CGe', I.* = CI.C' for C nonsingular.
7.31. (Sec. 7.8) Prove for optimal D L(IS) L(I,TDT') = = - =- P -1-(0-2+1)] log (p-1) 108 - (P-21+1)]. i=1 n p even, p odd.
7.30. (Sec. 7.8) Verify (17) and (18). [Hillt: To verify (18) let I = [([(', A"' [(A*[(', and A* = T* T*, where K and T* are lower triangular.]
7.29. (Sec. 7.8) Prove for p = 2 TDT d,A+(d-d) |A| all
7.28. (Sec. 7.8) Prove that Fe is not proportional to f by calculating FE.
7.27. (Sec. 7.7) Multivariate t-distributioll. Let y and u be independently distributed according to N(O, I) and the x;-distribution, respectively, and let .;nTri y = xJ.l..(a) Show that the density
7.26. (Sec. 7.6) Show that when I is diagonal the set 'ij are pairwise independent.
1.25. (Sec. 7.6) Show that when p = 3 and I is diagonal '12' '13' '23 are not mutually independent.
1.24. (Sec. 7.6) Reverse the steps in Problem 7.20 to derive (9) of Section 7.6.
r.23. (Sec. 7.6) Prove the conclusion of Problem 7.20 by using Problems 7.21 and 7.22.
'.22. (Sec. 7.6) Prove (without the use of Problems 7.19 and 7.20) that if I = I. then the set 'II', ... ,',,-I,p is independent of the set 'ji'p' i,j = l. .... p _. I. [Hillt:From Section 4.3.2 app'
.21. (Sec. 7.6) Prove (without the use of Problem 7.20) that if I = I, then'Ip"'" 'p-l, P are independently distributed. [Hint: 'jp = ajp/(,;a;; ,;a;;). Prove that the pairs (alp' all)' ... ,(ap-l,p,
7.20. (Sec. 7.6) Prove that the joint density of '12·3 ..... p"13.4 •..• p"2J.4 .... ,p' ... ,'Ip"""p-I.p is r{ Un - (p - 2)]} (1- 2 )±In-
7.19. (Sec. 7.6) Prove that if I. = 1, the joint density of 'ij.p' i, j = 1, ... , P - 1, and'Ip"""p-I,p is [(n-1)]R (p-1) p-1 (P-1)(p-2)/[(n-i)] - where R11p (p). [Hint: Tip=(rij 1. Use (9).] (in) I
7.1S. (Sec. 7.5) Consider the confidence region for .... given bywhere i and S are based on a sample of N from N( .... , I.). Find the expected value of the volume of the confidence region. N(*)'S()
7.17. (Sec.7.5) Find tS'IAlh directly from W(I.,n). [Hint: The fact that jw(AII.,n)dA == 1 showsas an identity in n.] 14-exp(tr)=2(n)
7.16. (Sec. 7.4) Verify that Theorem 7.4.1 follows from Lemma 7.4.1. [Hint: Prove that Qj having the distribution W(I.,,) implies the existence of (6) where 1 is of order 'j and that the independence
7.15. (Sec. 7.4) Show that tC ( XN2- I XN2-2) " = tC (X22 N.,4/4 )" , h ~O, by use of the duplication formula for the gamma function; X~ _I and X~- 2 are independent. Hence show that the distribution
7.14. (Sec. 7.4) Let Xu be an obseIVation from NCJ3z,,, I), a = I, ... , N, where za is a seal,,£. Let b = LaZaXa/LaZ';. Use Theorem 7.4.1 to show that [axax~bb'["z,; and bb' are independent.
7.13. (Sec. 7.3) Let ZI' ... ' Z" be independently distributed, each according to N(O, I). Let W = [:. Il- I ball Za Z~. Prove that if a' Wa = X~ for all a such that a' a = 1, then W is distributed
7.12. (Sec. 7.3.1) Find the first two moments of the elements of A by differentiating the characteristic function (11).
7.11. (Sec. 7.3.2) Prove Theorem 7.3.2 by use of characteristic functions.
7.10. (Sec. 7.3) Find the characteristic function of A from WeI, n). [Hint: From[w(AI I, n) dA = as an identity in ~.] Note that comparison of this result with that of Section 7.3.1 is a proof of the
7.9. (Sec. 7.2) The complex Wishart distribution. Let WI' ... ' w" be independently distributed, each according to the complex normal distribution with mean 0 and covariance matrix P. (See Problem
7.S. (Sec. 7.2) Independence of estimators of the parameters of the complex normal distribution. Let ZI' ... ' z,v be N obseIVations from the complex normal distribution with mean IJ and covariance
7.7. (Sec. 7.2) Use the proof of Theorem 7.2.1 to demonstrate Pr{IAI = O} = O.
7.6. (Sec. 7.2) Use (9) of Section 7.6 to derive the distribution of A.
7.5. (Sec. 7.2) X2-distribution. Use Problem 7.4 to show that if Yl'"'' Yn are independe?tly dist.ribut~d, each according to N(O, 1), then U = r.:= 1 Y; has the density U,"-Ie- ,Uj[2,nntn)], which is
7.4. (Sec. 7.2) Use Problems 7.1, 7.2, and 7.3 to prove that if the density of y' =yp".,y,,) is f(y'y), then the density of u = y'y is ~C(n)f(u)ul"-I.
7.3. (Sec. 7.2) Use Problems 7.1 and 7.2 to prove that the swface area of a sphere of unit radius in n dimensions is C(n) = 2 (n)
7.2. (Sec. 7.2) Prove that[Hint: Let cos2 (I = u, and use the definition of B(p, q).] S cosh-ede- -/2 r(h)r() T[(h+1)]
7.1. (Sec. 7.2) A transfonnation from rectangular to polar coordinates is(a) Prove w 2 = r.y~. [Hint: Compute in turn Y; + Y;_I'(Y;+Y;_I) +Y;-2' and so forth.](b) Show that the Jacobian is w" -I
6.20. (Sec. 6.5) Verify (33).
6.19. (Sec. 6.8) Let x\j), ... ,x~) be observations from NCJL(i),I.), i= 1,2,3, and let x be an observation to b~ classified. Give explicitly the maximum likelihood rule.
6.18. (Sec. 6.10) Show that b' x = c is the equation of a plane that is tangent to an ellipsoid of constant density of 7T, and to an ellipsoid of constant density of 7T2 at a common point.
6.17. (Sec. 6.8) In Section 8.8 data are given on samples from four populations of skulls. Consider the first two measurements and the first three sample>.Construct the classification functions
6.16. (Sec. 6.8) Let 7Ti be N(fL(i),l:), i = L ... , m. If the fL(i) arc on a line (i.e ..fLU) = fL + v i l3), show that for admissible procedures the Ri are defined by parallel planes. Thus show
6.15. (Sec. 6.6.2) Show (Z-D Pr{ Z - D su m) - Pr{ Z- su|Ti} - = 0 (u) { 2N [u + ( p 3)u Au + pA] A - - + 2 N [4 + 2 Au + ( p 3 + A )uA+pA] 1 + [34 + 4 Au + (2p 3 + A )u + 2 (p-1 1)A]} +
6.14. (Sec. 6.6) Show IC D2 is (4). [Hint: Let l: = I and show that IC(S -11:l = I) =[n/(n - p - 1)]1.]
6.13. (Sec. 6.6) Show that the derivative of (2) to terms of order 11- I is p- -(a){+[+- -(){ + [+ 4
6.12. (Sec. 6.5) Consider d'x(i). Prove that the ratio (d'F()-d'()) N N (d'x)-d'x) + (d'x) d') - a=1 a=1
6.11. (Sec. 6.5) Show that the elements of M are invariant under the transformation(34) and that any function of the sufficient statistics that is invariant is a function of M.
6.10. (Sec. 6.5.1) Show that the probabilities of misclassification of Xl •... ' x'" (all assumed to be from either 7T I or 7T 2) decrease as N increases.
6.9. (Sec. 6.5.1) Let W(x) be the classification criterion given by (2). Show that the T2-criterion for testing N(fL(I), l:) = N(fL(2),l:) is proportional to W(ill,) and W(i(2».
6.S. (Sec. 6.5.1) Find the criterion for classifying irises as Iris selosa or Iris versicolor on the basis of data given in Section 5.3.4. Classify a random sample of 5 Iris virginica in Table 3.4.
6.7. (Sec. 6.4) Let x' = (x(1)', X(2),). Using Problem S.23 and Problem 6.6, prove that the class of classification procedures based on x is uniformly as good as the class of procedures based on x(l).
6.6. (Sec. 6.4) Let P(21!) and POI2) be defined by (14) and (5). Prove if- ~~2 < C < ~~z, then P(21!) and POI2) are decreasing functions of ~.
6.5. (Sec. 6.3) When p(x) = n(xllL, ~) find the best test of IL = 0 against IL = IL*at significance level 8. Show that this test is uniformly most powerful against all alternatives fL = CfL*, C > O.
6.4. (Sec. 6.3) The Neymull-Peursoll!ulldumelllullemmu states that of all tests at a given significance level of the null hvpothesis that x is drawn from Pl(X)agaimt alternative that it is drawn from
6.3. ~Sec. 6.3) Prove that if the class of admissible procedures is complete, it is minimal complete.
6.2. (Sec. 6.3) Prove that every complete class of procedures includes the class of admissible procedures.
6.1. (Sec. 6.3) Let 1Ti be N(IL, I.), i = 1,2. Find the form of the admissible dassification procedures.
5.27. (Sec. 5.2) Prove (25) is the density of V = x;:j( xl + xt). [Hint: In the joint density of U = xl and W = xt make the transformation u = uw(1 - u)-l, w = w and integrate out w.j
5.26. (Sec. 5.5) Suppose x~g) is an observation from N(jJ.181,::£g). a = 1. .... Ny.g= 1, ... ,q. (a) Show that the hypothesis .... (l) = ... = .... (q) is equivalent to t!y~i) = 0, i = I, ...• q
5.25. Find the distribution of the criterion in the preceding problem under the null hypothesis.
5.24. Let Xli)' = (Yli )', Z(i)'), i = 1,2, where y(i) has p components and Zlil has q components, be distributed according to N(jJ.(i), ::£), whereFind the likelihood ratio criterion (or equivalent
5.23. (Sec. 5.4) LetProve jJ.'::£ -1jJ. ~ jJ.l l),::£ 111jJ.(I). Give a condition for strict inequality to hold.[Hint: This is the vector analog of Problem 5.21.] --(2) -(%)
5.22. (Sec. 5.3)(a) Using the data of Section 5.3.4, test the hypothesis 1-'-\1) = 1-'-(1 2).(b) Test the hypothesis I-'-V) = I-'-(?).I-'-~) = I-'-S2).
5.21. (Sec. 5.4) Prove that jJ.,::£-IjJ. is larger for jJ.' = (1-'-1,1-'-2) than for jJ. = 1-'-1 by verifyingDiscuss the power of the test 1-'-1 = 0 compared to the power of the test 1-'-1 = 0,
5.20. (Sec. 5.3) Let x~) be observations from N(jJ.(i), ::£,), a = 1, ... , Ni• i = 1.2. Find the likelihood ratio criterion for testing the hy')othesis jJ.(l} = jJ.(2).
5.19. (Sec. 5.3) Let x and S be based on N observations from N(fl., ::£), and let x be an additional observation from N(fl., ::£). Show that x - x is distributed according to
5.18. (Sec. 5.6.2) Prove that if the set A is convex, then the closure of A is convex.

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