Good Exact Confidence Sets for a Multivariate Normal Mean

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1 University of Pennsylvania ScholarlyCommons Statistics Papers Wharton Faculty Research 997 Good Exact Confidence Sets for a Multivariate Normal Mean Yu-Ling Tseng Lawrence D. Brown University of Pennsylvania Follow this and additional works at: Part of the Probability Commons Recommended Citation Tseng, Y., & Brown, L. D. (997). Good Exact Confidence Sets for a Multivariate Normal Mean. The Annals of Statistics, 5 (5), This paper is posted at ScholarlyCommons. For more information, please contact repository@pobox.upenn.edu.

2 Good Exact Confidence Sets for a Multivariate Normal Mean Abstract A class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudo-empirical-bayes construction. When the dimension is greater than, by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region. Sufficient conditions for the connectedness of the proposed confidence sets are also derived. In addition, our confidence sets could be used to construct tests for point null hypotheses. The resultant tests have convex acceptance regions and hence are admissible by Birnbaum. Tabular results of the comparison between the proposed region and other confidence sets are also given. Keywords multivariate normal mean, volume, coverage probability, confidence sets, James-Stein estimator, pseudoempirical-bayes construction Disciplines Probability This journal article is available at ScholarlyCommons:

3 The Annals of Statistics 997, Vol. 5, No. 5, 858 GOOD EXACT CONFIDENCE SETS FOR A MULTIVARIATE NORMAL MEAN BY YU-LING TSENG AND LAWRENCE D. BROWN Academia Sinica, Taipei and University of Pennsylvania A class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudoempirical-bayes construction. When the dimension is greater than, by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region. Sufficient conditions for the connectedness of the proposed confidence sets are also derived. In addition, our confidence sets could be used to construct tests for point null hypotheses. The resultant tests have convex acceptance regions and hence are admissible by Birnbaum. Tabular results of the comparison between the proposed region and other confidence sets are also given.. Introduction. One of the most frequently used statistical techniques is the linear model, which includes both the analysis of variance and linear regression as special cases. In the usual formulation of a linear model, the estimation problem can be reduced to that of estimating a multivariate normal mean. See, e.g., Hwang and Chen Ž In such a situation, the results of Stein Ž 956. and James and Stein Ž 96. lead immediately to a uniform, appreciable improvement in mean squared error over the least squares estimator when there are at least three parameters. Surprisingly, this phenomenon is not exceptional. Brown Ž 966. showed the same inadmissibility result for the best invariant estimator of location for a very wide variety of distributions and loss functions. In particular, his results implied that when the dimension is at least 3 the usual confidence region for the parameters in a linear model is inadmissible too. Since then considerable research has aimed at explicit constructions of dominating estimators and improved confidence sets for the mean vector of a multivariate normal distribution. Let X Ž X,..., X. p have a p-variate normal distribution with mean vector Ž,...,. p and identity covariance matrix I. There have been many breakthroughs in the theory of estimating. References to related Received April 995; revised January 997. This is a revised version of Dr. Tseng s thesis presented to Cornell University in partial fulfillment of the requirements for a Ph.D. degree. Research supported in part by NSF Grant DMS AMS 99 subject classifications. Primary 6F5; secondary 6C, 6C5. Key words and phrases. Multivariate normal mean, volume, coverage probability, confidence sets, JamesStein estimator, Stein-type estimator, pseudo-empirical-bayes construction. 8

4 GOOD EXACT CONFIDENCE SETS 9 works can be found in, for example, Faith Ž 976., Berger Ž 985. and Tseng Ž In contrast to the rich development in point estimation, until recently there has been comparatively little research on finding improved confidence sets for. The usual confidence set for is 4 Ž. C Ž X. : X c, where c is the Ž. th percentile of the chi-squared distribution with p degrees of freedom and is the Euclidean norm. It is an exact confidence set for, that is, P C Ž X., and it is a good confidence set estimator under several criteria. When p or, Brown Ž 966. and Joshi Ž 969. independently showed that under criterion Ž 6. Ž see below. C Ž X. is strongly admissible among all confidence sets. Stein Ž 96. showed that for all dimensions C Ž X. has the minimum volume among all level confidence sets which are invariant under the group of translations in R p, and it is minimax, that is, it satisfies Ž. Ž. sup E Vol C X inf sup E Vol C X, CC where C is the class of all confidence sets and VolŽ C. for any set C in R p is defined as H p VolŽ C. IŽ t C. Ž dt. R p with the Lebesgue measure in R. Stein Ž 96. also proved that the usual confidence set cannot be uniformly dominated in false coverage probability by any confidence sets. However, Stein Ž 96. also gave heuristic arguments claiming that the confidence sets associated with the now well known JamesStein estimators improve upon C Ž X. for large dimensions and conjectured that the same result holds for all p 3. It was independently proved in Brown Ž 966. and Joshi Ž 967. that, where the dimension is 3 or more, confidence spheres centered at a Stein-type estimator and having the same volume as C Ž x. for all x can have higher coverage probability than C Ž X. for all. They did not, however, provide the explicit form of better confidence regions. For about a decade since the work of Brown Ž 966. and Joshi Ž 967., no significant progress was made in finding specific improved confidence sets for a multivariate normal mean. This is not because the problem is not statistically important, but rather because it involves great technical difficulty. Fortunately, better understanding of Stein phenomena and far greater computer facilities in the past two decades have made some significant breakthroughs in set estimation possible. The literature on this development includes Faith Ž 976., Stein Ž 98., Berger Ž 98., Hwang and Casella Ž98, 984., Casella and Hwang Ž 983, 986. and Shinozaki Ž 989..

5 3 Y.-L. TSENG AND L. D. BROWN Faith Ž 976. developed Bayesian confidence regions and gave convincing numerical and theoretical arguments that they improve upon C Ž X. when the dimension is 3 or 5. Unfortunately, his regions had complicated shapes due to their Bayesian derivation. Stein Ž 98. gave basic formulas for unbiased estimation of the risk of an arbitrary point estimator and utilized them to suggest some approximate confidence sets. However, the validity of the approximate confidence regions and their properties were not further studied. Berger Ž 98. considered confidence ellipsoids associated with an admissible generalized Bayes estimator for the mean vector. He derived necessary and sufficient conditions under which his sets have uniformly smaller volume than the usual confidence region. Although uniform dominance results in coverage probability were not obtained, he gave asymptotic theorems and convincing numerical evidence that his sets maintain satisfactory coverage probabilities. A major problem with this confidence set estimation is in its implementation. It involves complicated calculations for the generalized Bayes estimator and the inverse of a posterior covariance matrix. Following in the spirit of Stein Ž 96., Hwang and Casella Ž 98, 984. successfully showed that recentering the usual confidence set to the positive part of a Stein estimator results in uniform improvement in coverage probability. The recentered sets have the same volume as the usual one, though. Casella and Hwang Ž 983, 986. considered recentered sets with variable radii by empirical Bayes arguments. Although they provided strong numerical results to support the superiority of their sets, analytical dominance results were not obtained. Shinozaki Ž 989. provided a class of confidence sets and showed that some of them have smaller volume than C Ž X. and have the same confidence coefficient as C Ž X.. The sets were constructed by shrinking the boundary points of C Ž X. toward the origin according to some functions which were not explicitly defined. The implicit functions used in the construction made the study concerning the geometry of the confidence sets and their associated acceptance regions extremely difficult and related questions remain unsettled. These are relevant because understanding the geometry of the associated acceptance regions of confidence sets is very important if the primary interest is in hypothesis testing. Our principal goal here is to construct confidence regions which have smaller volume than C Ž X. while retaining the constant coverage probability. Besides their theoretical importance, these kinds of improved confidence sets are more desirable in practice. Intuitively, smaller sets cover fewer points and, hence, are less likely to include false values. In other words, with the same preset acceptable confidence coefficient, sets with smaller volume provide higher precision for the practitioners. In fact, Ghosh Ž 96. and Pratt Ž 96. showed that the expected volume of a confidence region is equal to an integrated sum of its false coverage probabilities, now known as the GhoshPratt identity. This identity was used in Cohen and Strawderman Ž 973. and Brown, Casella and Hwang Ž 995. in establishing volume optimality results for certain problems.

6 GOOD EXACT CONFIDENCE SETS 3 In Section a class of exact confidence sets, denoted by C* Ž X., for is proposed through a pseudo-empirical-bayes construction. The sets have an unfamiliar shape, somewhat like an egg Ž see Figure, in Section 4.. Nevertheless they give spherical, hence, convex acceptance regions. As a result, they are associated with admissible tests. By its construction, the volume of C* Ž X. depends on X only through its Euclidean norm X. In Section 3 we calculate the asymptotic volume difference, as x tends to infinity, between C Ž x. and C* Ž x. and give conditions such that C* Ž x. has a smaller asymptotic volume. In Section 4 we derive sufficient conditions under which C* Ž x. is connected for all x. Theorem 5.3 in Section 5 analytically establishes a value of x after which the volume of C* Ž x. is always smaller than that of C Ž x.. With this theorem and with help from Section 4, we are able to provide, in Section 5, a computer-aided proof for the uniform dominance of C* Ž X. over C Ž X.. Finally, tabular results of the comparison between C* Ž X. and other confidence sets are given in Section 6.. Proposed confidence sets. If an improved confidence region is constructed by recentering C Ž X. at a Stein-type estimator, hence keeping the same volume, one can view the improvement as due to the effect of moving the usual confidence set toward the origin. This results in a uniform improvement in coverage probability with largest improvement near the origin in the parameter space. In order to maintain the same constant coverage probability while decreasing the volume, the same idea of a shrinkage effect is introduced, however, in a different fashion. For convenience, we first present the form of our confidence sets, C* Ž X.. Some immediate properties of C* Ž X. and the motivation for its construction are given in Sections. and., respectively. For any nonnegative number, let cž. be the Ž. th percentile of a noncentral chi-squared distribution with p degrees of freedom and noncentrality parameter. The proposed class of confidence sets for is ½ 5 Ž. C* Ž X. : X Ž t. cž., where Ž t. Ž A Bt., t, t Ž t. and A, B are positive constants whose values will be specified later... Coverage probability and associated acceptance regions. With this definition, automatically we have P Ž C* Ž X.. ; that is, like C Ž X., C* Ž X. is also an exact confidence set for. Another important feature of C* Ž X. is the level- acceptance regions, denoted by A* Ž., for testing the hypotheses H : versus H :

7 3 Y.-L. TSENG AND L. D. BROWN as given through the usual correspondence C* Ž x. iff x A* Ž.. For any fixed 4 A* Ž. x: x Ž t. cž., which is a p-dimensional ball which is centered at Ž Ž t.. and with radius cž., hence a convex set. The results of Birnbaum Ž 955. showed that for this testing problem a necessary and sufficient condition for a test to be admissible is that it have a convex acceptance region. Hence we have the following theorem. THEOREM.. Fix. For testing H : versus H :, the level- test with A* Ž. as its acceptance region is admissible... Motivation, the pseudo-empirical-bayes construction. The following theorem in Brown, Casella and Hwang Ž 995. motivates the construction of the improved confidence sets to be considered. It should be noticed that the proof of the BrownCasellaHwang theorem is essentially based on the GhoshPratt identity Ghosh Ž 96.; Pratt Ž 96.. THEOREM. Brown, Casella and Hwang Ž Let X fž x., Ž. and define the set C Ž x. to be 5 m Ž x. C Ž x. ½ : kž., f Ž x. where m Ž x. H fž x. Ž. d and kž. is chosen such that P C Ž X.. Then C Ž x. minimizes the expected Bayesian volume among all confidence sets. Notice from their proof that the confidence set with smallest expected volume under the chosen prior is found by inverting a class of most powerful tests of Ž 3. H : X f Ž x. versus H : X m Ž x., where m Ž x. is the marginal distribution of X under the prior. For the normal case X N Ž, I. consider N Ž, I. p p with some posi- tive constant ; that is, and p Ž. ž / ž / f x exp x p ž / ž / Ž. exp.

8 GOOD EXACT CONFIDENCE SETS 33 The marginal distribution m Ž x. of X is N Ž, Ž. I. p. Most powerful tests of Ž 3. accept H iff m Ž x. some constant f Ž x. p Ž Ž.. expž x. p Ž. exp Ž. x ž / ž 4 / x c. some constant Hence the confidence set which minimizes the expected Bayesian volume under is ž / ž 4 / ½ 5 Ž 4. C Ž x. : x c. We have been treating as a constant so far, but to utilize Ž 4. to produce confidence sets which have the potential to improve upon C Ž X. in terms of 6, needs to vary with either x or. Since we are aiming at constructing confidence sets which have coverage probability exactly and smaller volume than the usual confidence set, we need to replace by some function of instead of a function of x. In the usual empirical Bayes approach, hyperparameters are typically estimated from the observation x. As noted above we will instead use a function of the parameter to replace the hyperparameter. We call this kind of approach a pseudo-empirical-bayes construction. An intuitive justification for this type of approach is that is more directly related to than x. Earlier examples in which replacement of a nuisance parameter by a function of key parameters leads to better confidence sets than those using a function of x can be found in Hwang Ž 995. and Huwang Ž 99, Since C Ž X. is minimax in terms of expected volume, any confidence set having smaller volume than C Ž x. for all x is minimax itself. A minimax procedure tries to do as well as possible in the worst case. One might expect that minimax procedures would be Bayes with least favorable prior distributions. Along this line, to find reasonable functions of to replace, let us revisit the previous testing problem Ž 3.. Note that, under H, X N Ž, I. hence p x EŽ X. p Ž Ž.. and, under H, X N, I hence p x EŽ X. p pr. For fixed, if we equate p and p p we make the hypotheses of Ž 3. the most difficult to be distinguished, hence the resulting close to being

9 34 Y.-L. TSENG AND L. D. BROWN the least favorable distribution. This motivates the choice which is equivalent to p p p, Ž 5.. p As an added motivation, note that p is an unbiased estimator of since N Ž, I. p. Equation 5 motivates a particular linear function of as a reasonable replacement for. In this work, we obtain added flexibility by using A B with some positive constants A and B to replace in Ž 4.; note that Ž 5. corresponds to A, B p. This defines our new class of confidence sets for. 3. Asymptotic domination theorems. We consider one confidence set to be better than another if it has higher coverage probability at the same time it has no larger volume for every given observation. It was pointed out by Joshi Ž 969., however, that there is a technical difficulty to be avoided: by adding measure-zero sets to any confidence set C it is possible to increase its coverage probability without any increment in its volume. We should, therefore, compare confidence set estimators across different equivalence classes. Two confidence sets C Ž X. and C Ž X. are said to be equivalent if Vol C Ž x. C Ž x. x, where C Ž x. C Ž x. is the symmetric difference set of C Ž x. and C Ž x.. Let C be the equivalence class containing the usual confidence set C Ž X.. We say a confidence set C X dominates C X if C X C and the following conditions are satisfied: Ž 6. Ž i. P Ž CŽ X.. P Ž C Ž X.. Ž ii. VolŽ CŽ x.. VolŽ C Ž x.. x, with strict inequality either in Ž 6.Ž i. for some or in Ž 6.Ž ii. for all x in some set with positive Lebesgue measure. Since C Ž X. and C* Ž X. have the same constant coverage probability, to see if, for some choices of A and B, C* Ž X. dominates C Ž X. under criteria Ž 6., we need to calculate the volume of C* Ž x. for all x. In this section we calculate the asymptotic volume difference between C Ž x. and C* Ž x., and we give the range for choosing A and B such that C* Ž x. has a smaller asymptotic volume. To understand better the nature of C* Ž X. and the results of the asymptotic theorems to be presented, we first provide the following remark to clarify the relationship between C* Ž X. and the usual JamesStein estimator.

10 GOOD EXACT CONFIDENCE SETS 35 REMARK 3.. For large x, C* x can be related to the recentered set where 4 : * Ž x. c, ž / B a * Ž x. x and a, Aa x by the following approximating arguments. First note that x Ž. x Ž x x. Ž. x Ž x. * Ž x. since, as x, Also it is obvious that and Ž x.. A B x lim cž. c. x Hence C* Ž x. : * Ž x. c4 for large x. It is interesting to note that B a Ž p.. Ž p. Since the condition on a is necessary and sufficient for the domination of * Ž X. over X as point estimators, it is not surprising to see in the following theorems that this same condition on B implies asymptotic dominance of C* Ž X. over C Ž X.. The following lemma is useful in proving the asymptotic theorems and several other proofs in this paper. For its proof readers are referred to Tseng Ž LEMMA 3.. For any given, let c be the th percentile of a noncentral chi-squared distribution with p degrees of freedom and noncentrality parameter. Then d f cž. Ž 7. cž. d f cž. cž. Ž 8., p p, p, where f Ž. m, is the p.d. f. of a noncentral chi-squared distribution with m degrees of freedom and noncentrality parameter.

11 36 Y.-L. TSENG AND L. D. BROWN The following two theorems state ways to select the positive constants A and B in the proposed confidence set C* Ž X. such that it dominates the usual one in volume asymptotically. The proof for Theorem 3. is given in the Appendix and that for Theorem 3. is similar and hence omitted. For simplicity, let Ž x. VolŽC Ž x.. VolŽC* Ž x.. and define constants p3 j p k Ł H sin Ž. d and k p H sin Ž. d j in our presentation hereafter. THEOREM 3.. Suppose A and B in C* Ž X.. Then Vol C Ž x. x Ž x. BŽ p. 4 ož. as x, B Ž. p where Vol C x k k p c p, which is independent of x. If in addition p 3 and B Ž p., then C* Ž x. has an asymptotically smaller volume than that of C Ž x. in the sense that THEOREM 3.. lim x Ž x.. x If A and B in C* Ž X., then Vol C Ž x. A B x Ž x. BŽ p. 4 ož. B If in addition p 3 and B p, then lim A B x Ž x.. AB x as A B x. REMARK 3.. The second theorem says that if B is fixed such that B Ž p., we can find a constant A large enough that the proposed confidence set C* Ž X. dominates the set C Ž X. in volume, uniformly in x, if A A. In what follows we construct a moderate value of A under which C* Ž X. dominates C Ž X.. In this way C* Ž X. is significantly smaller than C Ž X. when X is not large. Ž See Section The geometry of C* ( x ). In the normality setting, one property that should reasonably be required of a confidence region is that it be connected. In this section, a sufficient condition under which C* Ž x. is connected for all x is derived. We are aware of the possibility of improvement upon the sufficient conditions to be presented below. However, numerical results show that they are reasonably good in practice, so we do not pursue further in this direction. The following theorem gives conditions under which C* Ž x. is a connected set for all x. The proof for this theorem is long and is given in the Appendix.

12 GOOD EXACT CONFIDENCE SETS 37 THEOREM 4.. The set C* X is connected if A and B are chosen such that A.5, B and where b 4 AB. A Ž A 3B. ½ B 5 ' c b p c min B 3 A B,, To understand the geometry of C* Ž X. better, we present in the following some interesting properties concerning its shape. It is clear that the shape of C* Ž x. depends on x only through x. Thus, to graph the shape of C* Ž x. at a given x, we identify the x-axis with the observation x and vary, the angle between x and x from to ; then Ž 9. will give a sector of C* Ž x.. The rest of the set is then generated by rotating the sector about the x-axis. Let P be the two-dimensional plane with x as one of its axes. The intersection of P and the boundary of C* Ž x., and also that of C Ž x., is graphed in Figure for various values of x when p 3 and.5 with A and B.5. Note that in this case b.5, c and cž b It is then easy to see that conditions for the connectedness of C* Ž x. for all x hold for this choice of A and B. In fact, C* Ž x. appears to also be convex in Figure. Note that when x, C* Ž x. is a smaller p-sphere contained in C Ž x., and for small to moderate values of x they are more like egg-shaped sets. As x gets large C* Ž x. becomes more like a p-sphere again; in fact as x tends to infinity C* Ž x. and C Ž x. tend to coincide. It is interesting to see that, in the direction of x, C* Ž x. is wider than C Ž x., while it is narrower than C Ž x. in the direction which is perpendicular to x. This is a desirable property for a confidence region of multivariate normal means, as is explained in Berger Ž 98., page 735. Another important result we have on the geometry of C* Ž x. is the following lemma, which is used in proving the uniform dominance of C* Ž X. over C Ž X. in Section 5. The lemma shows that the radius of the set C* Ž x. increases as the angle between x and x increases from to. For fixed,, let Ž. denote this radius; that is, solves Ž 9. Ž D cosž.. cž.. LEMMA 4.. For p 3, suppose A and B are selected such that B p, A, b c and the following inequalities hold: A BcŽ b. B' bcž b., 4 A cž b. A BcŽ b., p cž b. A 3 BcŽ b. B' bcž b. p 4 A

13 38 Y.-L. TSENG AND L. D. BROWN FIG.. The boundaries of C* Ž x. Ž solid line. and C Ž x. Ž dashed line. on the two-dimensional plane with x as one of its axes, when p 3 and.5 with A and B.5 in C* Ž X.. and Then B ž / c pž p b. Ž p. A cž b.. p bc BpŽ p b. Ž p. is a nondecreasing function of. See Tseng and Brown Ž 995. for the proof, which also shows that Ž. is increasing when A. Table gives the values of A and B satisfying the conditions of Theorem 4. and Lemma 4. for given p and. More remarks about the conditions are in order before we leave this section.

14 GOOD EXACT CONFIDENCE SETS 39 TABLE Values of A satisfying conditions of Theorem 4. Ž. and Lemma 4. Ž. for B Ž p. and B ŽŽ p.. ( ) ( ( )) B p B p p * * * REMARK 4.. To understand the conditions of Theorem 4. and Lemma 4. better, note that for fixed p, and B these conditions hold for all A A if they hold when A A. To see why, note that when B is fixed b Ž 4 AB. is a decreasing function of A. Hence cž b. is also decreasing in A. However, ' A ½ Ž A 3B. B 5 p c min B 3 A B, is an increasing function of A. Also note that all the left-hand sides of the four inequalities in Lemma 4. increase with A, while the right-hand sides decrease with A. REMARK 4.. It is interesting to see how the volume of C* Ž x. changes with various choices for the constants A and B. In Figure plots of the volume of C* Ž x. against that of C Ž x. are given for p 3 with several different values of A and B. The improvement in volume for small x is substantial when both A and B are small. The largest improvement in

15 4 Y.-L. TSENG AND L. D. BROWN FIG.. Plot of the volume of C* Ž x. Ž solid line. and that of C Ž x. Ž dashed line. when.5 and p 3, with various choices of A and B in C* Ž X.. volume is when x for all A and B. It is easy to see that, for a fixed A Ž B., smaller B Ž A. gives better performance for small x, at the minor price of slightly worse performance for large x. In fact, this is as expected if we recall that the variance parameter in the normal prior we used was replaced by A B in the construction of C* Ž X.. In the graph having B.4, which is smaller than Ž p..5, it is interesting to see that, even though the improvement at x is larger than that when B.5, C* Ž x. has larger volume than C Ž x. at moderate values of x. This is not surprising in view of Theorem 3. or Corollary 5.. From the two remarks above, for fixed B we will choose the smallest value of A which satisfies the required conditions in Theorem 4. and Lemma 4. in later development of this paper.

16 GOOD EXACT CONFIDENCE SETS 4 REMARK 4.3. Easy calculations show that the conditions of Theorem 4. imply the set C* x is a p-sphere, a convex set, when x. REMARK 4.4. It is interesting to note that the condition b c implies that x C* Ž x. for all x. Hence, for given p and, if the constants A and B are chosen such that b Ž 4 AB. c, then for all x the set C* Ž x. is not an empty set. 5. Uniform domination. We have seen in Section that C* Ž X. has a constant coverage probability at any given value. To show it dominates C Ž X. under Ž 6. one needs to prove that C* Ž x. has smaller volume than C Ž x. for all x in some set with positive Lebesgue measure. In this section, we present an algorithm for a computer aided proof that C* Ž x. has uniformly smaller volume than C Ž x.. First we describe the analytical results which are needed in the algorithm. To simplify the presentation hereafter, for given, and D x we use the following notation in this section: VŽ x. VolŽ C* Ž x.., V VolŽ C Ž x.., d JŽ D. VolŽ C* Ž x.., dd V Ž x. the computer output for Vol C* Ž x., c VŽ x., V, c the computer output for Vol C Ž x. V, where denotes the generic error term due to numerical integrations. Also Ž. let D be the solving the equality Ž. Ž D cos. cž. and d Ž D. Ž D.. dd We use the following results to prove the uniform dominance of C* Ž X. over C Ž X.: Ž i. at x, VolŽC* Ž.. VolŽC Ž.. Ž see Theorem 5..; Ž ii. a uniform upper bound U for JŽ D. Ž see Theorem 5... This number U gives, by the mean value theorem, the worst VŽ x. in a neighborhood of any given x one can possibly have; that is, we have V x V x U D D,

17 4 Y.-L. TSENG AND L. D. BROWN where D x and D x. Hence the worst value for VŽ x. is VŽ x. UŽ D D.. So if we know that VŽ x. V, we can be sure that VŽ x. V for those x s such that D D V V Ž x., c, c U by controlling the numerical errors such that V V Ž x. V VŽ x., c, c. Ž see Remark 5... However, from Ž i. we know that VŽ. V ; so we begin with x, that is, D, and we need to calculate V Ž x., c for the sequence of x s with Euclidean norms, D, D,..., where V, c V, cž x i. Di Di for i,,..., U and for any x with x smaller than the maximum D value we have calculated for V Ž x. we then know that VŽ x., c V. To complete the proof for the uniform dominance, however, we need to be able to stop calculating for V Ž x. at some point:, c Ž iii. a value D such that V VŽ x. for all x with x D Ž see Theorem Hence the maximum value of D for which we have to calculate V is D., c Ž x. REMARK 5.. To make sure that V V Ž x. V VŽ x., c, c, a GAUSS program is used to calculate V Ž x. with slightly larger radii Ž D., c for many s and to calculate V with a slightly smaller c. That is,, c VŽ x. V, cž x., V V, c, where both and are positive error terms due to numerical integrations. Hence if at some values of x our numerical results say V Ž x. V, then, c, c VŽ x. V Ž x. V V V ;, c, c that is, at those x s, the volume of C* x is truly smaller than C x. A macro called INTSIMP in GAUSS is used to calculate k p p VolŽ C* Ž x.. H sin Ž D. d, p Ž. using a slightly larger D found by a dichotomy method using. The program is run on a 486DX personal computer and the error term in INTSIMP is set to be at the order of 7. Now, we present our analytical results for Ž i. and Ž ii.. Notice that we assume the conditions of Lemma 4. in the theorem and lemma below; therefore the uniform dominance theorem holds only for C* Ž X. with properly selected constants A and B.

18 GOOD EXACT CONFIDENCE SETS 43 THEOREM 5.. Assume the conditions of Lemma 4.. Then C* Ž. is a subset of C Ž., that is, VolŽC* Ž.. VolŽC Ž... PROOF. At x, C Ž. : t c4 and C* Ž. : tž. cž.4; recall that t, Ž A Bt. and t. It suffices to show, for t c, Ž. tž. cž.. Note that we need only the condition since it implies A BcŽ b. cž b. A Bc cž b. p. This proves Ž. since cž. c cž b. p, t Ž A Bt. and hence Ž. holds if c Ž b t c A Bt. p for t c. ½ 5 THEOREM 5.. Let JŽ D. Ž ddd.žvolžc* Ž x.., where D x. Assume the conditions of Lemma 4. and A Ž p. c. Then ' ' where m cž b. b. p p p M p JŽ D., Ž p. Two preliminary results are useful in proving this theorem. Their proofs are given in the Appendix. LEMMA 5.. Assume the conditions of Lemma 4.. For given, and D, let Ž D. be the solving the equality Then ' ' where M c b. Ž D cos. cž.. M Ž D. M and D, LEMMA 5.. Assume the conditions of Lemma 4.. If in addition A p c, then Ž. Ž. where D ddd D. Ž D. Ž,. and D,

19 44 Y.-L. TSENG AND L. D. BROWN PROOF OF THEOREM 5.. We have k p p VolŽ C* Ž x.. H sin Ž D. d, p Ž. by the definition of D, and this implies H p p JŽ D. k sin Ž D. Ž D. d H p p k M sin Ž D. d Ž by Lemma 5.. k k p M p Ž by Lemma 5.. p p M p, Ž p. where the last equality is implied by an easy calculation for k k. p The next result we need is to find a D such that V VŽ x. for all x with x D. For given p and, we use the following theoretical approach to find a number for D: If we correctly bound all of the remaining terms of all the approximation equalities we have in the proof of Theorem 3., we have the following inequality: V VŽ x. D L D 4 L, where BŽ p. L k k p c p pb and L is a positive constant resulting from the bound for the remaining terms. Therefore, to find D such that V VŽ x. x s.t. x D, it is sufficient to find the smallest D such that D L D 4 L. ' From this, a theoretical value for D is L L. There are several untidy constants resulting from the bounds for all the 4 D or for smaller order terms when calculating VolŽC Ž x.. VolŽC* Ž x... The calculations are very long and tedious. Hence, for convenience, we present here only the consequent theorem and for the details readers are referred to Tseng Ž THEOREM 5.3. Assume the conditions of Lemma 4.. Then Vol C Ž x. Vol C* Ž x. x with x D, where D is a constant depending on A, B, p and.

20 GOOD EXACT CONFIDENCE SETS 45 Table gives the values D with A and B satisfying the conditions of Lemma 4. for given p and. It can be seen that Theorem 5.3 gives reasonably small values for D. This makes the described process for proving the uniform dominance of C* Ž X. upon C Ž X. computationally feasible. For example, when p 3 and.5, D is This means that, for all x with x 49.8, we know that C* Ž x. has a smaller volume than C Ž x.. Therefore, to finish the whole proof we only need to follow Ž ii. and calculate VolŽC* Ž x.. at various values of x smaller than Notice that both the constant U in Ž ii. and D in Ž iii. depend on the dimension p and the confidence coefficient. Therefore, for a given pair of p and we can calculate a sequence of V Ž x., c s in the way described in Ž ii. and Ž iii. and complete the proof that C* Ž x. has uniformly smaller volume than C Ž x.. Thus we have the following corollary. COROLLARY 5.. Assume the conditions in Lemma 4. and A Ž p. c. If the procedure described above is followed completely for a given pair of and p 3. Then C* Ž x. has uniformly smaller volume than C Ž x. for those values of and p. As a consequence, C* X is also minimax. TABLE Theoretical values of D.5. p B A D A D Note: B is taken to p here.

21 46 Y.-L. TSENG AND L. D. BROWN TABLE 3 Ratio of the effective radii RER of C* x and C Ž x. at x.5. p B A RER A RER Note: B is taken to p here. For an illustration of using Ž i. Ž iii. to prove the uniform dominance of C* Ž X. over C Ž X. for various values of p and, the reader is referred to Tseng Ž In short, the described algorithm is completely followed in Tseng Ž 994. for.5 and p 3, 4, 5, 9 or. Table 3 gives the ratio of the effective radii Ž this is defined in Section 6. of C* Ž x., with A and B satisfying the conditions of Corollary 5., and C Ž x. at x for.5 or. and p 3,...,. 6. Comparison with other confidence regions. As mentioned in the Introduction, several other confidence regions have been proposed since the works of Brown Ž 966. and Joshi Ž Unfortunately, it is impossible to make the comparison of C* Ž X. with all of them. Therefore, here we compare our confidence sets with the usual one, the ones given in Berger Ž 98. and those considered by Casella and Hwang Ž Comparisons between the two confidence sets are made in terms of the ratio of the effective radii of these sets. More precisely, the ratio of the effective radii of C Ž x. and C Ž x. is defined as Vol CŽ x. Vol C Ž x. This is proposed by Faith 976 to make the comparison between confidence sets more nearly independent of the dimension p. p.

22 GOOD EXACT CONFIDENCE SETS 47 TABLE 4 Ratio of the effective radii of C* X and C Ž X. Norm of x p Note:.5, A and B p for each p. In Table 4 the ratio of the effective radii of C* Ž x., with A and B Ž p., and C Ž x. is given for various values of x with p 3, 6 and and.5. It can be seen that C* Ž x. is clearly smaller than C Ž x.. In particular, the improvement can be substantial for small x and large p. This suggests that C* Ž X. is superior to C Ž X. for small area estimation problems. Table 5 gives the comparison of our confidence sets with the ones in Berger Ž 98. and Casella and Hwang Ž 983. for p 6 and with.. The constant A in C* Ž X. is taken to be and two choices of B are considered; B Ž p. and B Ž p.. Since their confidence sets have higher coverage probability, by trading coverage probability for volume it is not surprising that C* Ž x. has smaller volume. Table 5 shows that, for both B Ž p. and B Ž p., this is the case. In these comparisons we observe the following: for a fixed p, the improvement is greater for TABLE 5 Comparison of C* Ž X. with those in Berger Ž 98. and Casella and Hwang Ž 983.: ratio of the effective radii with C Ž x. Norm of x p 6 Casella and Hwang Berger C* Ž x., B C* Ž x., B p Casella and Hwang Berger C* Ž x., B C* Ž x., B Note:. and A.

23 48 Y.-L. TSENG AND L. D. BROWN TABLE 6 Ratio of the effective radii of C* X and C Ž X. when p 6 and. with A and B satisfying the conditions of Theorem 4. and Lemma 4. Norm of x Casella and Hwang Berger C* Ž x., B.5 A A C* Ž x., B.5 A A smaller x. When x is large, these three sets are not very different from each other; neither are they significantly different from the usual one. Note that, even though the numerical results show C* Ž X. is better than C Ž X. for the A and B used in Tables 4 and 5, we have not proved it dominates C Ž X. for those values since they do not satisfy the conditions of Lemma 4.. Table 6 gives the ratio of the effective radii of C* Ž x. and C Ž x. when p 6 and. with A and B satisfying the conditions of Theorem 4. and Lemma 4. Ž see Table.. With these choices of A and B, C* Ž x. still has satisfactory performance except for the case where A 3.79 and B.5. In that case, Berger s confidence region, which has, however, a probability of coverage smaller than.9 for a range of, has smaller volume when x is small. Nevertheless, this indicates that the sufficient conditions we have for the dominance of C* Ž X. over C Ž X. are not entirely satisfactory. There appears to be room for improvement and we defer this problem for future study. APPENDIX PROOF OF THEOREM 3.. First note that both Ž t. and depend on only through. Then C Ž x. Ž C* Ž x.. iff P C Ž Px. Ž C* Ž Px.., where P is any p p orthogonal matrix. Therefore, without loss of generality we can assume that x Ž x,,...,., where x. Let x, D x x,, s.t. Ž x. x D cos, and D cos. Then x Ž t. x Ž t. Ž t. Ž x. Ž t. t Ž t. Ž x. Ž x. Ž t. xž x..

24 GOOD EXACT CONFIDENCE SETS 49 Thus, for any fixed x, 4 :, s.t. cž. 4. C* Ž x. : x Ž t. cž. The idea is for any given x to approximate the term cž., for on the boundary of C* Ž x., by a function of D and, say RŽ D,.. Then the difference in volume between C Ž x. and C* Ž x. can be approximated as k p p p H p sin Ž. c R Ž D,. d, Ž which under the described condition is then shown to be a O D. term with positive coefficient if p 3. Define A B D cos W. BD A B D cos Ž Then it is easy to see that W O D. as D, since A and B are constants and is bounded. Now, by the definition of, A B D D cos Ž W. BD Ž. cos A B Ž A BŽ D cos.. W BD D BD BD ž / / cos 3 ož D. since W OŽ D.. BD ž D This implies that Hence t Ž t. Ž D D cos. Ž t. D cos Ž W W. Ž W.. 4 B D B D B D Ž 3. ož D.. B D

25 5 Y.-L. TSENG AND L. D. BROWN Now, by Ž. and easy calculations, we have Ž 4. Ž W. BD cos 3 ož D. by Ž. BD D 4 cos cos 3 ož D.. BD BD For the term c, the Taylor series approximation gives Ž 5. d cž. c cž. ož., Ž for. d f cž. c ož. Ž by Lemma 3.. f cž. p, p, c c ož D., pb D where the last equality is implied by Ž 3.. Therefore by Ž 3., Ž 4. and Ž 5., on the boundary of C* Ž x. we have the following equalities: cž. c 4 cos cos c ož D. pb D B D BD BD cos c ož D., BD which implies Applying this to 4 we have cos ( c ož D. BD cos ' c ož D. BD' c cos ' c ož D.. BD 4 cos cos ož D. BD BD Ž 6. cos ' c c D cos c cos ož D.. BD ž B /

26 Define GOOD EXACT CONFIDENCE SETS 5 Ž '. c p pb c c D cos p cos 4 pbc cos. pb D Ž Then O D.. Hence by applying Ž 3., Ž 5. and Ž 6. we have, for large D, where Therefore when D is large cž. RŽ D,., RŽ D,. c Ž. ož D C* Ž x. :, s.t. RŽ D,. :, s.t. c o D. Note that a Taylor series approximation gives p Ž. o Ž D ž. c / Ž 7. ž / ž / p Ž. ož D. p p Ž. ož D. c C ož D. ž / p p p ož D.. c 8 c Now we can calculate the volume difference for large D as H H x d d C Ž x.4 C* Ž x.4 k p p p sin Ž. c c Ž. ož D. d H p p p H ž / p c k o D c sin d p k c sin Ž. d ož D. p p p H c 8c k c sin Ž. d ož D. p 4c cos p p H c 8c B D k k p c p BŽ p. 4 ož D., pb D

27 5 Y.-L. TSENG AND L. D. BROWN where the fourth equality followed from Ž 7. from the facts that and the last equality followed That is, H H p k p p p sin Ž. cos d and sin Ž. cos d. since it is easy to see that But for p 3 and B Vol C Ž x. D Ž x. BŽ p. 4 ož., B k k p c p Vol C Ž x. I C Ž x. d. H p p R BŽ p. iff B. Ž p. 4 This completes the proof. Theorem 3. of Casella and Hwang 983 is used in the proof of Theorem 4.. The translated form of the CasellaHwang theorem in the present context is the following. THEOREM A. Casella and Hwang Ž The set C* Ž x. is connected for any given x iff the set CŽ D. is an interval for any given D, where 4 CŽ D. : HŽ D. cž., ž / HŽ D. D and. A B Ž A B. PROOF OF THEOREM 4.. Let Ž. Ž A B. and gž. Ž.. Then and HŽ D. g Ž. D, d 4 HŽ D. g Ž. D A A B B Ž A. Ž.. d An easy calculation shows that gž. is strictly increasing from to in,. provided that A.5; hence there is a unique Ž D. such that HŽ D.. Since Ž A B., we have d 3 A B A B, d which has zeros at and *, where * ' AB.

28 GOOD EXACT CONFIDENCE SETS 53 Ž Also it is easy to check that d d. *. Thus we have the following facts about the noncentrality as a function of : Ž. it is an increasing function on, * ; Ž. it is a decreasing function on *,.; and Ž 3. it has its maximal value b, which is defined to be * Ž 4 AB., at * ' AB. As a result, and applying Lemma 3., we have d d d cž. cž. d d d, on d f c, *., p, Ž., on Ž *,., d fp, Ž cž.., at *, since f ŽcŽ.. f ŽcŽ.. p, p, is a positive number. Hence the function cž. behaves similarly to the function ; that is, as a function of of, cž. increases on, *, decreases on *,., has its maximal value cž b. at * ' AB and also tends to c as tends to or since tends to in both cases. The goal is to prove the set CŽ D. is an interval for any given D, where 4 CŽ D. : HŽ D. cž.. Let the two sets C D and C D be defined as Then CŽ D. C Ž D. C Ž D.. Note that CŽ D. CŽ D. : 4, C Ž D. CŽ D. : 4. H D by the definition of, and cž. for all, which implies CŽ D. CŽ D. that is, CŽ D. CŽ D., and we have the following fact: FACT A.. C D is an interval iff C D and C D are intervals. CLAIM A.. C D is an interval. PROOF OF THE CLAIM. On C Ž D.,. There are two possibilities for the relationship between * and : Case. *. In this case, the function HŽ D. cž. is decreasing on,, thus C Ž D. is an interval. Case. *. We have the following subcases. Case.. HŽ D. c for all,. Obviously C Ž D. is an interval. Case.. HŽ D. c for some,. Then there is a unique number

29 54 Y.-L. TSENG AND L. D. BROWN Ž D. such that HŽ D. c since HŽ D. is strictly decreasing on,. and HŽ D.. Case... *. Obviously C Ž D. is an interval. Hence, we are left to consider the case that HŽ D. c for some in, and *. To prove that C Ž D. is an interval for the case that HŽ D. c for some, and *, it suffices to prove that the function HŽ D. cž. is decreasing on, ; in fact we only need to prove that on *,, since we can write C Ž D. C Ž D., 4, and HŽ D. cž. is decreasing on, *.. Define WŽ. Ž dd. HŽ D. Ž dd. cž., for. Then we want to show that WŽ. on *, if A and B are selected as described in the theorem. Ž Note that on *,, B A, where Ž. W Ž. 4 g Ž. D A A B B Ž A. Ž B A. f cž. A B f cž. p, p, Ž B A. cž b. 4 g Ž. D A A B B Ž A. A B p Ž B A. cž b. ' 4 c A A B B Ž A. A B p M, pž A B. M A cž b. B 3 cž b. ' 4 p c Ž AB. A A B B Ž A.. We have applied Lemma 3. for the first inequality, and used the fact that g Ž. D ' c on *,, by the definition of, for the second inequality. We want to show that M, hence WŽ., on *, if A and B are selected as described in the theorem. Two cases are to be considered: Case.. Hence 3 n, n 3. Thus, ' 4 if cž b. p' c BŽ 3 A B M Bc b p c AB B B A

30 GOOD EXACT CONFIDENCE SETS 55 Case.. Hence 3 n, n 3. Thus ' 4 3 M Bc b p c A A A AB B B A A A Ž A 3B. if cž b. p' c. B This proves that C Ž D. is an interval, for all D provided that A and B satisfy the condition described in the theorem. By similar arguments, we can prove that C Ž D. is an interval, for all D, under the same condition, by proving that W Ž. on, *, where uniquely solves H Ž D. c on Ž, *.. Therefore, CŽ D. is an interval, for all D provided that A and B satisfy the condition described in the theorem. Finally, applying Theorem A. we p have proved that the confidence set C* x is connected for any x R if A and B satisfy the condition described in the theorem. PROOF OF LEMMA 5.. From the proof of Lemma 4., we have, for all D, Ž D. Ž,., where Ž D. and Ž D.. Hence if we show that M and M, then we prove this lemma. First note that, when, t D D cos Ž. Ž. D, D and A Bt A B D. By the definition of we have t D c t. However, c cžt. cž b. for all D, we then have which implies t Ž D. cž b., 'c Ž b. Ž D. s inf sr A Bs since it is easy to show that s inf ' b sr A Bs Hence we have proved the inequality that ' ' b ' cž b. b M.

31 56 Y.-L. TSENG AND L. D. BROWN Now, when, t D D cos D, D and Ž A Bt. Ž A BŽ D... Then similarly we have which implies c t Ž D., ' c Ž D. s sup s A Bs ' b, since it is equally easy to show that s sup ' b. A Bs s Thus we have proved the other inequality that ' ' c b M. This completes the proof for Lemma 5.. PROOF OF LEMMA 5.. We will use the same notation as in Lemma 5.. Straightforward calculations show that KD Ž K. cos Ž D., Ž K. 3 Ž. Ž. where K dd c A Bt B and D. The conditions of Lemma 4. imply that Then we have Ž 8. Ž D. Ž K. and K. Ž K. KDŽ K. cos Ž K. Ž cos. KDŽ K. D cos. Note that since cos and K, we have Ž 9. Ž K. Ž cos. KD Ž K. D cos KD Ž K. D Ž D.. Note that, when D, we have Ž A Bt. A BŽ D. D Ž D. ' b, A BŽ D.

32 GOOD EXACT CONFIDENCE SETS 57 where the last inequality was proved in Lemma 5.. Also M 5.; hence by Lemma ½ ' D M, if D, M b, if D, where Ž p. M ' b ' c ' b since A by the assumption. c Then, by 8 and 9, we have proved Lemma 5.. REFERENCES BERGER, J. O. Ž 98.. A robust generalized Bayes estimate and confidence region for a multivariate normal mean. Ann. Statist BERGER, J. O. Ž Statistical Decision Theory and Bayesian Analysis, nd ed. Springer, Berlin. BIRNBAUM, A. Ž Characterizations of complete classes of tests of some multiparametric hypotheses, with applications to likelihood ratio tests. Ann. Math. Statist BROWN, L. D. Ž On the admissibility of invariant estimators of one or more location parameters. Ann. Math. Statist BROWN, L. D., CASELLA, G. and HWANG, J. T. G. Ž Optimal confidence sets, bioequivalence and the limacon of pascal. J. Amer. Statist. Assoc CASELLA, G. and HWANG, J. T. Ž Empirical Bayes confidence sets for the mean of a multivariate normal distribution. J. Amer. Statist. Assoc CASELLA, G. and HWANG, J. T. Ž Confidence sets and the Stein effect. Comm. Statist. Theory Methods COHEN, A. and STRAWDERMAN, W. E. Ž Admissibility implications for different criteria in confidence estimation. Ann. Statist FAITH, R. E. Ž Minimax Bayes set and point estimators of a multivariate normal mean. Technical Report 66, Univ. Michigan. GHOSH, J. K. Ž 96.. On the relation among shortest confidence intervals of different types. Calcutta Statist. Assoc. Bull HUWANG, L. Ž 99.. Good coverage probability confidence sets in linear errors-in-variables models. Ph.D. dissertation, Statistics Center, Cornell Univ. HUWANG, L. Ž Interval estimation in structural error-in-variables model with partial replication. J. Multivariate Anal HWANG, J. T. G. Ž Fieller s problems and resampling techniques. Statist. Sinica HWANG, J. T. and CASELLA, G. Ž 98.. Minimax confidence sets for the mean of a multivariate normal distribution. Ann. Statist HWANG, J. T. and CASELLA, G. Ž Improved set estimators for a multivariate normal mean. Statist. Decisions Ž Suppl HWANG, J. T. and CHEN, J. Ž Improved confidence sets for the coefficients of a linear model with spherically symmetric errors. Ann. Statist JAMES, W. and STEIN, C. Ž 96.. Estimation with quadratic loss function. Proc. Fourth Berkeley Symp. Math. Statist. Probab Univ. California Press, Berkeley. JOSHI, V. M. Ž Inadmissibility of the usual confidence sets for the mean of a multivariate normal population. Ann. Math. Statist JOSHI, V. M. Ž Admissibility of the usual confidence sets for the mean of a univariate or bivariate normal population. Ann. Math. Statist PRATT, J. W. Ž 96.. Length of confidence intervals. J. Amer. Statist. Assoc

33 58 Y.-L. TSENG AND L. D. BROWN SHINOZAKI, N. Ž Improved confidence sets for the mean of a multivariate normal distribution. Ann. Inst. Statist. Math STEIN, C. Ž Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab Univ. California Press, Berkeley. STEIN, C. Ž 96.. Confidence sets for the mean of a multivariate normal distribution. J. Roy. Statist. Soc. Ser. B STEIN, C. Ž 98.. Estimation of the mean of a multivariate normal distribution. Ann. Statist TSENG, Y. L. Ž Good exact confidence sets and minimax estimators for the mean vector of a multivariate normal distribution, Ph.D. dissertation, Statistics Center, Cornell Univ. TSENG, Y. L. and BROWN, L. D. Ž Good exact confidence sets for the mean vector of a multivariate normal distribution. Technical report, Inst. Statistical Science, Academia Sinica, Taipei. INSTITUTE OF STATISTICAL SCIENCE DEPARTMENT OF STATISTICS ACADEMIA SINICA UNIVERSITY OF PENNSYLVANIA TAIPEI 59, TAIWAN PHILADELPHIA, PENNSYLVANIA yltseng@stat.sinica.edu.tw lbrown@compstat.wharton.upenn.edu