Statistics & Decisions 7, (1989) R. Oldenbourg Verlag, München /89 $

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1 Statistics & Decisions 7, (1989) R. Oldenbourg Verlag, München /89 $ A NOTE ON SIMULTANEOUS ESTIMATION OF PEARSON MODES L. R. Haff 1 ) and R. W. Johnson 2^ Received: Revised version: April 3, 1989 Abstract. We exhibit an estimator of the mode of a vector of independent variates having densities in the Pearson (1895) class. This estimator dominates the componentwise minimum variance unbiased estimator under weighted squared error loss. 1. Introduction James and Stein (1961) presented a class of estimators which dominate the usual estimator of the mean vector of a multivariate normal distribution in three or more dimensions under quadratic loss. Since then much work has been directed at finding improved estimators of the mean in more general settings. Hudson (1978), for example, extended the work of James and Stein to an exponential class framework. Berger (1980) presented 1) Research supported by an NSF Grant 2) Work performed while at the Naval Ocean Systems Center, San Diego, Cal i forni a AMS 1980 subject classifications: Primary 62 C 15 Secondary 62 F 10, 62 C 25 Key words and phrases: Simultaneous estimation of modes, Pearson curves, Stein-like estimators, estimation of means

2 378 L. R. Haff, R. W. Johnson similar results for a variety of loss functions. Johnson (1984), and Haff and Johnson (1986) noted that Hudson's framework includes variates of the four parameter Pearson class provided only the mode is unknown. This structure was used by Haff and Johnson (1986) to obtain further results on estimating means. In this note, we provide alternative estimators for a vector of Pearson modes. These estimators dominate the componentwise minimum variance unbiased estimator (MVUE) under weighted squared error loss. Apparently simultaneous modal estimation has not been discussed in the literature until now. For certain asymmetric densities, the modes are more interesting than the means as objects of estimation. These problems are closely related in the present context however. Indeed, our main result shows that it is natural to formulate the general problem of estimating Pearson modes in terms of estimating means. We cite the applicability of this result to several examples which have appeared in the recent literature on simultaneous estimation. 2. The Dominance Result Let X= (Xp...,Xp) t be a vector of independent variates in which X. has p.d.f. f n (χ η IΘ 1 ) defined by Karl Pearson's (1895) equation on (k lis k 2i ), where ßj-j known quantities. We shall denote this by X i ~P(8 i,3 o i,on (k^.,k 2i ). Here we estimate θ= (θ^,...,6p) t, the mode vector, under loss function (l) L(M) = (<s-e)w-e), where Q = diag...»qp) is a diagonal matrix, the q^ (i = 1,2 p) being arbitrary fixed positive constants. The standard decision theoretic notions of "risk" and "dominance" are assumed as in Haff and Johnson (1986), a reference we henceforth abbreviate by H&J (1986). Additionally, the regularity conditions from Section 2 (pp. 46, 47) of that paper are assumed throughout.

3 L. R. Haff, R. W. Johnson 379 Let (2) a.(x) = (e oi + β χι χ + e Zi x 2 )/(i - 2B 2i ), ß 2i < 1/2, where a i (x)>0. Also, let (3) b.(x) = ; X dt/a.(t). As noted in H&J (1986), EX i = (B^ + 0 i )/(l i ). Since b. is one-toone and since b.(x^) is a minimal sufficient statistic for Θ. - see (1.2) and (1.3) of H&J (1986) - it is clear that (4) I = c î" d (Ρ X 1) is the componentwise MVUE of the mode θ where C=diag (c,,...,c ) with 1 t Ρ c i = 1-2ß 2i and d=(dp..d ) with d^ = ß^. Our main result incorporates equation (4) and the following lemma which shows how Pearson curves behave under affine transformations. Lemma. If U~P(m,r,s,t), then V ξ eu - f~p(em-f, e 2 r+efs + f 2 t, es + 2ft, t). Proof. See Kaskey, Krishnaiah, Kolman, and Steinberg (1980). From (2) and (3), set B= (bpb 2,...,b It will be necessary to subscript Β and its components by the variables under consideration. Thus, for example, let U and V be given as in the above lemma. Componentwise we need the relations (5) a v (v) = e 2 a (J (u) and b y (v) = b u (u)/e, which are easy to verify. Theorem. Let X i ~P(6.,ßo i,ß 2i )> i = 1.2,...,ρ, ρ s 3, and set Χ = (Xj,...Xp) t. Assume that the regularity conditions from H&J (1986), pp. 46, 47, hold. Now let Y = CX - d with C (pxp) a diagonal matrix of positive constants and d (pxl) a vector of constants. Then

4 380 L. R. Haff, R. W. Johnson (6) * =!" t Β χ C QC Β χ QC_1 Bv dominates Y as an estimator of Py = EY=CEX-d (which is the mode of X if c.j = 1 - and d^ = ) with respect to the loss function (1). Proof. Note that (7) Y = E Q-I/ZCX- t (P - Q^C-X-Q-^CEXII 2, B^ C A QC Α Β χ where is the euclidean norm. Set W = Q~ 1/2 CX SO EW = Q~ 1/2 CEX. From ~ ι/? -1 the above lemma and from (5) it follows that B, (, = Q C Β χ. Taking the expectation with respect to W, (7) becomes E > - -^Β,-EWII 2 B W B W ~ = E W (W - EW)V- EW) - (p - 2) 2 E W (bjb w ) _1 = E X (Y - EYîYV - ΕΥ) - (ρ - 2) 2 É W (B B W ) _1 < E X (Y- EYjVV- ΕΥ), where the first equality follows from Theorem 2.5 of H&J (1986) with Μ > - - ^ ν > Β 1 Λ and q i =1, i = 1,2,...,p. 3. Examples Here we present b v (X.) explicitly for X. a random variable from a x 1 ι ι number of distributions. For convenience, we drop the subscript i. Exampl e 1. (cf. James and Stein (1961)). If Χ~Ν(θ,σ 2 ), then Χ~Ρ(θ,σ,0,0) and b x (X) = X/a 2. Example 2. (cf. Hudson (1978)). If Χ~Γ(φ,λ) with density

5 L. R. Haff, R. W. Johnson 381 λφ Φ-i ρ-λχ f(x 4>)= Γ(φ) for x> 0, then Χ~Ρ((φ-1)/λ, 0,1/λ,0) and b(x) = λ In X. Example 3. (cf. Berger (1980)). If X= (φ - 1)/W where Ν~Γ(φ,λ), then Χ ~ Ρ((φ- 1)λ/(φ+ 1), 0, 0, 1/(φ+ 1)) and b(x) = -(φ - 1)/X. Example 4. (cf. Johnson (1984)). If X has the beta density f(x 0) «x c6 (l- x) c(1 ~ 9), 0< x< 1 (c = k-2 where k is the "concentration parameter", p. 233 of Brunk (1975)), then X~P(6,0,l/c,-l/c) and b(x)=(c + 2) In (X/(l - X)). Example 5. (cf. Johnson (1984)). If X has the Pearson type IV density (see Elderton and Johnson (1969)), then X~P(6,ß o,ß 1,ß 2 ) with D 2 ξ 4ß Q ß 2 - β 2 > 0 and b(x) = [2( )/D] arctan [(ßj+2ß 2 X)/D] References [1] Berger, J.: Improving on inadmissible estimators in continuous exponential families with applications to simultaneous estimation of gamma scale parameters. Ann. Stat. 8 (1980), [2] Brunk, H.: An Introduction to Mathematical Statistics, Third edition. John Wiley, New York, 1975 [3] Elderton, W. and Johnson, N.: Systems of Frequency Curves. Cambridge Univ. Press, London, 1969 [4] Haff, L. and Johnson, R.: The superharmonic condition for simultaneous estimation of means in exponential families. Canadian Jour. Statist. 14 (1986), [5] Hudson, H.: Empirical Bayes estimation. Technical report no. 58, Dept. of Statist., Stanford University, 1974

6 382 L. R. Haff, R. W. Johnson [6] Hudson, H.: A natural identity for exponential families with applications in multiparameter estimation. Ann. Statist. 6 (1978), [7] James, W. and Stein, C.: Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab., vol. 1, , Univ. of California Press, 1961 [8] Johnson, R.: Simultaneous estimation of generalized Pearson means. Ph. D. dissertation, Dept. of Mathematics, University of California, San Diego, 1984 [9] Kaskey, G., Krishnaiah, P., Kolman, B., and Steinberg, L.: Transformations to normality. Handbook of Statistics, vol. 1, , North Holland, 1980 [10] Pearson, K.: Memoir on a skew variation in homogeneous material. Phil. Trans. Roy. Soc. A 186 (1895), L. R. Haff Department of Mathematics University of California, San Diego La Joli a, California 92093, USA R. W. Johnson Department of Mathematics and Computer Science Carleton College One North College Street Northfield, Minnesota USA