Asymmetric quasimedians: Remarks on an anomaly

STATISTICS& PROBABILITY LETTERS ELSEVIER

Statistics & Probability Letters 32 (1997) 261-268

Asymmetric quasimedians: Remarks on an anomaly Govind S. Mudholkar*, Alan D. Hutson Department of Statistics, University of Rochester, Rochester, NY 14627, USA Received April 1995; revised April 1996

Abstract Let X1 <~X2<~ •. • <~X,, denote the order statistics of a random sample of size n and M the sample median defined conventionally as the middle Xm for n = 2m + 1 and the average (Xm +Xm+I)/2 for n = 2m. Hodges (1967) observed that for the normal populations the asymptotic efficiency 2In of the sample median is approached consistently through higher values for the even sample sizes, n = 2m, than for the odd samples sizes, n = 2m + 1. Hodges and Lehmann (1967) explained this even-odd anomaly in terms of the O(n -2)-term in the large sample variance of M, and extended it to quasimedians Mr of arbitrary symmetric populations. We obtain the large sample bias and variance of the asymmetric average Mr,Ah) = hXrn+l-r ÷ hXm+l+s, h : 1 - h , consider various tradeoffs, construct modifications Mr(l) and Mr(2), of Mr for asymmetric distributions. Also, the observation due to Hodges and Lehmann (1967), which is often interpreted as an anomaly, is examined in the asymmetric case.

Keywords." Asymptotic expansions; Bias-variance tradeoff

1. Introduction and summary Let Xl ~
(Xm+l-r +Xm+l+r)/2

if n = 2m + 1,

(Xm-r + Xm+l+r)/2

if n = 2m,

(1.1)

Mr =

where r = 0 yields the conventional sample median M. I f p = Q(½) denotes the population median then Hodges and Lehmann (1967) showed that for r = o(n), x / ~ ( M r - lO is asymptotically normal with mean 0, and for symmetric populations obtained the large sample approximation for the variance up to O(n - l )

* Corresponding author. 0167-7152/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII SO167-7 ! 52(96)00082-X

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262

as

Var[v~(M~

1 4f 2 1 4f 2

/~)]

1 16f2n( 9 ÷ 8r ÷ 8) 1 16fEn(O+8r+12)

if n = 2m + 1,

(1.2) i f n = 2m,

where f = f ( Q ( ½ ) ) and 0 = f ' / f . Note that if r in the variance (1.2) for n = 2m + 1 is replaced by r + g1 we get the variance for n = 2m. Hodges and Lehmann (1967) used the O(n -1 )-term in the large sample approximation to explain and extend to all symmetric distributions the finding of Hodges (1967), that it never pays to base the median on an odd number of observations since the next smaller even number of observations provides an equally accurate estimate. The curious anomaly in the above result was recently revived by Cabrera et al. (1994), who discussed the precision of medians in the asymmetric case, and Oosteroff (1994) gave an explanation in terms of the trimmed mean. If the population is asymmetric then there is no a priori reason to define quasimedians using equally weighted averages of the observations which are symmetrically spaced about the middle. Hence, in the same spirit of the quasi-quantiles discussed by Reiss (1980, 1989), we consider asymmetric averages M~,~(h) given by:

Definition 1.1. For the random sample with order statistics X1 <~X2 <~ • • • <~Xn let Mr, s( h )

hXm+l-~ + hXm+l+~ [ hXm-r + ~[rm+l+s

if n :- 2m + 1, if n : 2m,

(1.3)

whereh= 1-h. Then the basic theory of L-estimators and order statistics yields the following:

Theorem 1.2. I f r : o(n) and s = o(n) then x/n[Mr, s(h) - #] is asymptotically normal with mean 0 and variance 1/(4f2). In Section 2 we expand the expectation of Mr, s(h) and obtain the O(n -1 ) and O(n -2) terms in the bias. The O(n-2)-term in the variance is derived in Section 3. Some particular cases involving symmetric populations and symmetric averages are considered in Section 4. In Section 5 specializations of Mr, s(h), obtained by either removing the bias terms or minimizing the variance, or both, are examined, and a location and scale invariant measure MrO) = Mr, s(h), where h-

1 2

f'(#) 4f(/~) 2

and

s-

it + h(4r - 1 ) 4h

(1.4)

is identified as a modification Mr appropriate for asymmetric populations, MoO) being a refinement of the conventional median. The variation in Var(M~(1~) as n is changed from 2m to 2m + 1, and as r increases are discussed in Section 6. The anomaly is further examined by considering the bias and variance tradeoffs. Also, the variation in bias with respect to r and s is opposite in direction and typically, but not always, the bias increases as the variance decreases. Roughly speaking, it is seen that in typical situations involving right-skew distributions where the median is greater than the mode, the anomaly of the medians carries over to the asymmetric case and Var(M~(l~) decreases in r.

G.S. Mudholkar, A.D. Hutson I Statistics & Probability Letters 32 (1997) 261-268

263

2. The bias o f Mr, s(h)

In this section we examine the large sample bias of Mr, s(h) for a not necessarily symmetric population, with d.f. F(.) and p.d.f, f ( . ) , by expanding its expectation in a Taylor series. Towards this end it is convenient to use the quantile function Q(u) = F - l ( u ) , 0~ 4. Lemma2.1.

Proof. David and Johnson (1954), see also David (1981), give expressions for the cumulants, and mixed moments E(1--[Ura':,), of the joint distribution of c nonextreme order statistics U~,:n(i = 1,2 . . . . . c). The expressions given in Lemma 2.1 are particular cases of their results obtained in conjunction with the well-known relationships between mixed moments and cumulants, e.g. see Kendall and Stuart (1987). However, for the reasons discussed in Section 3.2 of van Zwet (1964), the rigor of their proofs requires the existence of E(X/:n) for some i and n. Theorem 2.2. For large samples the expectation o f Mr, s(h ) is

bl(h) b2(h) E[ME,s ( h ) ] = Q + ~ + ~ + O(n-3).

(2.1)

I f n = 2m + 1 then the coefficients bl(h) and b2(h) are given by bl(h) = 8(hs - h r ) Q ' + Q',

(2.2)

b2(h) = 16(hs - hr)(Q m - 8Q t) + 32(2hr 2 + 2hs 2 - 1)Q" + Q(4).

(2.3)

where h = 1 - h. I f n = 2m then the coefficients bl(h) and b2(h) can be obtained from (2.2) and (2.3) by replacing r and s by r + 1 and s + ½, respectively. Proof. Note that any order statistic Xj :n = Q ( U j : n ) in distribution, where Q denotes the population quantile

function. So expanding Q(Uj:n) in a Taylor series about i,i and suppressing the secondary subscript n, we get xj=O+

( U j - 1) Q, + ½ ( U j - ½)2 Q,, + 1 ( u j -

½)3 Q,,, + ' " ,

(2.4)

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where the equality is in distribution. Substituting the expressions for the noncentral moments E ( U j - 1/2) c in terms of the central moments given by Lemma 2.1, we get to order O(n -3), E(Xm+I+j) = Q +

8jQ' + Q" 1 6 j ( Q " - 8Q') + 32(2j 2 - 1)Q" + Q(4) + 8n 128n 2

Using the expansion (2.5) with n -- 2m + 1, j = - r and j = s and substituting in

E[Mr, s(h )] = hE(Xm+l-r ) + hE(Xm+l+s),

(2.5)

we get the expressions for bl(h) and b2(h) in (2.2) and (2.3), respectively. Using similar arguments it can be shown that the bias coefficients bl(h) and b2(h) of Mr,s(h) for n - - 2 m are those obtained from (2.2) and (2.3) by substituting r + ½ and s + 1 for r and s, respectively.

Corollary 2.3. For given r = o(n) and s = o(n), the O(n-1)-term in the bias vanishes for n = 2m + 1, /f h = (8sQ ~+ Q")/[8(r + s)Q~]. The weighting constant h is location and scale invariant, and in the case of symmetric populations reduces to h = s/(r + s). 3. The variance of M~,s(h) For r and s of order o(n) the asymmetric median Mr, s(h) is asymptotically normal with variance 1~(4nf2). The following proposition gives the O(n-2)-term for the variance.

Theorem 3.1. I f n = 2m + 1, the large sample var&nce Of Mr,s(h) is Var[Mr s ( h ) ] '

V~,s(h) ~32n + O ( n - 3 ) '

O'2 4n

(3.1)

where V~,s(h) = 1611 + 2hh(r + s)]Q '2 + 16[hr - hs]Q'Q" - Q,,2 _ 2Q,Qm.

(3.2)

I f n = 2m then the coefficients V~,s(h) can be obtained from (3.2) by replacing r and s by r + i a n d s + ½ , respectively. Proof. The nature of the proof is the same as that of Theorem 2.2 Again note that any order statistic Xj:, = Q ( U j : , ) in distribution. This leads to the relationship E(Xm+l+iSm+l+j) = E[Q(Um+I+i)Q(Um+~+j)]. Therefore, a Taylor series expansion of Q(Um+I+i)Q(Um+I+j) about I gives a series for E(Xm+l+iSm+l+j). Combining this with the results in Lemma 2.1 and E[Q(Um+I+i)] E[Q(Um+I+j)] from (2.5) we get to order

O(n-3), C°v(Xm+l+i'Xm+l+J)

-

0'24n

16(1 - i + j ) Q ' 2 _ 8(i 32n 2 + j ) Q ' Q ' ' _ Q"2 _ 2Q'Q'"

(3.3)

Substituting i = j = - r , i = j = s, and i = - r , j = s into (3.3) yields Var(Xm+~_r), Var(Xm+l+s) and Cov(X,n+l--r,Xm+l+s), respectively, giving (3.1) and (3.2).

Corollary 3.2. For n = 2m + 1 or n = 2m the weighting constant h which maximizes Vr,s(h), and hence minimizes Var[Mr, s(h )], is h = ~1 +Q"/(4Q'), which is independent of r and s. It is easily seen that h = 1 f o r symmetric distributions, i.e. Q" = O, which yields the classical result (1.2) given by Hodges and Lehmann (1967).

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265

4. Three special cases We describe the following three cases for n = 2m + 1. Those for the even sample size n = 2m are obtained by substituting r + ~l and s + 1 in place o f r and s, respectively.

4.1. Symmetric populations For symmetric populations the derivatives Q" and Q(4) both vanish, and the terms (2.2) and (2.3) in the bias and (3.2) in the variance reduce to

bl(h) = 8[hs - hr]Q',

b2(h) = 16(kts - hr)(Q'" - 8Q'),

(4.1)

and

Vr,s(h) = 1611 + 2hh(r + s)]Q '2 - 2Q'Q'".

(4.2)

4.2. Symmetrically situated Xr, Xs For r = s denote Mr(h) = Mr, r(h). Then for n = 2m + 1,

bl(h) = 8(h - h)rQ' + Q",

(4.3)

b2(h) = 16(h - h)r(Q'" - 8 Q ' ) + 32(2r 2 - 1)Q" + Q(4),

(4.4)

vr,r(h) = 1611 + 4hhr]Q '2 + 16(h - h)rQ'Q" - Q"2 _ 2Q'Q'".

(4.5)

and

4.3. Mr(h) for symmetric populations W h e n both r = s and Q" = Q(4) = 0 then for n = 2m + 1,

b,(h) = 8(h - h)rQ',

b2(h) = 16(h - h)r(Q'" - 8Q'),

(4.6)

and Vr, r ( h ) = 1611 +

4hhr]Q '2 - 2Q'Q m.

(4.7)

5. The extended quasimedians Mr(l) and Mr(2) In this section we introduce some extensions o f Mr which can be obtained from Mr, s(h). For a given r, Mr(h) has two unknowns. These can be determined by using the earlier results to reduce the bias o f the statistic to O ( n - 3 ) , or by making the bias O(n -2) and minimizing the variance. Definition 5.1. For a given r, let Mr(l) denote the reduction o f Mr, s(h) obtained by minimizing the variance and making the bias o f order O(n - 2 ) .

Theorem 5.2. For n = 2m + 1, MS1) of Definition 5.1 is 9iven by 1

Q"

h=~+~-Q-

7

and

s--

h+h(4r 4h

- 1)

(5.1)

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G.S. Mudholkar, A.D. Hutson I Statistics & Probability Letters 32 (1997) 261-268

I f n = 2m then the value o f h is the same, but s = [f~ + h(4r + 1 )]/(4fO - ½. Both h and s are location and scale invariant.

Remark 5.3. In general, s is not an integer, but provides qualitative information. For symmetric populations, (5.1) leads to the form of the variance (1.2) given by Hodges and Lehmann (1967). Also, M0(1) gives an extension of the commonly defined median M0 given by (1.1) to asymmetric populations. Theorem 5.4. I f n = 2m + 1 then the bias o f M~ 1) is O(n -2 ) and its variance is 9iven by Var(M~ 1)) _ Q'24n

[8(1 + r)Q '2 + (4rl6n 2- 1)Q'Q" - QtQm] + O(n -3 ).

(5.2)

Also, the bias o f M(r 1) can be reduced to order O(n -3) by takin9

r -

h - h + ~/hhQ"[32([~ - h)Q t + 16hf~Q" + 28Q" - 4(h - h)Q"' - Q(4)] 4 8hQ" '

(5.3)

for r real. I f n = 2m then the value o f r is obtained by replacing r by r + ½ on the left-hand side o f

Eq. (5.3). Proof. By straightforward substitution we get (5.2). To obtain (5.3) maximize vr,~(h) from (3.2) with respect to h and then solve bl(h) = 0 = b2(h) from (2.2) and (2.3) with respect to r and s. Definition 5.5. For a given r, let M (2) denote the weighted average Mr, s(h) with bias of order O(n-3). Theorem 5.6. I f n = 2m + 1 then M~2) o f Definition 5.5 is 9iven by the location and scale invariant s = [ - 16Q'Q" + Q,Q~4) + 8Q,2r _ 2Q"Q'"]/[8Q"(Q" - 8Q'r)] and h = (8sQ' + Q")/[8(r + s)O'], for O" ¢ O. The expressions for n = 2m are obtained by modifyin9 r and s appropriately.

6. The anomaly and bias-variance tradeoffs The anomaly discovered by Hodges (1967) is that the precision of the median of a normal sample decreases as the even sample size n = 2m increases to n -- 2 m + 1. Cabrera et al. (1994) termed it an "odd property" of second-order deficiency and discussed the medians of asymmetric populations. Oosteroff (1994) examined the trimmed means with a view to explaining the anomaly. Hodges and Lehmann (1967) clarified and extended the anomaly by computing the O(n-2)-terms in variance of the quasimedians Mr, M0 being the usual median, of arbitrary symmetric populations. It was noted in Section 1 that the replacement of r by r + 1 in the Var(Mr) for n = 2m+ 1 gives Var(Mr) for n = 2m. The anomaly is thus a special case of a not so anomalous an observation, that in large samples Var(Mr) decreases as r increases. In Section 5, Mr is extended by considering the nonleading terms in the biases and variances of Mr,~(h), M~ l) being the simplest and possibly the most natural of the extensions. In this section we examine the anomaly further in terms of Mr,~(h) and M~ 1). First we consider the asymmetric average Mr, s(h). Theorem 6.1. I f the population is unimodal with its median # 9reater than its mode, then f o r r = o(n), the variance o f Mr, s(h ) 9iven by Theorem 3.1 decreases as r increases. Furthermore, i f hQ ~ - Q" > 0 then for s = o(n) the variance ofMr, s(h) decreases as s increases.

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267

Proof. Follows from Theorem 3.1, where for unimodal populations with the median 12 > mode, O ,, ( 2t ) = _f(12)/f(12)3, which is positive. Corollary 6.2. Under the conditions of Theorem 6.1 the Var[Mr, s(h)] increases when n is increased from 2m

to 2m + 1. Also for a symmetrical population the variance of Mr,s(h) is decreasing in r and s. I f r = s, and the population is unimodal with its median 12 greater than its mode then the variance of Mr, r(h) decreases as r increases. Theorem 6.3. /~ the population is unimodal with its median 12 greater than its mode, then for r = o(n), the

variance of M(~ ) given by Definition 5.1, decreases as r increases. Proof. Follows from Theorem 5.4, where for unimodal populations with 12 > mode, Q ,,( 2l ) = _f,(12)/f(12)3, which is positive. Corollary 6.4. Under the conditions of Theorem 6.3 the Var(M~ 1)) increases when n is increased from 2m to 2m + 1; r = 0 is the case noted by Hodges (1967). I f the population is unimodal with its median 12 greater than its mode then the variance of M(o1), which is analogous to the median Mo given by ( 1 . 1 ) f o r asymmetric populations, increases as n goes from 2m to 2m + 1. I f the median 12 is less than its mode and the magnitude f(12)2/f,(12) > ½ then the Var(M~(1)) decreases when n is increased from 2m to 2m + 1. Remark 6.5 (The case of bias). As noted in Theorem 2.2 the bias coefficients bl(h) and bz(h) for n = 2m can be obtained from those for n = 2m + 1 by substitution of r and s by r + ½ and s + ½, respectively in Eqs. (2.2) and (2.3). For large n, the bias coefficient hi(h) dominates b2(h). Hence, we get the following:

Theorem 6.6. The O(n-l )-term in the bias of M~,~(h) is increasing positively in s and increasing negatively inr. Proof. The O(n - l ) term in the bias of M~,s(h) is [8(hs - hr)Q' + Q"]/[8n]. Corollary 6.7. For symmetric populations, Q" = 0, the O(n-1)-term in the bias of Mr, s(h) is increasing positively in s and increasing negatively in r. Also, symmetric and asymmetric populations the O(n -1 )-term 1 in the bias of Mr, r(h) is increasing in r if h < 2" Theorem 6.8. The bias of M(~l) is of order O(n-2). I f the median # is greater than the mode then b2

increases as r increases. Proof. Substituting the values of h and s for Mr(1) from (5.1), respectively, into b2(h) from (2.3) yields the

O(n -2) term of the bias, [ K - 16(2Q t + Q tt ) ( r + 4 r 2 Q t Q tt )]/[128(2Q 2 - Q t Q tt )n 2 ], where K is some constant that is independent of r. But for unimodal populations with 12 > mode, Q ,, (2) 1 = _ f , ( # ) / f ( # ) 3 , which is positive. Corollary 6.9. I f the median 12 is greater than the mode then b2 increases as n goes from 2m + 1 to 2m. Remark 6.10 (Bias variance tradeoffs). It is reasonable to expect that there would be some tradeoff for a

decrease in variance when n is increased from n = 2m to n = 2m + 1, or as r is increased. It is easy to show from the results for the asymmetric populations given in foregoing sections that a decrease in the variance is, in general going to result in an increase in the bias. Thus, for asymmetric distributions the bias in the quasimedian increases with r, and increases as n is goes from 2m to 2m + 1.

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Remark 6.11. If the median p is greater than the mode, bias increases and variance increases as n goes from 2m to 2m ÷ 1. The observation is not valid for symmetric populations. Obviously, much more can be derived using the results in the earlier sections. However, such comments are not directly in the spirit of the principal observation, generally considered an oddity, by Hodges (1967) and Hodges and Lehmann (1967).

Acknowledgements We are thankful to the referee for constructive comments. Alan Hutson's research is supported by National Institutes of Health grant NIH T32 E507271. References Cabrera, J., G. Maguluri and K. Singh (1994), An odd property of the sample median, Statist. Probab. Lett. 19, 349-354. David, F.N. and N.L. Johnson (1954), Statistical treatment of censored data, Biometrika 41, 228-240. David, H.A. (1981), Order Statistics (Wiley, New York, 2nd ed.). Hodges, J.L., Jr. (1967), Efficiency in normal samples and tolerance of extreme values for some estimates of location, Proc. 5th Berkeley Syrup., Vol. I, pp. 163-186. Hodges, J.L. Jr. and E.L. Lehmann (1967), On medians and quasimedians, J. Amer. Statist. Assoc. 62, 926-931. Kendall, M.G., A. Stuart and J.K. Ord (1987), Kendall's Advanced Theory o f Statistics, Vol. 1 (Oxford Univ. Press, New York, 5th ed.). Oosteroff, J. (1994), Trimmed mean or sample median? Statist. Probab. Lett. 20, 401-409. Reiss, R.-D. (1980), Estimation of quantiles in certain nonparametric models, Ann. Math. Statist. 8, 87-105. Reiss, R.-D. (1989), Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics (Springer, New York). van Zwet, W.R. (1964), Convex Transformation of Random Variables, Mathematical Centre Tracts, Amsterdam.