the mean, median, and mode of unimodal …...3. mean, median, and mode for univariate unimodal...

Math-Net.RuAll Russian mathematical portal

S. Basu, A. DasGupta, The mean, median, and mode of unimodaldistributions: a characterization, Teor. Veroyatnost. i Primenen.,1996, Volume 41, Issue 2, 336–352

DOI: https://doi.org/10.4213/tvp2942

Use of the all-Russian mathematical portal Math-Net.Ru implies that you have read and

agreed to these terms of use

http://www.mathnet.ru/eng/agreement

Download details:

IP: 54.70.40.11

December 2, 2018, 21:31:49

Т о м 41 Т Е О Р И Я В Е Р О Я Т Н О С Т Е Й

И ЕЕ П Р И М Е Н Е Н И Я 1996

В ы п у с к 1

© 1996 г. BASU S.*, DasGUPTA А.**

THE MEAN, MEDIAN, AND MODE OF UNIMODAL DISTRIBUTIONS:

A CHARACTERIZATION

Для одновершинного распределения на вещественной прямой знаменитое mean-median-mode неравенство (неравенство, связывающее среднее, медиану и вершину распределения) утверждает, что они часто «идут» в алфавитном или обратном алфавитном порядке (на английском языке). Известны различные достаточные условия выполнения этого неравенства. В данной статье полностью описано трехмерное множество средних, медиан и вершин одновершинных распределений. Установлено, что это множество линейно связно, но не является выпуклым. Выведены некоторые фундаментальные неравенства, связывающие среднее, медиану и вершину одновершинных распределений. Эти неравенства применяются для (i) доказательства неодновершинности некоторых распределений и (и) получения оценок для медианы одновершинного распределения. В многомерном случае используется обобщенное понятие а-одновершинности и даны характеризации множества средних векторов, когда вершина фиксирована или «бегает» по сфере. В частности, обнаружено, что множество средних векторов обобщенных одновершинных распределений с заданными вершиной и ковариационной матрицей есть в точности эллипсоид и этот эллипсоид полностью описан.

Ключевые слова и фразы: а-одновершинность, связный, выпуклый, эллипсоид, среднее, mean-median-mode неравенство, медиана, вершина, проблема моментов, сфера, звездная одновершинность, равномерное распределение, одновершинность.

Let X be a real valued random variable with a cumulative distribution function (c.d.f.) F, and mean ц — fi[F] = E^(X) < oo. A number m = m[F] is said to be a median of X if P{X ^ m} ^ \ and P{X ^ m} ^ | . A median m, thus denned, always exists, although in general, a random variable X may have several medians. Suppose, furthermore, that X is unimodal about some point M = M[F] (called a mode of X ) , i.e., F(x) is

*Department of Mathematical Sciences, University of Arkansas, Fayetteville, A R 72701, USA.

**Purdue University, USA.

1. Introduction

The mean, median, and mode of unimodal distributions: a characterization 337

convex for x £ ( - 0 0 , M ) , and concave for x £ (M,oo). It is easy to see that a median m is uniquely defined for a unimodal random variable X.

The well known mean-median-mode inequality states that for a unimodal distribution F, often, the mean, median, and mode occur in an alphabetical or reverse alphabetical order, i.e.,

M < m < ( i or M ^ m ) / t . (1)

It is well known that this inequality, however, is not always true as can be seen from the following very simple example.

E x a m p l e 1. Let X ~ F, where F is a mixture of three distributions: F{x) = ((1 - 6)/4)Fl(x) + ((3 - 6)/4)F2(x) + (6/2)F3(x), where 0 < S < 1, and Fi, F2, and F3 are, respectively, c.d.f.'s of the Uniform[-3-f^,0], Uniform[0,1 - S], and the degenerate distribution at 0. Clearly, F is unimodal about M = 0, and /x. = ~Ep(X) = 0. However, the median m equals (1 - S)2/(3 - 6) > 0, violating both inequalities in (1). Other examples can be found in [5].

Various sufficient conditions for the validity of (1) are given in [6], [14], [15], [10]. Dharmadhikari and Joag-Dev [4], [5] give a sufficient condition using stochastic ordering of distributions; this encompasses the works of the previous authors. In Section 2, we briefly review the literature on the validity of the mean-median-mode inequality.

Our approach in this article is different. We consider the class of all unimodal distributions with a fixed variance a2,

Ta = {F: F is unimodal and Var F (X) = a2 } ; (2)

and look at the three-dimensional set

Ma = {(ti[F], m[F], M[F}) :FeFa}cR3, (3)

the collection of all possible triplets of mean, median, and mode for unimodal distributions on the real line with a given variance. Knowledge about the set Ma is valuable on its own right; furthermore, it points out the extent to which inequality (1) may get violated, and also may help to single out distributions that cause such violations. The set Ma turns out to be a connected but nonconvex set; in Section 3, we analytically describe the envelope of the set. The derivation of the set Ma is a nontrivial mathematical exercise. By moment theory techniques we first reduce an appropriate infinite dimensional problem to finite dimensions; some further calculations of a rather difficult nature then yield the exact boundaries of Ma

lt is well known that \M[F] - fi[F]\ < y/3cr[F] for any unimodal distribution F (cf. [7]). Using the exact formulae for the boundary of the set Ma, we generalize this inequality to all paired combinations of fi[F], m[F], and M[F] (see Corollary 4). The use of these inequalities in: (a) establishing nonunimodality, and (b) for obtaining bounds on m[F] or a[F] (when F is

4 Теория вероятностей и ее применения, № 2

338 Basu S., DasGupta A.

known to be unimodal), is described in Section 3.5. We also give an explicit quantification of the extent to which the mythical inequality in (1) can get violated (Theorem 5).

Section 4 focuses on multidimensional random variables. Here, we use the generalized notion of a-unimodality due to Olshen and Savage [12], and prove that for an a-unimodal distribution, an inequality similar to the one-dimensional case holds between the mode M and the mean y , . For a fixed mode M , Theorem 7 shows that the set of mean vectors y . for a-unimodal distributions with a specified covariance matrix is an ellipsoid around M. This ellipsoid is explicitly described. We consider this exact ellipsoidal representation very satisfying. If the covariance matrix is identity and we let M vary in a sphere, the mean vectors then form a sphere (see Theorem 8).

The principal achievements of this article are the following: (i) to explicitly characterize the three-dimensional set Ma of all possible

means, medians, and modes of univariate unimodal distributions; (ii) to use the set Mc m quantifying the extent and nature of violations

of the celebrated mean-median-mode inequality; (iii) to use the set Ma in establishing non-unimodality of certain distri

butions; (iv) to derive new sharp inequalities relating the mean, median, and

mode; and (v) to obtain some neat characterizations in the multivariate case for

generalized unimodal distributions.

2. Conditions for validity of the inequality: a review

The search for sufficient conditions under which the mean-median-mode inequality (cf. (1)) holds for a continuous unimodal distribution F dates back to Groeneveld and Meeden [6]. They assume F to be absolutely continuous (with respect to the Lebesgue measure) with density / , and their sufficient condition requires that / (m 4 - x) - / (m - x) changes sign once for x > 0 and. that / ( M + x) - / ( M - x) does not change sign (van Zwet [15] and MacGillivray [10] point out that their restriction to non-negative random variables is superfluous).

Van Zwet [15] shows that a more general sufficient condition is given by (assuming F has a density / )

F(m - x) + F(m + x) > 1 for all x. (4)

The following result is available. Theorem 1 (van Zwet). If condition (4) holds, then ц < m ^ M . If,

moreover, m ф M , then y . < m < M . An even more general sufficient condition, based on stochastic ordering,

is given in [4].


Theorem 2 (Dharmadhikari and Joag-Dev [5]). Let X be a unimodal random variable. If (X - m ) + is stochastically larger than (X — m)~, then X has a mode M satisfying M < m < fi.

Notice that Dharmadhikari and Joag-Dev do not assume existence of densities; neither do they assume that X has a unique mode. They further show that Theorem 1, and the sufficient condition of Groeneveld and Meeden follow as corollaries from Theorem 2; so also does the following result.

Corollary 1 (van Zwet). Let X be a unimodal random variable with density f and c.d.f. F. Iff(F~1(t)) sC - 1 ) ) for allO<t<%, then fi ^ m < M . / / , moreover, m ф M, then ц < m < M .

For various other ramifications of these results, we refer the reader to [6], [15], [4], [5].

3. Mean, median, and mode for univariate unimodal distributions

3 .1 . Preliminaries. Let X be a unimodal random variable with c.d.f. F £ T a. The goal is to explicitly characterize the set Ma of the mean fi, the median m, and the mode M for unimodal distributions with variance o2. Let T»a - {F: F is unimodal and EF(X) = fi, V&TF(X) = a2}, and let M% = {{ti,m[F],M[F]): F £ ?Ч) С { д } x R 2 . Clearly, M„ = (ti,fi,fi) + <тМ\; further, the set Ma is the union of the M„ sets, i.e., Ma — M*. Hence, from now on, we assume that ц = 0 and о — 1.

We first use the known fact that for any F € ?\, the set of modes form the interval [-\/3, \ /3] . Determination of the set M\ now goes along the following two principal steps:

(i) Fix M e [-\Д,\Д], and determine

m = inf m[F], m— sup m[F],

where ?\'M = {F e ?\: F is unimodal at M } . (ii) For every m < m < m, show that 3 F e т\,м such that m is the

median of F. Step (i) determines the boundaries of M\, whereas step (ii) proves that

it is path connected (as m ^ 0 ^ m for every fixed M, see Theorem 4 and Figure 1), and hence connected. Practically all the work goes into accomplishing step (i); the proof of step (ii) is easy on comparison.

P r o o f of s t e p (ii). Fix m £ (m,m). Since the infimum m and the supremum m are, in fact, attained (as we will show while accomplishing step (i)), 3Fi and F2 £ T°{M such that m[fi] = m and m[F2] = m. For 0 ^ a ^ 1, define the real valued function g(a) = aFi(m) + (1 - a)F2(m). The function g(a) is continuous in a £ [0,1]; further 5(0) = F2(m) < \ (since m < m) and 5(1) = F\(m) > \ (as m > m). Hence, За* e [0,1] such that g(a*) = \. Thus, m is the median of F* = a*Fr + (1 - a*)F2. Clearly,

4*

340 Baau S., DasGupta A.

Fip*(X) = 0, Varp»(X) = 1, and F* is unimodal about M (since both F\ and Fi are so); hence, F* G т\'м', and step (ii) is obtained.

V5 -

I—, ! , , r , r — I

Mxtg

Fig. 1. The region bounded by solid lines is the set M.\: the set of modes and medians of

univariate unimodal distributions when the mean is fixed at 0, and the variance

is fixed at 1. The dashed area is the set / ° , where the mean-median-mode

inequality holds

3.2. Reduction to mixtures of two uniforms. To find the infimum median m over the family т\,м (step (i) above), we use the following easily proved lemma (Lemma 2 in [1]) to show that m, in fact, can be described in terms of suprema of probabilities of intervals.

Lemma 1. m = i n f ^ ^ o . M m[F] = inf T e R {r: s u p F e : F o , M F(T) ^ j}.

P r o o f . See [1]. Determination of m, thus, can be done along the following steps: (i) for

т G R, find sup F e ^. 0 ,M F(T); (ii) find T = {r: s u p F € J F o , M F(T) > \ } ; and (iii) find m = inf{r: r G T}. A similar statement holds for

rri — sup m[F]. F 6 J r o , M

The next theorem shows that, for any r G R, sup F 6 ^-o ,M F(T) and i n f F 6 ^ - 0 , M F(T) are, in fact, attained at distributions F* G F ® , M which are mixtures of at most two uniforms.


Theorem 3. For any r £ R,

inf F(r) = inf H(T) and sup F(T) = sup Н(т),

гоЛеге W M = {II £ ^ ? ' M : Я = (1 - p)f/^ + pU™, 0 < p < 1, 771 ^ щ £ R}, and is <Ле c.d.f. of the Uniform[min(M,M + n ) , m a x (M ,M + 7 7 ) ] distribution.

R e m a r k . From existing moment theory techniques (see, for example, [ 11 ] , [9 ] ) , it follows that the problem of finding extrema of F(r) over the class T\,M can be reduced to finding the extrema over mixtures of at most three uniforms. The above theorem reduces the dimensions further only to mixtures of two uniforms, without which the exact analytic calculations we do in Theorem 4 perhaps would have been impossible.

P r o o f of T h e o r e m 3 . We will only describe the reduction for the infimum problem. Let X ~ F £ т\'м. By familiar arguments, we can

write X = M 4 - UZ, where U ~ Uniform [0 ,1], Z ~ G is a real random variable, and U and Z are independent. Further, ~EF(X) = 0 о E G ( Z ) — - 2 M and Var F (X) = 1 о E G ( Z 2 ) = 3 ( 1 + M 2 ) . Without loss of generality, we assume M ^ 0 (the conclusion for the case M < 0 follows by symmetry). We have to treat the following two cases separately.

C a s e I: r ^ M . Since т - M ^ 0 , a straightforward argument shows that PF{X ^ r) = PF{UZ ^ т - M} = f{z)dG(z), where f(z) equals 1 for 2 £ ( -oo ,r -M] , and equals ( T - M ) / Z for z £ ( r -M , o o ) . The problem at hand now reduces to finding inf f(z)dG(z) subject to E G ( Z ) = - 2 M and E G ( Z 2 ) = 3 ( 1 + M 2 ) . The assertion of Theorem 3 will follow from general moment theory (see, for instance, Theorem 2.1 and Remark 2 . 3 in Chapter XII in [ 8 ] ) if we can show that a quadratic a + bz + cz2 such that f(z) ^ a + bz + cz2 for all z can cut / at at most two points. However, this is easy to see. For such a quadratic, it easily follows that (i) с must be ^ 0 , and (ii) сф 0 . Let h(z) = f(z) - (a + bz + cz2). h(z) ^ 0 , and strictly convex on each of the subintervals ( - 0 0 , r - M ] and (r - M ,oo); thus proving that at each subinterval h(z) can have at most one zero.

C a s e II: r < M . The argument for Case II is quite similar and we skip the details.

Corollary 2.

inf m[F] = m = inf m[#] and sup m[F] — Ш— sup m[#]. F£T°-M ненм Fer°'M непм

Proof . It follows trivially from Lemma 1 and Theorem 3 .

3.3 . Formulae for extremal medians. Our next objective is to determine exact expressions for m and m (in terms of the mode M ) .


Theorem 4. For M ^ 0,

, M 3 - 27M + ( M 2 + 9 ) 3 / 2

m = mm m\F\ = ^ „ — - r - — — F e J F o , M

J 27(M 2 - 3)

and

' M 3 - 27M - ( M 2 + 9 ) 3 / 2

m — max m[F] = < F £ _ F O , M

27(M 2 - 3) 3 - МУ6 - M 2

I V6-M2-M

if О ^ М ^ ^ О . б ,

if VM ^ M < V3.

P r o o f . Since the proof is entirely technical, we defer it to the appendix.

The following consequences of Theorem 4 are worth noting separately. Corollary 3. (i) {m[F]: F is unimodal at 0, E F ( X ) = 0, Var F (X) = 1} = [ - £ ] . (ii) [m[F]: F is unimodal at V3, EF(X) = 0, Var F (X) = 1} = {0}. Proo f . Both (i) and (ii) follow trivially from Theorem 4. An alterna

tive proof of (ii) is that the only unimodal F with E F ( X ) = 0, Var F (X) = 1 and M[F] = V3 is F = U[-V5, л/3].

3.4. Subsequent results. In Figure 1, we plot the boundaries of the set Mi, i.e., m and m against M (M e [-V3,V3]). By step (ii) of Section 3.1, Mi is a connected set. The following corollary to Theorem 4 can readily be seen from Figure 1.

Corollary 4. For a unimodal distribution F with E F ( X ) = fi, medi-an[F] = m, mode[F] = M, and Var F (X) = a2,

(i)

(ii) i H ^ t ! < v^6,

(Ш) M ^ i ^ Vs. о

Moreover, each inequality is attained. Proof . Result (i) is well known. It was first obtained in [7]. Also see

[5, p. 9]. For (ii)," without loss of generality, we assume a = 1, ft — 0, and

M > 0. For fixed M > 0, let m{M} = т т ^ ^ м m[F], and m{M} =

max F 6 ^-o ,M m [ ^ ] , the lower and upper boundary points of M° corresponding to M. From Theorem 4, it can be proved that (d/dM)m{M} > 0 for M 6 [0,л\/3), i.e., m{M} is t in M. Also, m{M} is nondecreasing for M € [0,л/б], and nonincreasing for M б (Vb,V3\. Since m{M} < 0 and m{M} > 0, it follows that |m| ^ тах[тахм^о m{M}, - т т м ^ о m{M}]


= max[m{v/0T6}, -m{0}] = тах[л/(Гб, 5 ] = \АПб. This completes the proof of(ii). _

Towards proving (iii), let <pi(M) = |m{M} - M| = m{M} - M for 0 ^ M < VM, and ipi(M) = M - m{M} for \ЖЕ < M < л/3. Notice <p'i(M) ^ 0 for M e [0, л/бТб] and > 0 otherwise; thus max^i(M) = max[y>i(0),v>i(\/5)] = <fi(\/3) = \/3. Similarly, y>2(M) = |m{M} - M | = M - m{M} is nondecreasing in M, thus тах<^г(М) = угСл/З) = V3. This proves (iii) and completes the proof of the corollary.

Figure 2 is a three-dimensional plot of the set Ma=i — {(/i[F],m[F], M[.F]): F is unimodal and Varp(X) = er2 = 1}. As we mentioned before, Ma=\ is simply the three dimensional set obtained by translating the origin of the set M\ along the vector (ц,ц,ц).

Fig. 2. The set M.a=\\ the set of modes, means, and medians of univariate unimodal

distributions

Corollary 5. The set Ma is connected hut not convex. Proof . Nonconvexity is trivial (M® is not convex). We will show that


Ma is pathwise connected. Note that, for every fi £ R, the point (fi,fi,fi) £ Ma (because there is a symmetric unimodal distribution with mean = /z, median = \i, mode = fi, and variance = a2). The proof now follows from the fact that each /i-section of Ma is path connected (see Section 3.1).

In Example 1 and Figure 1, we have observed that there are unimodal distributions for which the alphabetical ordering of mean, median and mode does not hold. It is of natural interest to quantify the amount of maximum possible deviation from this ordering. Towards this end, let / = {(fi, m, M) £ R 3 : inequality (1) or its reverse holds}, and let S& = Ma Л I, the subset of Ma, where inequality (1) holds. For any 9 = (fi, m,M) £ Ma, let d(9,Sa) =

inf l 6 S o. \\в - n\\ be the i2-distance between the point 9 and the set Sa. The

quantity d(Ma,Sa) — s u p S e ^ d(9,Sa) is a reasonable quantification of the

maximum possible deviation from the mythical ordering. Without loss of generality, let a = 1. Instead of looking at the 3-

dimensional sets Ma=\ and Sa=\, we first look at a ti-section of them, in particular, the fi = 0 section. Recall, M\ was our notation for the fi = 0 section of Ma=\. Let 1° = {(fi,m,M) £ / : ц = 0}, and S® — M\ П 1°. The set 1° is the dashed area in Figure 1.

Theorem 5. d = 0.294931. P r o o f . For brevity, we will consider the case M ^ 0 (the proof for

M < 0 is similar). From Figure 1, it follows that

d(M4,S?)= sup inf | | a -£ | |

is given by the maximum of [A] the distance of the farthest point on the upper boundary m{M) of

M% from the 45° line m = M (for 0 < M ^ VM), and [B] sup M ^ 0 { min of (i) and (ii)}, where for each fixed M ^ 0 (i) distance of m{M} from the horizontal axis, and (ii) distance of m{M} from the 45° line m = M.

Towards [A], for each fixed M, the distance between the point (m{M} ,M) and the 45° line is the same as the distance between the two points (m{M}, M) , and ([m{M} + M ] /2 , [m{M} + M ] /2 ) , which equals > / 5 3 | M - m { M } | . Since |M - m{M} | is nonincreasing in M £ [0, V / 0J6] , the maximum attains at M = 0, and the maximum distance = 0.235702.

In [B], (ii) is similar to [A]; the distance is given by (for each fixed M) the distance between the points (m{M},M) and ([m{M} + M ] /2 , [m{M} + M ] /2) . Hence, for each fixed M, minimum of (i) and (ii) is equals to min [ - m { M } , v C 5 | M - m { M } | ] = v / 0 3 | M - m { M } | if 0 < M < 0.122164, and = - m { M } if 0.122164 ^ M ^ \Д (after some simplifications). But |M - m{M} | and m{M} are both nondecreasing in M; thus the supremum over M is attained at M = 0.122164 and the maximum distance = 0.294931. This proves the theorem.


From Theorem 5, it follows trivially that the Z2-distance between the two 3-dimensional sets Ma and Sa satisfies d(Ma,Sa) ^ 0.294931 a. We strongly suspect that, in fact, equality holds, i.e.,

Conjecture 1. d(Ma,Sa) = 0.294931 a.

Unfortunately, we were not able to prove this conjecture analytically.

3.5. Examples. Our objective in this section is to point out the possible directions of use of our obtained results. For clarity, we look at two simple examples, rather than considering complex (probably more realistic) examples. The first example shows how inequality (ii) in Corollary 4 can be used to prove nonunimodality of certain distributions. The second example outlines a method for obtaining useful bounds on the median of a fixed distribution when the actual median may be hard to obtain.

E x a m p l e 2. Let F be a mixture of two point masses and a uniform: F = PiS{-n}+P2U [-n,l]+P3<5{i},Pi ^ 0, Р1+Р2 +рз = 1, and the p^s are so chosen that E F ( X ) — 0 (8{ey denotes the degenerate distribution at в). F is clearly bimodal with modes at -n and 1, the two endpoints of its support. Let Хк denote the sample median of a sample of (2k - 1) observations from F and let Fk denote the distribution of Xk- It is known that median of Fk

= median of F, i.e., m[.Ffc] = m[F] (see [13]). For small values of k, say к = 2 or 10, though the form of F). can easily be written down, proving unimodality or nonunimodality of Fk seems nontrivial. However, violation of the inequality in part (ii) of Corollary 4 (V* = 0.6Var(Xfc) - [m[F] -E(Xfc)]2 < 0) is a sufficient condition for nonunimodality of the distribution Fk. Evaluation of E(X^) and Var(Xjt) needs some numerical work, but is much more straightforward compared to a direct check of unimodality. For example, for n = 1.1 and p 3 close to 0.5 (but < 0.5), V* turns out to be < 0 for all к ^ 39, thus proving that Xk has a nonunimodal distribution for к < 39.

E x a m p l e 3. The inequality of Corollary 4, namely (median-mean)2 ^ 0.6 Variance, can also be used to obtain useful bounds on mean or median or variance, depending on which two of the three are easier to obtain.

As a simple verification of how it works, let X ~ exp(A) with ц = E(X) = 1/Л. Further, F(x) = 1 - e~Xx,x > 0, and solving F(x) = \ gives median m = log(2)/A. From the above inequality, we obtain Var(X) ^

( м - m ) 2 = 0.5114213/A2, which is approximately half of Var(X) = 1/A2.

For another simple example, let X ~ F = Beta(a,/3). Thus, \i — a/(a + /3) and a2 = af3/((a + /3)2(a + (3 + 1)). Moreover, if a > 1, and P > 1, then F is unimodal at M = (a - l)/(a + /3 - 2); if a < 1, /3 > 1 (a ^ 1, /3 ^ 1), F is unimodal at M = 0 (M = 1). But evaluation of the median m of F requires solving an equation involving incomplete Beta integrals. Corollary 4, however, gives the following useful bounds on m

346 Baau S., Das Gupta A.

(for a < 1 or /3 jt 1):

max | / * - \AL6<T, M - v^trj ^ m ^ min j/i + л/оТбсг,М + v^crj . (5)

These bounds can easily be evaluated without any numerical work. Table 1 shows the bounds obtained from (5) along with the actual values of the median rn for different combinations of a and fi.

Table 1 . Bounds on the median m of Beta(a,/3) distribution

a 0.5 1.0 1.5 2.0

1.0 Bounds m

(0.484,0.898) 0.750

(0.500,0.724) 0.500

(0.197,0.454) 0.370

(0.151,0.408) 0.293

1.5 Bounds m

(0.567,0.944) 0.837

(0.546,0.803) 0.630

(0.306,0.694) 0.500

(0.248,0.609) 0.414

2.0 -Bounds m

(0.634,0.966) 0.879

(0.592,0.849) 0.707

(0.391,0.752) 0.586

(0.327,0.673) 0.5

3.0 Bounds m

(0.729,0.985) 0.921

(0.665,0.900) 0.794

(0.511,0.822) 0-693

(0.445,0.755) 0.614

4.0 Bounds m

(0.785,0.993) 0.941

(0.717,0.926) 0.841

(0.592,0.863) 0.756

(0.529,0.805) 0.686

4. Multivariate unimodality

Our objective in this section is to generalize some of the results of Section 3 in the setting of multivariate unimodal distributions. Unlike in one dimension, there are several definitions of unimodality in higher dimensions such as star unimodality, block unimodality, central convex unimodality, and log concavity (and more; see [5]). We wilkrestrict ourselves to the generalized notion of a-unimodality, introduced by Olshen and Savage [12], of which star-unimodality is a special case.

D e f i n i t i o n 1 (Olshen and Savage). A p-dimensional random vector X is said to have an a-unimodal (a > 0) distribution about M if, for every bounded, nonnegative, Borel-measurable function g on R p , ta E[o(t(X-M))] is nondecreasing in t e (0,oo).

The definition given by Olshen and Savage is, in fact, more general; it applies for X taking values in any p-dimensional vector space. It can be seen that ordinary unimodality on R is equivalent to Definition 1 with p = 1 and a = 1. In general, Definition 1 corresponds to the property of starshaped


level sets for the distribution of X. Theorem 6 (Olshen and Savage). A p-dimensional random vector X

is a-unimodal (about M) if and only if X = U1/a Z + M, where U ~ Uniform[0,1] and Z is a p-dimensional random vector independent of U.

A p-dimensional and p-unimodal (about M) random vector X is called star unimodal (about M). Generally, star unimodality is defined through a natural extension of the idea of one-dimensional unimodality, and the above definition follows as an equivalent version. The concept of star unimodality has recently been successfully used in inference problems, see [2].

Since the use of a median in the multivariate case is less pervasive, we focus our efforts on extending part (i) of Corollary 4 given in Section 3.4. Let us make it more precise.

Let 0 denote the set of all possible mean vectors у corresponding to p-dimensional a-unimodal random vectors X with mode[X] = M and

covariance matrix D(X) = E, i.e.,

® M , E — {ё = E(X): X is a-unimodal about M and D(X) = E}. (6)

We have the following neat ellipsoidal representation of &M,E-Theorem 7. Assume |E| ф 0. Then

© M , £ = {& (ё - ¥ ) т Е - г ( ё - M) < a(a + 2 ) } .

Proof . Without loss of generality, we take M = 0. Further, if X is

a-unimodal (we use a-unimodal to imply a-unimodal about 0) with E(X) =

у and D(A') = E, then for ApXp nonsingular, Y = AX has E(F) = Ay.

and T>(Y) = AT,AT. It follows from Theorem 6 that Y is also a-unimodal.

Taking A = E - 1 / 2 shows that it is enough to prove the result for E = / . The proof will be divided into the following steps: (i) у € ©о,/ yTy ^a(a + 2) = a* (say). (ii) у £ 0o,/ => Py G 0o,/ for any orthogonal matrix P.

(iii) For every scalar 7 satisfying -Va* ^ 7 < \/a*, у = (j,0,..., 0)T G

в 0 , / .

We claim that these three steps show 0O,/ = {у: уту ^ a*} and proves

the theorem. That 0О / С {у. уту ^ a*} follows from step (i). Next, for

any у with уту = 7 2 ^ a*, 3 P orthogonal such that Py — ( 7 , 0 , . . . ,0)T.

By (iii), Py G 0o,/> and hence by step (ii), у G ©o,/-

Step (ii) is an easy consequence of Theorem 6. For proving (i), let X

be a-unimodal with D(X) = / . By Theorem 6, X = V Z, where V is a

348 Basu S., Das Gupta A.

scalar random variable, V — U1!01 with U ~ Uniform[0,1], and V and Z

are independent. Note that \iv = E(V) — a/(a + 1 ) arid o2, = Var(V) = a/((a + 2 ) ( a + l) 2 ); thus a* = рЦо2. Now,

J = D(X) = D(FZ) = alE(ZZT) + /4D(Z) ^ alE(ZZT). ( 7 )

Thus, уту ^ nX/ol, which proves (i).

To prove (iii), fix -\fa* < 7 ^ Va*. Consider p independent scalar random-variables Z\, Z 2 , . . . , Zv such that E(Zi) = f/fiv, E ( Z 2 ) = ( I + 7 2 ) / (a2 + p2), and for i = 2 , . . . , p , E(Z;) = 0 , E ( Z 2 ) = (<r2 + /л 2)" 1. Take Z = ( Z i , . . . , ZP)T and now pick a scalar random variable V independent of Z

such that V = U1^" with U ~ Uniform[0,1]. X = V Zis clearly a-unimodal,

moreover, it can be seen that D(X) = / and E(X) = ( 7 , 0 , . . . , 0 ) T . The

proof of the theorem is therefore complete. R e m a r k . Notice the ellipsoids of Theorem 7 are nested, i.e., the

same ellipsoid is obtained if D(X) ^ E. Corollary 6. Let X be a p-dimensional random vector, star unimodal

about M, with ЩХ) = y, D{X) = E. Then ( g - M ) T E _ 1 ( g - M ) ^ p ( p + 2 ) .

P r o o f . Immediate from Theorem 7. Thus, according to Theorem 7, the means у of a-unimodal distributions

lie in an ellipsoid with center at the mode M, and the axes of the ellipsoid are multiples of the eigenvectors of the dispersion matrix E.

In the above, we assume that the mode M is fixed at a certain point in R p . Next, we look at the set of mean vectors у of a-unimodal distributions, when the mode M is also allowed to vary in a sphere in R p .

Theorem 8. Let 0д/,£ be as defined before, and let

On the other hand,

yyT = ЩХХТ) - ЩХ) = E(V2) E ( Z Z T ) - I

^I^P--1}1 (from(7)) =%I-О у J uv

M e n X is a-unimodal about M with M G fi

where the fi is the sphere {M: (M - M 0 ) T ( M - M„) ^ /32}.

Then the set of mean vectors у is again a sphere, and equals

0 , = {у: (y - M0)T(y - M 0 ) <(Vo7+(3)2}.


Proof . The proof is notationally slightly complex, but is indeed nothing more than a proof of the fact that the Minkowski sum of two spheres is again a sphere.

Without loss of generality, we take M 0 = 0. For a random vector X,

a-unimodal about M with ЩХ) = у and D(X) = / , by Theorem 7, we have

(y-M)T(y-M) ^ a*. If, moreover, M £ ft, then \\у\\ ^ ||g-M|| + ||M|| ^ y/a*+0; this shows that 0 7 С {у. ут у ^ (у/а* + /З)2}.

For proving that в / equals the entire sphere, pick any у with \\y\\ =

7 ^ y/a* + /3. Find an orthogonal matrix P such that Py = (7 ,0 , . . . , 0)T,

and next find 0 < 71 < y/a* and 0 ^ 72 ^ P such that 7 = 7 1 + 7 2 . Clearly, M* = ( 7 2 , 0 , . . . , 0)T £ ft, and for such an M*, 3 a random vector X,

a-unimodal about M*, with D(X) = / and E(X) = y* such that у* - M* =

(7i ,0, . . . ,0) T . Thus, у* = (у* - M*) 4 - M*, and hence y* £ 0 7 . Now,

у = (PTy* - PTM*) + PTM*, and PTy* - PTM* £ 0 p 7 - M V ; PTM* £ 0, hence ^ £ 0/. This completes the proof.

5. Summary and discussions

Past work in this area was directed towards finding more and more general sufficient conditions for the validity of the mean-median-mode inequality. In contrast, we concentrate our attention to exact characterizations of the three-dimensional set of means, medians and modes. Derivation of the set involves some novel use of moment theory, and, most surprisingly, we were able to analytically describe this set. Furthermore, this exact description enabled us to derive some fundamental inequalities among the mean, median and mode of unimodal random variables. We were also able to specify the region where the inequality gets violated, and found the maximum extent to which a unimodal random variable may violate the mean-median-mode inequality. In the multivariate case, we obtain a very pleasant ellipsoidal characterization of the set of means of a-unimodal distributions with a fixed mode M. When the mode M is also allowed to vary in a sphere, we prove that the mean vectors form a larger sphere.

Several open questions remain. We conjectured that the maximum deviation from the inequality is 0.294931(7 and we proved it for each fixed mean y, but were unable to prove it in its fuU generality. In Theorem 8 we assume T, = I, and that the mode M is in a sphere. The case of a general £

and when the mode varies in other nicely structured convex sets (such as an ellipsoid, or a rectangle) are hard geometric problems, and again, open for exploration. We believe the geometric results in [3] are of probable relevance here.


6. Appendix P r o o f of T h e o r e m 4. The derivation of m and m looks intim

idating, but it is really straightforward on patient verifications. First note that, by Corollary 2, we only need consider distributions of the form Я = ( 1 -p)U% + pU™ (ту! < 772) satisfying EH(Z) = - 2 M and BH(Z2) = 3(1+ M 2 ) . Without loss of generality, we assume M > 0. Since E # (Z) = - 2 M ^ M , it follows that 771 must be < 0.

C a s e I [771 < 0, 772 ^ 0]. For notational simplicity, we write 771 = -2a, 772 = 26, where a, 6 > 0. Thus, the distribution Я is of the form

Я = (1 - p) U[-2a + M,M] + p U[M, M + 26]. (8)

Now, E # (Z) = - 2 M => p = (a - M)/(a + 6), p ^ 0 a ^ M. Further, E H ( Z 2 ) = 3(1 + M 2 ) allows us to write

4M6 + 3(1 + M 2 ) 3(l + M 2 ) - 4 a M a ~ 4(6+ M) " ° r * ~ 4 ( a - M ) ( 9 }

(we will use either expression as necessary). For ease of calculations, we further subdivide Case I to the following two subcases.

S u b с a s e IA [m[#] > М]. From (8), m[H) >Mop>\<^b< - M + 1 / 3 - M 2 / 2 on simplification (using (9)). The condition 6 ^ 0 requires - M + V3-M2/2 ^ 0 о M 2 ^ 0.6; hence, Subcase IA is possible only if 0 ^ M ^ vCe.

Since Я has no jumps, we have 1 - Я ( т ) = \. Solving for m (and use of (9)) gives m = 26 + M -6 (2M6 + 62 + 0.75(1 + M 2))/(0.75 - 0.25M 2) = ф\{Ь) (say). 6* = ( - 4 M + VM2 + 9)/6 is the only non-negative solution to Ф[{Ь) = 0. Moreover, il>"(b*) < 0, and 6* satisfies the required boundary conditions. Hence,

шл , fu*\ M 3 - 2 7 M - ( M 2 + 9)X- 5

S U S S I A Т [ Я ] = М Ь ) = 2 7 ( M 2 - 3 )

Notice that, for each fixed M ^ 0, this subcase restricts the median m [ # ] to be > M , whereas Subcase IB and Case II below only allows т [ Я ] ^ M . Thus, for those M where Subcase IA is possible (i.e., for 0 ^ M < л/ОТб), m = max S ubcaseiA т [ Я ] . For the same reason (since other feasible cases allow т [ Я ] ̂ M) , we refrain from evaluating inf subcase IA т [ Я ] .

S u b c a s e IB [ т [ Я ] ^ М]. For ease of calculations, we formulate this case in terms of «a». The domain of «a» is bounded by: ( i ) p ) 0 o a ) M , (ii) 6 ̂ 0 о a < 3(1 + M 2 ) / ( 4 M ) = A b and (iii) т [ Я ] ^ M о a < (2M + v / 3 - M 2 ) / 2 = A 2 . Thus, a e [M, min(A b A 2 )] , and min(A b A 2 ) = A2 if 0 ^ M 2 < 0.6, and - Ai otherwise.

Solving Я ( т ) = \ gives m = M - 2a + a (4a2 - 8aM + 3 + 3M 2 ) / (3 -M 2 ) = ф2(а) (say). ф'2(а) = 0 has only one solution a* = (4M+VM2 + 9)/6 in the domain of «a»; further, ф2(а*) > 0, i.e., a* is a local minimum.


Towards determining maxm[/i], note that we are really only interested in the case M 2 > 0.6 (see Subcase IA), which implies a e [M, Ai]. Clearly, max т [Я] is attained at the boundaries of the domain of «a». Thus, for л/Мб < M < %/3, maxsubcasem т [Я] = max{V>2(M),^2(Ai)} = ^ ( A i ) = ( 3 - M 2 ) 2 / ( 1 6 M 3 ) .

min ,02(a) is clearly attained at a = a*, and

r „ , . . M 3 - 2 7 M + ( M 2 + 9) 1- 5

s . f f i IB m [ H ] = M a } = 27(M2 - 3) •

It can be shown that minsubcase IB m [ # ] < 0 for 0 ̂ M < л/3, and = 0 at M =

C a s e II [rji < 0, т?2 ^ 0]. Again, for notational convenience, we write rji = -2a,щ = -26, where a > 6 ̂ 0. Thus,

Я = (1 - p) U[-2a + M, M] + p [-26 + M, M], (10)

and from the conditions EH(Z) = - 2 M and E#(Z 2 ) = 3(1 + M 2 ) , we have

a-M , , 4 a M - 3 ( l + M 2 ) , 1 1 4

p = г - and 6=——-7—^—p- ^. (11) a - 6 4(a - M) 4 '

Now, 0 ^ p < l = > O M ^ 6 > 0 , and ^ f l o a ) (3(1+ M 2 ) ) / (4M) = A* (from (11)). Thus, a ^ max(M, A*) = A\. Depending on the position of т [Я] , we subdivide Case II into following subcases.

S u b c a s e ПА [т[Я] < -26 + М]. From (10), т [Я] < -26 + M о (l-p)(2a-26)/(2a) ^ \ о a < (M+V6 - M 2 ) / 2 = A 2 . Thus, A* < a < AJ, however, for M 2 < 0.6, A* < A*; hence, this subcase is possible only if 0 . 6 < M 2 < 3 .

As before, solving Я ( т ) = \ gives m = M - 2a - a (4a2 - 8aM + 3 + 3M 2 ) / (3 - M 2 ) = фз(а) (say). фз(а) turns out to be increasing for ae[AJ,A;],(^(o)>0);hence,

гн-i / гл*1 3 - MV6 - M 2

max m Я = = , Subcase НА L J L J y/6 - M2 - M

and

S u b c a s e IIB [т[Я] > -26 + М]. т [Я] ^ -26 + M => a ̂ AJ; thus, a ̂ max[A*, A*] = A* if M 2 ^ 0.6, and = A\ for M 2 ^ 0.6. Following similar steps as in Subcase IIA, we get m = M - a (3 + 3 M 2 - 4aM)/(4aM -4a2 - M 2 + 3) = ф4(а) with ф'4(а) < 0 Va. Thus, max S u b c a s e I I B т [Я] = ф4[А*] = т а х 8 и Ь с а 8 е 1 1 А т [ Я ] (if M 2 ^ 0.6) and min S u b c a S eнв т [Я] = l i m a - ю о ф4(а) - 0.

Combining all the cases, we know from Subcase IA that for 0 < M <

Tn = max т [Я] . Subcase IA


For \Ш> < M < л/3,

m = max \ max m[H], max m[H] > = max m[H]. I. Subcase IB Case II J Case II

For m, we note that minsubcase ПА т[Щ and minsubcase ив m[H] a r e both > 0, whereas minsubcase IB m[H] ^ 0. The proof of Theorem 4 is therefore complete.

Acknowledgement. We have benefitted from valuable discussions with Johannes Kemperman, Friedrich Pukelsheim, Yosi Rinott, and Willem van Zwet. The first author is indebted to Rajeeva Karandikar, Mathew Penrose, and Srikanth Iyer for numerous helpful discussions during this work. They all have our sincerest appreciation.

REFERENCES

1. Basu S., DasGupta A. Bayesian analysis with distribution bands: the role of the loss function. Technical Report, 208, Departement of Statistics and Applied Probability, University of California, Santa Barbara, 1992.

2. DasGupta A., Ghosh J. K., Zen M. A new general method for constructing confidence sets in arbitrary dimensions, with applications. — Ann. Statist., (в печати).

3. DasGupta A., Studden W. J. Robust Bayesian Analysis and optimal experimental designs in normal linear models with many parameters. I. Technical Report, 88-14, Department of Statistics, Purdue University, 1988.

4. Dharmadhikari S. W., Joag-dev K. Mean, median, mode III. — Statist. Neerl., 1983, v. 37, p. 165-168.

5. Dharmadhikari S. W., Joag-dev K. Unimodality, Convexity, and Applications. New York: Academic Press, 1988.

6. Groeneveld R. A., Meeden G. The mode, median and mean inequality. — Amer. Statistician, 1977, v. 33, p. 120-121.

7. Johnson N. L., Rogers C. A. The moment problem for unimodal distributions. — Ann. Math. Statist., 1951, v. 22, № 3, p. 432-439.

8. Karlin S., Studden W. J. Tchebycheff systems: with applications in analysis and statistics. New York: John Wiley, 1966.

9. Kemperman J. H. B. The general moment problem, a geometric approach. — Ann. Math. Statist., 1968, v. 39, № 1, p. 93-122.

10. MacGillivray H. L. The mean, median, mode inequality and skewness for a class of densities. — Austral. J. Statist., 1981, v. 23, № 2, p. 247-250.

11. Mulholland H. P., Rogers C. A. Representation theorems for distribution functions. — Proc. London Math. Soc , 1958, v. 8, № 30, p. 177-223.

12. Olshen R. A., Savage L. J. A generalized unimodality. — J. Appl. Probab., 1970, v. 7, № 1, p. 21-34.

13. Reiss R.-D. Approximate Distributions of Order Statistics. Berlin: Springer-Verlag, 1989.

14. Runnenburg J. Th. Mean, median, mode. — Statist. Neerl., 1978, v. 32, p. 73-80 . 15. van Zwet W. R. Mean, median, mode, II. — Statist. Neerl., 1979, v. 33, p. 1-5.

Поступила в редакцию 1 4 . X I I . 1 9 9 3

the mean, median, and mode of unimodal …...3. mean, median, and mode for univariate unimodal...

Documents