Further some inequalities are given for the moments of bivariate distributions

http://jipam.vu.edu.au/

Volume 5, Issue 4, Article 86, 2004

SOME INEQUALITIES BETWEEN MOMENTS OF PROBABILITY DISTRIBUTIONS

R. SHARMA, R.G. SHANDIL, S. DEVI, AND M. DUTTA DEPARTMENT OFMATHEMATICS

HIMACHALPRADESHUNIVERSITY

SUMMERHILL, SHIMLA-171005, INDIA

shandil_rg1@rediffmail.com

Received 23 February, 2004; accepted 19 July, 2004 Communicated by T. Mills

ABSTRACT. In this paper inequalities between univariate moments are obtained when the ran- dom variate, discrete or continuous, takes values on a finite interval. Further some inequalities are given for the moments of bivariate distributions.

Key words and phrases: Random variate, Finite interval, Power means, Moments.

2000 Mathematics Subject Classification. 60E15, 26D15.

1. INTRODUCTION

Therth order momentµ⁰_r of a continuous random variate which takes values on the interval [a, b]with pdfφ(x)is defined as

(1.1) µ⁰_r =

Z b

a

x^rφ(x)dx.

For a random variate which takes a discrete set of finite values x_i (i = 1,2,. . ., n)with corre- sponding probabilitiesp_i (i= 1,2,. . ., n), we define

(1.2) µ⁰_r =

n

X

i=1

pix^r_i. The power mean of orderris defined as

(1.3) M_r = (µ⁰_r)^1/r for r6= 0,

and

(1.4) M_r = lim

r→0(µ⁰_r)^1/r for r= 0.

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

040-04

It may be noted here thatM−1,M₀andM₁respectively define harmonic mean, geometric mean and arithmetic mean.

Kapur [1] has reported the following bound for µ⁰_r when µ⁰_s is prescribed, r > s, and the random variate, discrete or continuous, takes values in the interval[a, b]witha≥0,

(1.5) (µ⁰_s)^r/s ≤µ⁰_r ≤ (b^r−a^r) µ⁰_s+a^rb^s−a^sb^r b^s−a^s .

Inequality (1.5) gives the condition which the given moment values must necessarily satisfy in order to be the moments of a probability distribution in the given range [a, b]. Kapur [1]

was motivated by the consideration of maximizing the entropy function subject to certain con- straints. But before maximizing the entropy function one has to see whether the given moment values are consistent or not i.e whether there is any probability distribution which corresponds to the given values of moments. If there is no such distribution then the efforts of finding out the maximum entropy probability distribution will not produce any result and hence we should not proceed to apply Lagrange’s or any other method to find the maximum entropy probability distribution, [2].

Here we try to obtain a generalization of inequality (1.5) for the case where r and s can assume any real value. This shall help us in deducing bounds between power means. This will also provide us with an alternate proof of inequality (1.5) and enable us to tighten it when the random variate takes a finite set of discrete valuesx₁, x₂,. . ., x_n.

In addition some inequalities between the moments of bivariate distributions are also ob- tained.

2. SOMEELEMENTARYINEQUALITIES

We prove the following theorems:

Theorem 2.1. Ifris a positive real number andsis any non zero real number withr > sthen fora ≤x≤b; witha >0, we have

(2.1) x^r ≤ (b^r−a^r) x^s+a^rb^s−a^sb^r b^s−a^s , and forxlying outside(a, b)we have

(2.2) x^r ≥ (b^r−a^r) x^s+a^rb^s−a^sb^r b^s−a^s .

Ifr is a negative real number with r > sthen inequality (2.1) holds for xlying outside(a, b) and inequality (2.2) holds fora≤x≤b.

Proof. Consider the following functionf(x)for positive real values ofx:

(2.3) f(x) =x^r− b^r−a^r

b^s−a^sx^s+a^sb^r−a^rb^s b^s−a^s ,

whererandsare real numbers such thatr > sands 6= 0. The functionf(x)is continuous in the interval[a, b]witha >0. Thenf⁰(x)is given by

(2.4) f⁰(x) = x^s−1

rx^r−s−s

b^r−a^r b^s−a^s

. f⁰(x)vanishes atx= 0andc, where

(2.5) c=

s r

b^r−a^r b^s−a^s

_r−s¹ .

By Rolle’s theorem we have thatclies in the interval(a, b).

Ifr is a positive real number ands is a negative real number withr > sthenf⁰(x) ≤ 0iff x≤c. This means thatf(x)decreases in the interval(0, c)and increases in the interval(c,∞).

Further, sinceclies in the interval(a, b)andf(a) = f(b) = 0, it follows that

(2.6) f(x)≤0 for a≤x≤b,

and forxlying outside(a, b)

(2.7) f(x)≥0.

On substituting the value off(x)from equation (2.3) in inequalities (2.6) and (2.7), we obtain inequalities (2.1) and (2.2) respectively.

Ifr is a negative real number with r > s then f⁰(x) ≤ 0 iff x ≥ c. This means that f(x) increases in the interval(0, c)and decreases in the interval (c,∞). Sinceclies in the interval (a, b)andf(a) =f(b) = 0it follows that inequality (2.7) holds fora ≤x≤bwhile inequality (2.6) holds forxlying outside(a, b)and thus we get inequalities for the case whenris negative

real number.

Theorem 2.2. Fora≤x≤bwitha >0, we have

(2.8) x^r ≤ (b^r−a^r) logx+a^rlogb−b^rloga logb−loga , and forxlying outside(a, b), we have

(2.9) x^r ≥ (b^r−a^r) logx+a^rlogb−b^rloga logb−loga , whereris a real number.

Proof. Consider the following functionf(x)defined for positive real values ofx, (2.10) f(x) =x^r− (b^r−a^r)

logb−logalogx+ b^rloga−a^rlogb logb−loga .

The functionf(x)is continuous in the interval[a, b]wherea >0. Thenf⁰(x)is given by

(2.11) f⁰(x) = 1

x

rx^r− b^r−a^r logb−loga

, and we havef⁰(x) = 0atx=cwhere

(2.12) c=

b^r−a^r r(logb−loga)

¹_r .

By Rolle’s Theorem we have that clies in the interval(a, b). Alsof⁰(x) ≤ 0iff x ≤ c. This means that f(x) decreases in the interval(0, c) and increases in the interval (c,∞). Further, sinceclies in the interval(a, b)andf(a) = f(b) = 0it follows that

(2.13) f(x)≤0 for a≤x≤b,

and forxlying outside(a, b)we have

(2.14) f(x)≥0.

On substituting the value of f(x) from equation (2.10) in inequalities (2.13) and (2.14), we

obtain inequalities (2.8) and (2.9) respectively.

3. INEQUALITIESBETWEENMOMENTS

Theorem 3.1. Letrbe a positive real number andsbe any non zero real number withr > s. If a positive random variate takes valuesxi(i= 1,2,. . ., n)in the interval[a, b], witha >0, then we have

(3.1) µ⁰_r ≤ (b^r−a^r) µ⁰_s+a^rb^s−a^sb^r b^s−a^s , and

(3.2) µ⁰_r ≥ x^r_j −x^r_j−1

µ⁰_s+x^r_j−1x^s_j −x^s_j−1x^r_j x^s_j−x^s_j−1 , wherej = 2,3,. . ., n.

If a continuous random variate takes values in the interval[a, b],witha > 0, then the upper bound for µ⁰_r is given by the inequality (3.1) whereas the lower bound is given by following inequality

(3.3) µ⁰_r ≥(µ⁰_s)^r/s.

Proof. It is seen thatµ⁰_rcan be expressed in terms ofµ⁰_sin the following form : (3.4) µ⁰_r = x^r_β−x^r_α

x^s_β−x^s_α

!

µ⁰_s+x^s_βx^r_α−x^s_αx^r_β x^s_β −x^s_α +

n

X

i=1

p_i

"

x^r_i −x^r_β −x^r_α

x^s_β −x^s_αx^s_i +x^r_βx^s_α−x^r_αx^s_β x^s_β−x^s_α

# ,

whereαandβ take one of the values among1,2,. . ., nwithα 6=β. Without loss of generality we can arrange values of the variate such thata =x₁ ≤x₂ ≤ · · · ≤x_n =b. If we takeα= 1 andβ = nthen x₁ ≤ x_i ≤ x_n fori = 1,2, ,. . ., n. It follows from (2.1) that the last term in equation (3.4) is negative and we conclude that the upper bound forµ⁰_r is given by inequality (3.1). Further ifx_α = xj−1 andx_β = x_j, j = 2,3,. . ., nthen each x_i lies outside (xj−1, x_j) and it follows from (2.2) that the last term in equation (3.4) is positive and we conclude that the lower bound forµ⁰_r is given by inequality (3.2). It is also clear that equality in the inequalities (3.1) and (3.2) holds iffn = 2.

If the value ofµ⁰_scoincides with one ofx^s_j−1orx^s_j,then from inequality (3.2) we have

(3.5) µ⁰_r ≥(µ⁰_s)^r/s.

Also if xj−1 approaches xjwe get inequality (3.5) and we conclude that for a continuous random variate the lower bound forµ⁰_ris given by inequality (3.5). The upper bound forµ⁰_rcan be deduced from Theorem 2.1. Multiplying both sides of inequality (2.1) by pdfφ(x)we get, on using the properties of definite integrals, inequality (3.1).

Theorem 3.2. Let rand sbe negative real numbers withr > s. If a positive random variate takes valuesx_i (i= 1,2,. . ., n)in the interval[a, b], witha >0, we have

(3.6) µ⁰_r ≥ (b^r−a^r) µ⁰_s+a^rb^s−a^sb^r b^s−a^s , and

(3.7) µ⁰_r ≤ x^r_j −x^r_j−1

µ⁰_s+x^r_j−1x^s_j −x^s_j−1x^r_j x^s_j−x^s_j−1 , wherej = 2,3,. . ., n.

If a continuous random variate takes values in the interval [a, b], with a > 0, the lower bound forµ⁰_r is given by inequality (3.6) whereas the upper bound forµ⁰_ris given by following inequality:

(3.8) µ⁰_r ≤(µ⁰_s)^r/s.

Proof. We again consider equation (3.4). If we takeα = 1andβ = nthenx₁ ≤ x_i ≤ x_nfor i = 1,2,. . ., n.It follows from Theorem 2.1 that the last term in equation (3.4) is positive and we conclude that the lower bound for µ⁰_r is given by inequality (3.6). Also ifxα = xj−1 and xβ = xj,j = 2,3,. . ., nthen eachxi lies outside(xj−1, xj). It follows from Theorem 2.1 that the last term in equation (3.4) is negative and we conclude that the upper bound forµ⁰_r is given by inequality (3.7). Also ifx_j−1 approachesx_j we get inequality (3.8). The lower bound for µ⁰_rcan be deduced from Theorem 2.1. Multiplying both sides of inequality (2.2) by pdfφ(x)we get, on using the properties of definite integrals, inequality (3.6).

Theorem 3.3. For a random variate which takes valuesx_i(i= 1,2,. . ., n)in the interval[a, b], witha >0, we have

(3.9) µ⁰_r ≤ (b^r−a^r) logM0+a^rlogb−b^rloga logb−loga , and

(3.10) µ⁰_r ≥ x^r_j −x^r_j−1

logM₀+x^r_j−1logx_j −x^r_jlogx_j−1 logx_j−logx_j−1 , wherej = 2,3,. . .n,ris a real number and

(3.11) M₀ =x^P₁¹x^P₂²· · ·x^P_nⁿ.

For a continuous random variate which takes values in the interval [a, b] witha > 0 the up- per bound for µ⁰_r is given by inequality (3.9) whereas the lower bound for µ⁰_r is given by the following inequality

(3.12) µ⁰_r ≥(M₀)^r.

Proof. It is seen thatµ⁰_rcan be expressed in terms oflogM₀ in the following form:

(3.13) µ⁰_r = x^r_β−x^r_α logxβ−logxα

logM₀+ x^r_αlogx_β −x^r_βlogx_α logxβ−logxα

+

n

X

i=1

Pi

x^r_i − x^r_β −x^r_α

logx_β −logx_α logxi+x^r_βlogxα−x^r_αlogxβ

logx_β−logx_α

. Without loss of generality we can arrange values of the variate such thata=x₁ < x₂ <· · · <

x_n = b. If we take α = 1and β = n then x₁ ≤ x_i ≤ x_n fori = 1,2,. . ., n. It follows from Theorem 2.2 that last term in equation (3.13) is negative and we conclude that the upper bound forµ⁰_r is given by inequality (3.9). Also ifx_α = xj−1 andx_β = x_j, j = 2,3,. . ., nthen each x_i lies outside (xj−1, x_j). It follows from Theorem 2.2 that the last term in equation (3.13) is positive and we conclude that the lower bound forµ⁰_r is given by inequality (3.10).

If the value ofM₀ coincides with one ofxj−1orx_j then from inequality (3.10) we have

(3.14) µ⁰_r ≥(M0)^r.

Also if xj−1 approaches x_j we get inequality (3.14) and we conclude that for the continuous random variate the lower bound forµ⁰_r is given by inequality (3.14). The upper bound for µ⁰_r can be deduced from Theorem 2.2. Multiplying both sides of inequality (2.8) by pdfφ(x)we get, on using the properties of definite integrals, inequality (3.9).

4. INEQUALITIESBETWEEN MOMENTS OF BIVARIATEDISTRIBUTIONS

The moments of a bivariate probability distribution are the generalizations of those of uni- variate one and are equally important in the theory of mathematical statistics. For a discrete probability distribution, if p_i is the probability of the occurrence of the pair of values(x_i, y_i) i= 1,2,. . ., n, the momentµ⁰_rsabout the origin is given by

(4.1) µ⁰_rs =

n

X

i=1

P_ix^r_iy_i^s. We obtain a bound onµ⁰_rsin the following theorem:

Theorem 4.1. Letµ⁰_rsbe the moment of orderrinxand of ordersiny, about the origin(0,0), of a discrete bivariate probability distribution. The random variatesxandyvary respectively over the finite positive real intervals [a, b] and [c, d]. If µ⁰_{k m} is the corresponding moment of orderkinxandminysuch thatr≥k,s≥mandrm=ksthen we must have by necessity, (4.2) (µ⁰_{k m})^k+mr+s ≤µ⁰_{r s} ≤ (b^rd^s−a^rc^s) µ⁰_{k m}+a^rc^sb^kd^m−a^kc^mb^rd^s

b^kd^m−a^kc^m .

Proof. Ifu,v,αandβ are positive real numbers withα+β = 1then from Hölder’s inequality [3],

(4.3)

n

X

i=1

u^α_iv_i^β ≤

n

X

i=1

u_i

!α n

X

i=1

v_i

!β

. We make the following substitutions,

(4.4) u_i =p_ix^r_iy^s_i , v_i =p_i and α= k+m r+s . This gives,

(4.5) u^α_iv^β_i =p_ix^k_iy^m_i . Also,

(4.6)

n

X

i=1

ui

!α

=

n

X

i=1

pix^r_iy^s_i

!^k+m_r+s , and

(4.7)

n

X

i=1

v_i

!β

= 1.

From (4.3), (4.5), (4.6) and (4.7), we get

(4.8) µ⁰_{r s} ≥(µ⁰_{k m}) ^k+mr+s.

Fora≤x≤b,c≤y≤d,r≥k,s≥mandrm=ks, inequality (4.3) will remain valid if we substituten = 2,u₁ =p₁a^rc^s,u₂ =p₂b^rd^s,v₁ =p₁,v₂ =p₂,α= ^k+m_r+s,

p₁ = b^kd^m−x^ky^m b^kd^m−a^kc^m, and

p₂ = x^ky^m−a^kc^m b^kd^m−a^kc^m.

These substitutions give

(4.9) x^ry^s ≤ (b^rd^s−a^rc^s) x^ky^m+a^rc^sb^kd^m−a^kc^mb^rd^s b^kd^m−a^kc^m .

Without loss of generality we can have that the random variate take values a = x₁ < x₂ <

· · · < x_n = b and c = y₁ < y₂ < · · · < y_n = d therefore a ≤ x_i ≤ b and c ≤ y_i ≤ d, i= 1,2,. . ., n. From inequality (4.9), it follows that

x^r_iy_i^s ≤ (b^rd^s−a^rc^s) x^k_iy_i^m+a^rc^sb^kd^m−a^kc^mb^rd^s b^kd^m−a^kc^m , or

n

X

i=1

P_ix^r_iy^s_i ≤ (b^rd^s−a^rc^s) Pn

i=1P_ix^k_iy_i^m+ a^rc^sb^kd^m−a^kc^mb^rd^s Pn i=1P_i

b^kd^m−a^kc^m ,

or

µ⁰_{r s} ≤ (b^rd^s−a^rc^s) µ⁰_{k m}+a^rc^sb^kd^m−a^kc^mb^rd^s b^kd^m−a^kc^m .

Inequality (4.2) also holds for the continuous bivariate distributions. The upper bound in in- equality (4.2) is a consequence of inequality (4.9). Multiplying both sides of inequality (4.9) by joint pdf φ(x, y)and integrating over the corresponding limits, we get the maximum value of µ⁰_rswhere

µ⁰_rs= Z b

a

Z d

c

x^ry^sφ(x, y)dxdy and

Z b

a

Z d

c

φ(x, y)dxdy= 1.

Now consider, Rb a

Rd

c f^αg^βdxdy Rb

a

Rd

c f dxdyα Rb

a

Rd

c g dxdyβ = Z b

a

Z d

c

f Rb

a

Rd

c f dxdy

!α

g Rb

a

Rd

c g dxdy

!β

dxdy

≤ Z b

a

Z d

c

"

αf Rb

a

Rd

c f dxdy + βg Rb

a

Rd

c g dxdy

# dx dy

= 1,

whereα+β = 1andf andgare positive functions. We therefore have (4.10)

Z b

a

Z d

c

f^αg^βdx dy≤ Z b

a

Z d

c

f dxdy

^αZ b

a

Z d

c

g dxdy ^β

, and make the following substitutions,

f =x^ry^s φ(x, y), g =φ(x, y) and α= k+m r+s .

Inequality (4.10) then yields the minimum value ofµ⁰_rs.

5. APPLICATIONS OFRESULTS

On using the results derived in Section 3 and giving particular values torandsit is possible to derive a host of results connecting the Harmonic mean (H), Geometric mean (G), Arithmetic mean (A) and Root mean square (R) when one of the means is given and the random variate takes the prescribed set of positive valuesx₁, x₂, . . . , x_n.

If we putr = +1ands = −1we get inequalities between Aand H, if we putr = 0and s =−1we get inequalities betweenGandH, and so on. Root mean squareR corresponds to r= 2. In particular the following inequalities are obtained from the general result,

(5.1) [(xj−1+x_j)A−xj−1x_j]¹² ≤R ≤[(a+b)A−ab]¹² ,

(5.2) ab

a+b−A ≤H ≤ xj−1xj

xj−1+x_j −A,

(5.3) b^A−aa^b−A_b−a¹

≤G≤

x^A−x_j ^j−1x^x_j−1^{j−A 1}

xj−xj−1

,

(5.4)

x²_j−1+xj−1x_j +x²_j − xj−1xj(xj−1+xj) H

¹₂

≤R ≤

a²+ab+b²− ab(a+b) H

¹₂ ,

(5.5) (xj−1+xj)− x_j−1x_j

H ≤A≤a+b− ab H,

(5.6)

h

x^x_j^{j(H−xj−1)}x^x_j−1^{j−1(xj−H)}i_H(xj−¹_xj−1)

≤G≤

b^b(H−a)a^a(b−H)H(b−a)¹

,

(5.7)

log G

xj−1

x²_j xj

G

x²_j−1

log _x^xj

j−1

≤R² ≤ log ^G_ab² b G

a²

log_a^b ,

(5.8) log _a^bab

log ^G_aa b G

b ≤H ≤

log _x

j

xj−1

xj−1xj

log

G xj−1

xj−1 xj

G

xj

,

(5.9)

log G

xj−1

xj xj

G

xj−1

log _x^xj

j−1

≤A≤ log ^G_ab b G

a

log_a^b ,

(5.10) R²+ab

a+b ≤A≤ R²+xj−1x_j xj−1+x_j ,

(5.11) ab(a+b)

a²+ab+b²−R² ≤H ≤ xj−1x_j(xj−1 +x_j) x²_j−1+xj−1xj+x²_j −R², and

(5.12) b^R

2−a2 b2−a2 a^b

2−R2

b2−a2 ≤G≤x

R2−x2 j−1 x2

j−x2 j−1

j x

x2 j−R2 x2

j−x2 j−1

j−1 ,

wherej = 2,3,. . ., n.

We now deduce the result that the power mean M_r is an increasing function of r. If r is positive andsis any real number withr > sthen from inequality (3.3) we have

(5.13) (µ⁰_r)^1r ≥(µ⁰_s)^1s , orM_r ≥M_s.

Ifris a negative real number withr > swe again get inequality (5.13) from inequality (3.8).

From inequality (3.12) we have M_r ≥ M₀ for r > 0, and M_r ≤ M₀ for r < 0. Hence we conclude that the power mean of orderris an increasing function ofr. In particular, we get that

M₋₁ ≤M₀ ≤M₁ ≤M₂. REFERENCES

[1] J.N. KAPUR AND A. RANI, Testing the consistency of given values of a set of moments of a probability distribution, J. Bihar Math. Soc., 16 (1995), 51–63.

[2] J.N. KAPUR, Maximum Entropy Models in Science and Engineering, Wiley Eastern and John Wiley, 2^ndEdition, 1993.

[3] G.H. HARDY, J.E. LITTLEWOODANDG. POLYA, Inequalities, Cambridge University Press, 2^nd Edition, 1952.