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volume 5, issue 4, article 86, 2004.

Received 23 February, 2004;

accepted 19 July, 2004.

Communicated by:T. Mills

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Journal of Inequalities in Pure and Applied Mathematics

SOME INEQUALITIES BETWEEN MOMENTS OF PROBABILITY DISTRIBUTIONS

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

Department of Mathematics Himachal Pradesh University Summer Hill, Shimla -171005, India EMail:shandil_rg1@rediffmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 040-04

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Some Inequalities Between Moments of Probability

Distributions

R. Sharma, R.G. Shandil, S. Devi and M. Dutta

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J. Ineq. Pure and Appl. Math. 5(4) Art. 86, 2004

Abstract

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

2000 Mathematics Subject Classification:60E15, 26D15.

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

1. Introduction

Therth order momentµ⁰_rof 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 valuesx_i(i= 1,2,. . ., n) with corresponding probabilitiesp_i(i= 1,2,. . ., n), we define

(1.2) µ⁰_r =

n

X

i=1

p_ix^r_i.

The power mean of orderris defined as

(1.3) M_r= (µ⁰_r)^1/r for r 6= 0, and

(1.4) M_r= lim

r→0(µ⁰_r)^1/r for r= 0.

It may be noted here thatM−1,M0 andM1respectively define harmonic mean, geometric mean and arithmetic mean.

Kapur [1] has reported the following bound forµ⁰_rwhenµ⁰_sis 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 .

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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 constraints. 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 wherer andscan 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 obtained.

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2. Some Elementary Inequalities

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 .

Ifris a negative real number withr > sthen inequality (2.1) holds forxlying 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 > sands6= 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

.

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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).

Ifris a positive real number andsis a negative real number withr > sthen f⁰(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) and f(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 of f(x)from equation (2.3) in inequalities (2.6) and (2.7), we obtain inequalities (2.1) and (2.2) respectively.

If r 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,∞). Since clies in the interval(a, b)and f(a) = f(b) = 0 it follows that inequality (2.7) holds for a ≤ x ≤ b while inequality (2.6) holds for x lying outside (a, b)and thus we get inequalities for the case when r is 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 ,

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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 of x,

(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 thatclies in the interval(a, b). Alsof⁰(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

(2.13) f(x)≤0 for a≤x≤b,

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and forxlying outside(a, b)we have

(2.14) f(x)≥0.

On substituting the value off(x)from equation (2.10) in inequalities (2.13) and (2.14), we obtain inequalities (2.8) and (2.9) respectively.

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3. Inequalities Between Moments

Theorem 3.1. Letrbe a positive real number andsbe any non zero real num- ber withr > s. If a positive random variate takes valuesx_i (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 forµ⁰_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_r can 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

# ,

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where α andβ take one of the values among 1,2,. . ., nwithα 6= β. Without loss of generality we can arrange values of the variate such that a = x1 ≤ x₂ ≤ · · · ≤ x_n = b. If we take α = 1 and β = n then x₁ ≤ x_i ≤ x_n for i = 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−1andx_β =x_j,j = 2,3,. . ., nthen eachx_ilies 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 for µ⁰_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 ifxj−1approachesx_jwe get inequality (3.5) and we conclude that for a continuous random variate the lower bound forµ⁰_r is given by inequality (3.5).

The upper bound for µ⁰_r can 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 s be 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 ,

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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], witha >0, the lower bound forµ⁰_ris given by inequality (3.6) whereas the upper bound for µ⁰_ris given by following inequality:

(3.8) µ⁰_r ≤(µ⁰_s)^r/s.

Proof. We again consider equation (3.4). If we take α = 1 and β = n then x₁ ≤ x_i ≤ x_n for 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 forµ⁰_r is given by inequality (3.6). Also ifx_α = x_j−1 and x_β = x_j, j = 2,3,. . ., n then each x_i lies outside (xj−1, x_j). It follows from Theorem2.1 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−1approachesx_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 ,

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and

(3.10) µ⁰_r ≥ x^r_j−x^r_j−1

logM₀+x^r_j−1logx_j −x^r_jlogxj−1

logx_j −logxj−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] with a > 0the upper bound for µ⁰_r is given by inequality (3.9) whereas the lower bound forµ⁰_ris given by the following inequality

(3.12) µ⁰_r ≥(M0)^r.

Proof. It is seen that µ⁰_rcan be expressed in terms of logM₀ in the following form:

(3.13) µ⁰_r = x^r_β −x^r_α

logx_β −logx_α logM₀+x^r_αlogx_β−x^r_βlogx_α logx_β −logx_α +

n

X

i=1

P_i

x^r_i − x^r_β−x^r_α

logx_β−logx_αlogx_i+x^r_βlogx_α−x^r_αlogx_β logx_β−logx_α

.

Without loss of generality we can arrange values of the variate such that a = 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 Theorem2.2that last term in equation (3.13) is negative and we conclude that the upper bound forµ⁰_r is given by inequality

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(3.9). Also ifx_α =xj−1 andx_β = x_j, j = 2,3,. . ., nthen eachx_i lies outside (xj−1, xj). It follows from Theorem 2.2 that the last term in equation (3.13) is positive and we conclude that the lower bound for µ⁰_r is given by inequality (3.10).

If the value of M₀ coincides with one of xj−1 or x_j then from inequality (3.10) we have

(3.14) µ⁰_r ≥(M₀)^r.

Also ifxj−1approachesx_jwe get inequality (3.14) and we conclude that for the continuous random variate the lower bound forµ⁰_ris 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).

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4. Inequalities Between Moments of Bivariate Distributions

The moments of a bivariate probability distribution are the generalizations of those of univariate 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 moment µ⁰_rs about the origin is given by

(4.1) µ⁰_rs=

n

X

i=1

P_ix^r_iy^s_i.

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 variates x and y vary respectively over the finite positive real intervals [a, b]

and[c, d]. Ifµ⁰_{k m}is the corresponding moment of orderkinxandminysuch thatr≥k,s≥mandrm=ksthen we must have by necessity,

(4.2) (µ⁰_{k m})

r+s

k+m ≤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. If u, v, α and β are positive real numbers with α+β = 1 then from Hölder’s inequality [3],

(4.3)

n

X

i=1

u^α_iv^β_i ≤

n

X

i=1

u_i

!α n

X

i=1

v_i

!β

.

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We make the following substitutions,

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

(4.5) u^α_iv_i^β =p_ix^k_iy_i^m. Also,

(4.6)

n

X

i=1

u_i

!α

=

n

X

i=1

p_ix^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+m^r+s.

For a ≤ x ≤ b, c ≤ y ≤ d, r ≥ k, s ≥ m and rm = ks, inequality (4.3) will remain valid if we substitute n = 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,

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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 valuesa = x1 < x2 <· · ·< xn =bandc=y1 < y2 <· · ·< yn =dthereforea≤xi ≤b andc≤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_i^s ≤ (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 inequality (4.2) is a consequence of inequality (4.9). Multiplying both sides of inequality (4.9) by joint pdfφ(x, y)and integrating over the corresponding limits, we get the maximum value ofµ⁰_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.

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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 andg are 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.

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5. Applications of Results

On using the results derived in Section 3and giving particular values to r and s it 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 betweenA andH, if we putr = 0ands = −1we get inequalities betweenGandH, and so on. Root mean square R corresponds tor = 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 ≤ x_j−1x_j 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_xj₋¹_xj−1 ,

(5.4)

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

¹₂

≤R ≤

a²+ab+b²− ab(a+b) H

¹₂ ,

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(5.5) (xj−1+x_j)−xj−1x_j

H ≤A ≤a+b−ab H,

(5.6) h

x^x_j^j^(H−x^j−1⁾x^x_j−1^j−1^(x^j^−H⁾i_H( ¹

xj−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^x^j

j−1

≤R² ≤ log ^G_ab² b G

a²

log ^b_a ,

(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^x^j

j−1

≤A≤ log ^G_ab b G

a

log _a^b ,

(5.10) R² +ab

a+b ≤A≤ R²+xj−1x_j x_j−1+x_j ,

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(5.11) ab(a+b)

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

and

(5.12) b

R2−a2 b2−a2 a

b2−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 meanM_ris an increasing function of r. Ifr is positive ands is any real number withr > sthen from inequality (3.3) we have

(5.13) (µ⁰_r)¹^r ≥(µ⁰_s)¹^s , 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 forr < 0.Hence we conclude that the power mean of orderris an increasing function ofr. In particular, we get that

M−1 ≤M0 ≤M1 ≤M2.

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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^nd Edition, 1993.

[3] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cam- bridge University Press, 2^nd Edition, 1952.