Stability of a characterization of normal distributions based on the first two conditional moments

© Hindawi Publishing Corp.

STABILITY OF A CHARACTERIZATION OF NORMAL DISTRIBUTIONS

BASED ON THE FIRST TWO CONDITIONAL MOMENTS

TRUC T. NGUYEN and KHOAN T. DINH

(Received23 March 1998 andin revisedform 27 July 1998)

Abstract.A characterization of normal distributions of two independent random vari-ables X andY with a finiteE[X2_{] basedon the linearity of E[X|X+Y ]andthe}

ho-moscedasticity of var[X|X+Y ]given by Rao (1976) is provedto be stable.

Keywords and phrases. Conditional density function, normal close, characteristic function, small parameter.

2000 Mathematics Subject Classification. Primary 62E10.

1. Introduction and basic results. The problem of characterization of normal dis-tributions in empirical Bayes view basedon the linearity of the posterior mean andthe homoscedasticity of the posterior variance was studied by several authors [2, 4, 6].

LetU1,...,Unbe a random sample from the distributionF(u|θ)such thatE[Uj| θ]=θexists, θis a location parameter ofF(u|θ), then the observedvalues ofUj

can be written asuj=θ+wj, j=1,...,n, wherewj does not depend onθ, andis

the observedvalues of a random sampleW1,...,Wnfrom some distribution function.

Assume thatθis a given value of an unobservedrandom variableΘ, thenΘandthe random sampleW1,...,Wn are independent. Set Z =Σnj=1Uj/nand X =Σnj=1Wj/n,

thenZ=Θ+X, whereΘandXare independent. It is clear that to study the normality of the random variablesΘandWis equivalent to study the normality of the random variablesΘandX. Then, without loss of generality, we can assume thatn=1 in the random sampleU1,...,Un. Assumedthe linearity of the posterior meanE[Θ|Z=z]

with respect toz, andthe homoscedasticity of the posterior variance var[Θ|Z=z], Rao (1976) gave the following result on the normality ofY1andΘ.

LetX andY be two independent random variables having the nondegenerate dis-tributionsFandG, respectively. SetZ=X+Y, then the following characterization of normality ofF andGwere obtained.

Theorem1.1. Suppose thatE[X2_{]is finite. The following conditions:}

E[X|Z]=ao+a1Z, var[X|Z]=bo, (1.1)

where0< a1<1, and ao,a1,bo are constants, hold if and only if bothF andG are

normal distributions.

Corollary1.2. LetX1,...,Xn, n >1, be a random sample from a nondegenerate

distributionF, and letZ=Σn

j=1Xj. The condition

var[X1|Z]=bo, (1.2)

holds if and only ifFis a normal distribution.

Definition1.3. A distributionF isε-normal if there existµ∈Randσ >0 such that supX|F(x)−Φ((X−µ)/σ )| ≤ε, whereΦis the distribution function of a standard

normal distribution.

Let a distributionF be characterizedby a propertyS. This characterization of F basedonSis stable if the propertyS is replacedby a weaker propertySε, whereεis

a “small parameter”, then the distributionFεis close toFin term ofε[5]. Sapogov [7]

showedthe stability of Cramer’s theorem, that is, with some supplementary con-ditions, he showed that if F =F1∗F2 is ε-normal, then F1 is CσF1−3(ln(1/ε))−1/2 -normal, andF2is CσF−23(ln(1/ε))−1/2-normal. Hoang [3], with supplementary condi-tions, showedthatε-independence ofX+Y andX−Y, whereXandY are indepen-dent, impliesβ1(ε)-normality ofF1andF2, andhe also showedthatε-independence of ¯X=(1/n)Σn

j=1XjandS2=Σnj=1(Xj−X)¯ 2, whereX1,...,Xnare i.i.d.F, impliesβ2(ε) -normality ofF. In his results,β1(ε)andβ2(ε)both are of the order of(ln(1/ε))−1/2. In this paper, we show the stability of the characterizations of normal distributions given by Theorem 1.1 andCorollary 1.2. To show these stability results, we needthe following result on a measure of closeness of two distribution functions using their characteristic functions given by Esseen [1]. (See also [5]).

Theorem1.4. LetFandGbe distribution functions andϕandψbe their respective characteristic functions. IfGhas the derivativeG_{, then}

sup

X |F(x)−G(x)| ≤

1

Π

T

−T

ϕ(t)−_tψ(t)dt+24

ΠT supX |G

_{(x)|. (1.3)}

In the proofs of the theorems in Section 2 andSection 3 below, the inequality

|ez_{−1| ≤ |z|e}|z|_{, (1.4)}

for any complex numberz, is usedseveral times.

2. Stability of the characterization given by Theorem 1.1. In this section, we show that the characterization of normal distributions given by Theorem 1.1 is sta-ble, that is, we show that if the posterior mean isε-linear andthe posterior mean is ε-homoscedastic, thenF andGboth areC[ln(1/ε)]−1/2_-normal.

Theorem2.1. LetXandY be two independent random variables with nondegener-ate distribution functionsF and G, respectively. Assume thatE[X2_{]is finite and set} Z=X+Y. If

E[X|Z]=ao+a1Z+b1(Z), (2.1)

wherea1,bo are constants, 0< a1<1. Assume thatsupZ|b1(Z)|, supZ|b2(Z)|< ε, andε < e−2(bo+2√σX2+σY2+5)/(1−a1)_{, where σX}2 _andσY2 _{are the variances of X and} Y, respectively, then bothF and G areC(ln(1/ε))−1/2_{-normal, where C depends on} a1,bo,σX2, andσY2.

Before proving Theorem 2.1, the following question on the set of the distributions F andGis raised: does the set of the distributionsFandGthat satisfy (2.1) and(2.2) contain only normal distributions? In other words, is there anyF andGsuch thatZ is not normal and(2.1) and(2.2) hold? The answer to this question is given by the following example.

Example2.2. Suppose thatXhas anN(0,1)distribution andYhas an(1/2)N(0,1) +(1/2)N(0,1−ε)distribution, 0< ε <1, are two independent random variables. Then Z=X+Y has an(1/2)N(0,2)+(1/2)N(0,2−ε)distribution. Hence,X|Z=zhas an αN(z/2,1/2)+(1−α)N(z/(2−ε),(1−ε)/(2−ε))distribution, where

α= 1/

√ 2e−x2_/4

1/√2e−x2_/4+1/√_2−εe−x2_/2(2−ε). (2.3) Therefore,E[X|Z=z]=z/2+b1(z), where

b1(z)=√ εz

2(2−ε)√2+√2−ε eεz2_/4(2−ε). (2.4) Hence,

|b1(z)|< ε|z|e−εZ2/4< ε, var[X|Z=z]=1₂+b2(z), (2.5) where

b2(z)=−ε

4√2(2−ε)1/2_e−z2_/(2−ε)

+2(2−ε)+z2_e−(4−ε)z2_/4(2−ε)

4√2(2−ε)2_e−z2_{/2(2−ε)+4(2−ε)5/2e−z}2_/4 . (2.6) Then var[X|Z=z] <1/2 and

|b2(z)|<ε(12+z

2_)e−(4−ε)z2_/4(2−ε) 4√2e−z2_/2(2−ε) =

ε(12+z2_)e−z2_/4 4√2 <3

2ε

2. (2.7)

Proof of Theorem2.1. Assume thatε <1. By (2.1) anda finiteE[X2_],E[Y2_]is also finite andb2

o>0. Without loss of generality, assume thatE[X]=E[Y ]=0, and E[var[X|Z]]=bo. This impliesao=0, E[b1(Z)]=E[b2(Z)]=0 and(2.1) and(2.2) can be written as

E[X|Z]=a1Z+b1(Z), E[X2_|Z]=b

o+a21Z2+2a1Zb1(Z)+b21(Z)+b2(Z). (2.8) Multiply both sides of (2.8) byeitZ_{andtake the expectedvalues. The following}

equa-tions inϕandψ, the characteristic functions ofF andG, respectively, are obtained.

1−a1ϕ(t)ψ(t)=a1ϕ(t)ψ(t)+S1(t), ϕ_{(t)ψ(t)= −b}

where

S1(t)=EieitZb1(Z), S2(t)=E−eitZ2a1Zb1(Z)+b21(Z)+b2(Z). (2.10) Hence, for anyt,

|S1(t)| =EieitZb1(Z)≤E[|b1(Z)|]≤ε. (2.11) By Cauchy-Schwarz inequality

_E eitZ_Zb

1(Z)≤E[|Z||b1(Z)|]≤EZ21/2Eb21(Z)1/2≤

σX2+σY2ε,

|S2(t)| =E−eitZ2a1Zb1(Z)+b12(Z)+b2(Z) ≤2a1EZb1(Z)+Eb21(Z)+Eb2(Z) ≤2a1

σX2+σY2ε+ε2+ε≤2 a1

σX2+σY2+1

ε.

(2.12)

Sinceϕ(0)=ψ(0)=1, ϕ_{(0)=ψ(0)=0, andthe continuity of characteristic}

func-tions, let

δ(ε)=supδ:|ϕ(t)|,|ψ(t)| ≥ε1/4_, |ϕ(t)| |ϕ(t)|,

|ψ_(t)|

|ψ(t)| ≤1,|t| ≤δ

. (2.13)

For|t| ≤δ(ε), divide both sides of (2.9) by ϕ(t)ψ(t), the following equations are obtained

1−a1ϕ

_(t)

ϕ(t) =a1 ψ_(t)

ψ(t)+R1(t), (2.14)

1−a2

1ϕ

_(t)

ϕ(t) = −bo+a21 ψ_(t)

ψ(t) +2a21 _ϕ(t)

ϕ(t)

_ψ(t)

ψ(t)

+R2(t), (2.15)

where

R1(t)=_ϕ(t)ψ(t)S1(t) , R2(t)=_ϕ(t)ψ(t)S2(t) . (2.16)

From (2.11)

|R1(t)| =_|ϕ(t)|S1(t)_ψ(t)| _|≤_ε1/4ε_ε1/4=ε1/2, R1(0)=S1(0)=0, (2.17)

andfrom (2.12)

|R2(t)| =_|ϕ(t)|S2(t)_ψ(t)| _|≤2a1

σX2+σY2+1ε

ε1/4_ε1/4 =2a1 σX2+σY2+1

ε1/2_{. (2.18)}

Set

h(t)=ϕ(t)

ϕ(t), k(t)= ψ_(t)

Then

ϕ_(t)

ϕ(t) =h(t)+h2(t),

ψ_(t)

ψ(t) =k(t)+k2(t). (2.20)

From (2.14),

k(t)=1−a1 a1 h(t)−

R1(t)

a1 (2.21)

andthen,

k_(t)=1−a1 a1 h

_(t)−R1(t)

a1 . (2.22)

Substitute in (2.15) the following equation is obtained

h_{(t)= −} bo

1−a1− a2

1 1−a1

_R

1(t) a1 +

R2 1(t) a1

1 +

21−2a1 a2

1 R1(t)h(t)

+₁₋R2(t)_a

1. (2.23)

FromE[X]=0,

h(t)=ϕ_ϕ(t)(t)=−₁₋bo_a 1t+

1 1−a1 ×

t

0

−a2 1

_R

1(u) a1 +

R2 1(u)

a2 1 +

21−2a1 a2

1 R1(u)h(u)

+R2(u)

du, (2.24)

andfromϕ(0)=1,

ϕ(t)=exp− bo 21−a1t

2₊ 1 1−a1 ×

_t

0 _v

0

−a2 1

_R

1(u) a1 +

R2 1(u)

a2 1 +

21−2a1 a2

1 R1(u)h(u)

+R2(u)

dudv

=exp− bo 21−a1t

2_+R(t),

(2.25)

where

R(t)=₁₋1_a 1

t

0 v

0

−a2 1

_R

1(u) a1 +

R1(u)2 a2

1 +

21−2a1 a2

1 R1(u)h(u)

+R2(u)

For

|t| ≤δ(ε), |Re{R(t)}|

≤

−₁₋a1_a 1

t

0R1(v)dv +₁₋1_a

1 t 0 v 0

R2(u)−R21(u)−2

1−2a1R1(u)h(u)dudv

≤ a1 1−a1

t

0R1(v)dv

+₁₋1_a 1 t 0 v 0

|R2(u)|+R21(u)+2|1−2a1R1(u)h(u)|dudv

≤₁₋a1_a 1ε

1/2_|t|+ 1 1−a1

2 a1

σX2+σY2+1

ε1/2_{+ε+2|1−2a1|ε}1/2|t|2 2

≤₁₋ε1/_a2 1

a1|t|+1₂ 2a1

σX2+σY2+1+2|1−2a1|

|t|2

≤₁₋ε1/_a2 1

|t|+1₂ 2σX2+σY2+3

|t|2_.

(2.27)

Hence,

|ϕ(t)| =exp

−₂ bo 1−a1|t|

2

+Re

−₁₋a1_a 1

_t

0R1(v)dv+ 1 1−a1 × t 0 v 0

R2(u)−R12(u)−21−2a1R1(u)h(u)dudv

≥exp

−₂ bo 1−a1|t|

2₋ ε1/2 1−a1

|t|+1₂ 2σX2+σY2+3

|t|2_.

(2.28)

Since the function

exp

−₂ bo 1−a1|t|

2₋ ε1/2 1−a1

|t|+1₂ 2σX2+σY2+3

|t|2 _(2.29)

is decreasing in |t|, and|ϕ(t)|> ε1/4 _{for|t| ≤δ(ε), δ(ε)is not smaller than the} solutiont0>0 of the equation

exp

−₂ bo 1−a1|t|

2₋ ε1/2 1−a1|

|t|+1₂ 2σX2+σY2+3

|t|2_=ε1/4_{, (2.30)}

or equivalently, the equation

bo+ε1/2 2

σX2+σY2+3

|t|2_+2ε1/2_{|t| =}1−a1 2 ln

₁ ε

. (2.31)

The function on the left-handside of (2.31) is an even function, andfort >0, it is increasing. Then equation (2.31) has two opposite solutions. Fort >0, equation (2.31) becomes

bo+ε1/2 2

σX2+σY2+3

t2_+2ε1/2_t=1−a1 2 ln

₁ ε

Fort >1,

bo+ε1/2 2

σX2+σY2+5

t≤bo+ε1/2 2

σX2+σY2+3

t2_+2ε1/2_t ≤bo+ε1/2 2

σX2+σY2+5

t2_, (2.33)

andfort≤1,

bo+ε1/2 2

σX2+σY2+5

t2_≤b

o+ε1/2 2

σX2+σY2+3

t2_+2ε1/2_t ≤bo+ε1/2 2

σX2+σY2+5

t. (2.34)

Hence,

to≥min   

1−a1ln(1/ε) 2bo+ε1/22σX2+σY2+51/2

,

1−a1ln(1/ε) 2bo+ε1/22σX2+σY2+5

  . (2.35)

If we chooseε < e−2(bo+2√σX2+σY2+5)/(1−a1)_{, then}

1−a1

2bo+2σX2+σY2+5>1, δ(ε)≥to≥δo(ε)=

1−a1ln(1/ε) 2bo+2σX2+σY2+5

1/2 . (2.36)

Using [1, Theorem 1.2], whereGis the cumulative distribution function of a normal distribution with zero mean and the varianceb0/(1−a1), andby choosingT=δo(ε),

sup

X

F(x)−Φ 

 x

bo/1−a1  

≤_Π1 δo(ε)

−δo(ε)

_ϕ(t)−e−[bo/2(1−a1)]t2 |t| dt+

24

Πδo(ε)supX Φ



 x

bo/1−a1  

= 1

Π

δo(ε)

−δo(ε)e

−[bo/2(1−a1)]t21−eR(t) |t| dt+

1221−a1/bo

Π3/2_δo(ε)

≤_Π1 δo(ε)

−δo(ε)e

−[bo/2(1−a1)]t2|R(t)|e|R(t)| |t| dt+

1221−a1/bo

Π3/2_δo(ε) ≤_Π2

δo(ε)

0

ε1/2_t+1 2

2σX2+σY2+3t2

1−a1t

×e−bo+ε1/22√σX2+σY2+3t2+2ε1/2t2(1−a1)_dt+ 12 √

2 bo1/2Π3/2δo(ε),

by using (2.27),

≤ 1

Π

δo(ε)

0

ε1/2_2+2σ

X2+σY2+3t

1−a1

×e−bo+2ε1/2√σX2+σY2+3t2+2t2(1−a1)_dt+ 12 √

2 bo1/2Π3/2δo(ε)

≤_Π1e−bo+2ε1/2√σX2+σY2+3δo(ε)2+2δo(ε)2(1−a1)

× δo(ε)

0

ε1/2_2+2σ

X2+σY2+3t

1−a1 dt+

12√2 b1o/2Π3/2δo(ε)

=ε1/2

4δo(ε)+2σX2+σY2+3δo(ε)2

2Π1−a1

×e−bo+2ε1/2√σX2+σY2+3δo(ε)2+2δo(ε)2(1−a1)₊ 12 √

2 boΠ3/2δo(ε)

≤ε1/2δo(ε)2

7+2σX2+σY2

2Π1−a1

×e−bo+2ε1/2δo(ε)27+2√σX2+σY22(1−a1)₊ 12 √

2 boΠ3/2δo(ε)

=ε1/2ln(1/ε)

7+2σX2+σY2

4Πbo+2σX2+σY2+5 e

bo/2(1−a1)

×eε1/2_ln(1/ε)7+2√_σX2_+σY2_4bo+2√_σX2_+σY2₊₅ +24

bo+2σX2+σY2+5

1−a1boΠ3/2

ln ₁

ε −1/2

≤ 7+2

σX2+σY2

4Πbo+2σX2+σY2+5e

[bo/2(1−a1)]+

7+2√σX2+σY2

4bo+2√σX2+σY2+5

+24

bo+2σX2+σY2+5

1−a1boΠ3/2

ln

₁ ε

−1/2 ,

(2.38)

since for small ε, ε1/2_{ln(1/ε) <1, andε}1/2_{ln(1/ε) < [ln(1/ε)]}−1/2_{. ThereforeF is} C[ln(1/ε)]−1/2_{-normal, whereC is a constant depends only ona}

1,bo,σX2 andσY2.

FromZ=X+Y , X=Z−Y, substitute in (2.1) and(2.2), the following conditions on Y are obtained:

E[Y|Z]= −ao+1−a1Z−b1(Z), var(Y |Z)=bo+b2(Z). (2.39)

Similarly,Gis alsoC[ln(1/ε)]−1/2_{-normal. This completes the proof.}

3. Stability of the characterization given by Corollary 1.1. In this section, we show that the characterization of normal distribution given in Corollary 1.2 is stable.

Theorem3.1. LetX1,...,Xn, n >1, be a random sample from a nondegenerate

distribution functionF with a finite second moment, and letZ=Σn j=1Xj. If

wherebo>0is a constant not depending onZ,supZ|(n/(n−1))c(n,Z)| ≤εn, ε <

min{1,b2o/n}, thenFisC[ln(1/ε)]−1/2-normal.

Proof. Without loss of generality, assume thatE[var(X1|Z)]=((n−1)/n)bo.

Hence,E[c(n,Z)]=0. To simplify the notation in the proof, assume that supZ|c(n,Z)|

≤ε, andε <1. From (3.1),

EX2 1|Z

=Z_n2₂+n_n−1bo+c(n,Z). (3.2)

Multiply both sides of (3.2) byeitZ_{andtake the expectedvalue, the following equation}

in the characteristic functionϕofF is obtained:

ϕ_(t)ϕn−1_(t)=ϕ2

(t)ϕn−2_(t)−b

oϕn(t)+Sn(t), (3.3)

where

Sn(t)= −_nn₋₁EeitZc(n,Z). (3.4)

Then

sup

t |Sn(t)| ≤ n

n−1EeitZ|c(n,Z)|

≤_nn₋₁E[|c(n,Z)|]≤ε. (3.5)

Letδ(ε)=sup{τ:|ϕ(t)|> ε1/2n_{, |t|< τ}. Then for|t|< δ(ε), equation (3.3) can be}

written as

ϕ_{(t)ϕ(t)−ϕ2} (t)

ϕ2_(t) = −bo+ Sn(t)

ϕn_(t). (3.6)

Take integration both sides of (3.6) with respect tot, andassuming that the mean of F isµ,

ϕ_(t)

ϕ(t) = −bot+iµ+ _t

0 Sn(u)

ϕn_(u)du. (3.7)

Take integration both sides of (3.7) with respect to t, together with the condition ϕ(0)=1, we get

ϕ(t)=e−(1/2)bot2 +iµt+

t

0 v

0 Sn(t)

ϕn_(u)dudv. (3.8)

Let

Rn(t)= t

0 v

0 Sn(u) ϕn_(u)dudv.

|ReRn(t)| ≤ _t

0 _v

0

|Sn(u)|

|ϕn_(u)|dudv ≤

_t

0 _v

0 ε

ε1/2nndudv

≤ε1/2t2 2.

(3.9)

Hence,

|ϕ(t)| =e−(1/2)bot2+Re(0t 0v(|Sn(u)|/|ϕn(u)|)dudv)

Since e−(1/2)(bo+ε1/2)t2 _{is a decreasing function in |t|, and |ϕ(t)| > ε}1/2n _{for |t|<} δ(ε), δ(ε)is not smaller than the maximum of the absolute values of the solutions of the equation

e−(1/2)(bo+ε1/2)t2_=ε1/2n_{, (3.11)} or equivalently, the equation

bo+ε1/2t2=_n1ln ₁

ε

. (3.12)

Equation (3.12) has two opposite solutions. Let

to=

_ln(1/ε) n(bo+ε1/2)

1/2

(3.13)

be its positive solution. Then

δ(ε)≥to=

_ln(1/ε) n(bo+ε1/2)

1/2

≥δo(ε)=

_ln(1/ε) n(bo+1)

1/2 , _ϕ(t)−e−(1/2)bot2+iµt_=e−(1/2)bot2_×eR_n(t)₋₁

≤e−(1/2)bot2_×Rn(t)e|Rn(t)|_, ≤1₂e(1/2)(−bo+ε1/2)t2_ε1/2_t2_≤1

2ε1/2t2,

(3.14)

if we assume thatε < b2

o. Using [1, Theorem 1.2] once again, for any 0< T < δ(ε),

sup

X

F(x)−Φx−µ bo

≤ 1

Π

T

−T

ϕ(t)−e−(_t1/2)bot2+iµtdt+ 24

ΠTsupX

Φx−µ

bo

≤_Π1 _T

−T

1

2ε1/2|t|dt+

12√2 TboΠ3/2=

ε1/2_T2

Π +

12√2

Π3/2_Tbo. (3.15)

If we chooseT=δ0(ε), (3.15) becomes

sup

X

F(x)−Φ

_x−µ

bo

≤ε_Π1_n(b/2ln(1/ε)

o+1)+

122n(bo+1)

Π3/2_bo

ln ₁

ε −1/2

≤ !

(3/e)3/2√_n

Π(bo+1) +

122n(bo+1)

Π3/2_bo "

ln ₁

ε −1/2

,

(3.16)

since forε >0,ε1/2_{ln(1/ε)≤(3/e)}3/2_[ln(1/ε)]−1/2_{. In order for the last term of the} inequality (3.16) does not depend onn, insteadof choosing supZ|c(n,Z)|< ε, we

choose supZ|c(n,Z)|< εn, then (3.16) becomes

sup

X

F(x)−Φx−µ bo

≤

!

(3/e)3/2

Π(bo+1)+

122(bo+1)

Π3/2_bo "

ln1 ε

−1/2

, (3.17)

assuming thatε <min{1,b2o/n}. Therefore,F isC[ln(1/ε)]−1/2-normal, whereC

Acknowledgement. We are grateful to the referees for their valuable comments which ledto a substantial improvement in the presentation of this paper.

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Nguyen: Department of Mathematics and Statistics, Bowling Green State Univer-sity, Bowling Green, Ohio43403-0221, USA

E-mail address:[email protected]