Asymptotics in Poisson order statistics

Asymptotics in Poisson order statistics

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Brands, J. J. A. M. (1988). Asymptotics in Poisson order statistics. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 88-WSK-08). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Eindhoven University of Technology

Department of Mathematics and Computing Science

ASYMPTOTICS IN POISSON ORDER STATISTICS

by I.I.A.M.Brands

EDT-report 88-08

July

1988

AMS subject classification: 41A60, 62E20

ASYMPTOTICS

IN

POISSON

ORDER

STATISTICS

by

JJA.M. Brands

Department of Mathematics, Eindhoven University of Technology,

The Netherlands ABSTRACf

In order statistics sums involving incomplete gamma functions are met. The asymptotic behaviour of such sums is studied, going beyond the results obtain-able by the central limit theorem.

1. Introduction

Some colleagues*) of the author have posed the following problem: Determine the asymptotic behaviour forJ.l.-+ooof the sums

(1) S (J.1,m, n):= ~ /m(J.1, k)(l-/(J.1,k»/1 ,

k=O 00

(2) T(J.1,m,n):= ~ (J.1-k)r(J.1, k)(1-/ijI.,k»/1 ,

k=O

where m and n are positive integers and Ii

(3) /(J.1,k):=(k!rl

I

e-'tkdt (J.1>O,kel'fo)

o

This problem arose in the study oftheexpectation and variance of the order statistics in a random sample from the Poisson distribution with large meanJ.l. In section 2 we present the results. A brief description of the derivation is given in section 3. The details of the derivation are given in sections 4 to 8.

- 2-2. Results

The sums S and T have the following asymptotic behaviour:

(4) SijJ.,m,n)=A(m,n)J,1112+B(m,n)+OijJ.-112) ijJ.~-) , (5) TijJ.,m,n)=C(m,n)IJ.+D(m,n)J,1112+0(1) (IJ.~-) , where GIG (6) A (m,n)

=..J2

J

r(x)r(-x)dx , (7) B (m,n)

=

-2/3

J

xr(x)r(-x)dx , (8) C(m,n)=-3B(m,n) , 00 (9) D(m,n)

=..J2

J

r(x)l'(-x)( 213-x2)dx , -eo and x (to)

f

(x):= n-l12

J

e-s2ds (XEJR.) .

Clearly f can be expressed in the errorfunction but fonnulas (6) to (10) do not become simpler in doing so. Some coefficients are

(11) A (1,1)

=

2A(1,2)

=

2A(2, 1)

=

n-l12 B (n,n)

=

0 (nEIN)

{3

B(1,2)=-B(2,1)=B(I,3)=-B(3,1)= 12 n-I D(1,1)=2D(1,2)= 2D{2,1)=

t

n-1I2 . 3. Sketch of the derivation

The results are obtained by taking the following steps.

(i) The sums are approximated by sums over Iu-kISIJ.213 with an error of 0 (e-Cflll3)

ijJ.~oo)

where c is some positive constant

(ii) ForkE [IJ.-C IJ.213, IJ.+c IJ.213]the asymptotic behaviour of1ijJ., k)forIJ.~oois detennined. Letx E JR. be defmed by

- 3-whereh(s):=-s+log(l+s) (s>-l) .

Roughlyx =UL-k) (2J1)-112. ThenIUL,k) has a complete asymptotic expansion in powers ofll-1I2wichis unifonn withrespecttox e JR,X=OULI/6) UL~oo).

I

UL,

k) :::

f

(x)-21/2T 11t-I12e-x1J1-112

+ ....

(J1~00),

where fis defined by (10), i.e.

Ik -J11

~

c J12J3

~

I

I(J1, k) - {f(x)

+

e-x1

f

ql(x)J1-l/2}

I

~

B 1l-(N+l)12.

1=1

(iii) The sums ~ are approximated by integrals

J ...

diewith an error which, for

IIl-kISIl213 IIl-kISIl213

every positive number r, is OUL-r) UL~oo). Then these integrals are transfonned into

00

integrals over x and then approximated by integrals

J

dx with errors of the kind

4. The truncation of the sum

The functionIUL, k)interpreted as a function of the real variable ke [0,00) is decreasing on [0,00). This statement follows from

11 00

~

I

UL,

k)=(r(k+1»-2

J

dt

J

e-t--ttk~

log(tt-1)dt<O o 11

Let~1l-J.L2J3:=a. Then substitutingt=k(1+s) we have

l-JUL,

kJ:;;1-JUL,a)=(r(a+l)r1e-aaiol₊₁

J

_{eah(s)ds ,}

a-I_11-1

where h(s):=-s+log(l+s) . The function h is concave and negative on (-1,00) , whence

h (sJ:;;h(J1a-l_1)

+

(s-J1a-1

+

1)h'(J1a-l-l) (s;::J,J.a-l-1).

It follows that

00

0< e-a aa+l (r(a

+

1))-1

J

eah(s)~e-aaa+l(r(a+1))-1eah (a-lll-l) (a h' (J1a-l-1)r1

a-I_11-1

-4-:lit 1/3 II]

0<

I:

<~. ~312e-TJ.l.

=

O(e""1l ) ~-+oo)

Q!;kS;a

k-l_ll-l

Let ~~+J1213

.

Then I(~,k)=e-kkk+l(r(k+I)rl

J

ekh(s)tis. Since h(s~-1I2s2 on (-1, 0] -1

we have

k-l_ll-l

I~, k)~e-kkk+l(r(k+l))-1

J

e-1/2ks2 tis -1

00

~e-kkk+l(r(k+I»-1

J

e-1J2ks2

tis

l-k-I_ll

-~ekkk+l(r(k+1»-1

J

e-1/2k(l_k-1Il)stis l-k-Ill

Hence, for both of the sums

Hence,inboth cases

where

c

is a positive number.

S. The incomplete gamma function As we have seen already the substitution

(12) t=k(I+s)

in the integral representation ofI(~,k)gives k-'1l-1

(13) I~,k)=(k

!r

1e-kkk+1

J

ekh(s)tis , -1

where

(14) h(s):= -s+log(I+s) (s>-I). We introduce a new integration variable

y

by

5 -(15) y:=k1l2Ih(s)1 112 sgn(S) (s>-I).

Then we get (16)

where

(17) X:=kll2lh(k-l~-I)lll2sgn(JJ.-k).

To study the transformation (15) we introduce first in (13) (18) t=lh(s)l l12sgn(s).

Then

(19)

..J2

t=S(1-2/3s+2/4s2-2/5s3+....)112 (IsI <1) ,

=s-1I3s2+7/36s3+ ... (lsl<l) ,

the radius of convergence being one since h(s)/s2 has no zeros inside the unit circle. By the Biirmann-Lagrange theorem we can expand s as a powerseries in 't with a positive radius of con-vergence, say p.

We calculate (20)

(21)

The transformation (18) changes the integral (13) into

k-l12_x

(22) I (JJ.,k)=(k!)-1

e-

k_kk₊1

I

_e-

k_'t2 _{: d't}

-00

We shall study the asymptotic behaviour of this integral fork~ooand fixed xeR. Therefore we

shall denote the right hand of (22)byi(x,k).

Inorder to use (21) we must truncate the interval of integration. Now it is easily shown that

where

c

is a positive number.

Inside the circle I'tI:~1I2p the powerseries in (21) is also an asymptotic series, i.e. for every Ne lNwehave

(23)

Hence

-

6-ds _C"

..fi'

d't ="'42 +4/3't

+

6~-81135~+'.. +CN-l1'-1+0(1') (l'tIS1I2p)

(24) i(x,k)

=

(k

!r

le-kkk+l

J

e-k"'(..fi'+

~

t+....+cN_ll'-l+O(-eN))d't

l'tISll2p, 't<k-lI2x

Now we change the lower bound into - 0 0and, evenually, the upper bound into k-l12x, thereby

.1:1/3

making an error of the kind0(e-C

) (k~-)uniformly in XEJR.

Then we substitutet=k-l12yand we get

(25) i(x,k)=(k

!r

le-.l:kk+1I2 (e-l (..fi'+4/3k-112y+

~

k-l y 2+ . ..

--kl/3

+0(e-e ) (k~oo) uniformly inXE JR.

x

Since

J

e-l O(k-Nl2y N) dy

=

o

(k-Nl2) (k~-)unifonnly inxER, we have

--(26)

i

(x,k)=.8J!Q

&

x

J

e-y2(..fi'+4/3k-l/2_{y+' .. +cN_} 1k-{N-l)l2yN-l) dy uniformly in XE

m.,

where Hence N-l

(28) i(x,k)=g(k) ,,£ctfi(x)k-l12+O(k-N/2) (k~oo)uniformlyin XE JR, 1=0

where the

c/s

are given by (21) and x

7

-Integration by parts gives (30) Ji(X)=e-X

2

PI(X)+

l~li

r(1;1 )(21Cr1flf(x) (IeIN) . wherefisdefined by (10) and

l

(/-l~

(31) PI(x):=-2-3fl_1C-1fl_r(1+1)

L

_{(r(1+1 _s»-lxl-I-2s (IeIN) .}

. 2 s=O 2

Letting

x~oo

wefind thatJi(oo)=O if1is odd and Ji(oo)=r{1;1)(21Cr1fl ifIis even.

Since limIUJ.,k)=l we get the asymptotic series for (g(k»-1 by lettingx~ooin (28). We have

Il--t""

(32) 1::::g(k) iClJi(oo)k-l'2

(k~oo)

.

1=0

Comparing (28) (30) and (32) we see that the factor withwhichf (x) occurs in (28) has the same asymptotic expansion as the factorg(k) in (32). Hence

(33) I(x,k»::::f(x)+g(k)- LCle-- x2PI(x)k-lfl (k~oo)uniform1yinxeJR.

1=1

(Le. after truncation the (absolute) error is smaller thanCk-(N+l)12 withCindependent ofx).

According to the results of section 4 we restrict ourselves to values of ke[1J.;1213,IJ.+I1213].Then x=0UJ.l/6) UJ.~oo).

Inorder to get an asymptotic series forIUJ.,k) forlJ.~oowe have to express k as a function ofIJ. andx. From(17) we havekh{lJ.k-l_1)=-x2,sgnx=sgnUJ.-k). Puttingk=IJ.(1-'I') and 1J.-1flx=z we

get (34) 'I'+(1-'I')10g(1-'I')=z2 whence (35) Then (36)

i

~

=z2 (I'l'l <1), sgnz=sgn'l'. ;=2 (i-1)i

- 8-00 (37) 'F 1:dizi (Iz I<r) . i=1 Hence 00 (38) k9J.- 1:difJ.l-iI2Xi (Ix I<r~). i=1

Clearly, forfJ. sufficiently large,xis within the range of convergence sincex=O(fJ.1/6) (fJ.~oo).

A few coefficientsdiare calculated. (39)

(42)

We need also expansions fork-1J2 , k-1andk-312 :

Dearly, there is a positive number'0 such that for all

ae

JR the function (1-'I')Q has a power-series expansion in z which converges for Iz I<r0; if

00

(40) (l-'I')Q= 1:a/a)zj (I z I<ro)

j:4J

then

00

(41) kQ9J.Q(1-'l')Q= 1:aj(a)fJ.a-il2xj (Ix I<rofJ.1/2) .

j:4J We calculate k-l12

=

fJ.-ll2+rl12fJ.-l

x+~fJ.-3I2x2+13

-fi

_{...,-2x 3+ ...} 12 36 (43) (44) k-I_{9J.-1+21J2fJ.-3/2x+S/3fJ.-2x2+ ...} k-312= fJ.-3/2+3.2-1J2fJ.-2x+ ...

Anasymptotic expansion forg(k)can be determined from (32).

00

(45) g(k) :::: "Lglk-l12 (k~oo)

1=0

Some coefficientsglare

(46) go=I, g2=-1I12andg/=Oif 1

=

odd.

-

9-(47) I(Il,k)

-=f

(x)+e-:i

L

QI(X)Il-112 (IJ.-+oo) 1=1

Theql'sare polynominals. We calculate

(48) (49) q1(x)=_21I2Tlx-112 ()= 5 -112 q2 X

- - x

x

12

6. The replacement of the sum by an integral We have already

(50)

By induction one can prove easily that

(51) I II. 00

~J(J.l,k)=(r(k+l»-l-l

I

dt

I ...

· 0 0

J

e-t--'tl_{- ' " -'t,(ttl· .. td'L(t,tit ...,tl)dtl ... dtl ,}

o

where the functionsL(to,tl' ...,t/)are defined by

(52) L(to)=1

With methods simular to those used in the treatment ofI (IJ.,k) in section 5 we can prove easily that

(54)

IJ3

with an error of the kind

o

(e-CII. ) (IJ.-+oo). Furthennore, if we restrict ourselves to values ofksuch that IJ.l.-kISJ.l.213, then the integrals in

II. 1I.+2l3 1I.+211.2IJ

(51) canbereplaced by

J

I· ..

I

1I.-211.2IJ 11.+211.:113 1I.-211.2IJ

Then it follows from (51) that for IIl-k ISJ.l.213

(56)

10

-From (52) and (53) it follows that

[ 11+211213J 1+1 M ' +l~ M/log 213 (IeIN0), 11-21ol

whence, by (54) and (55), we have for all ke IV

d '

I - , I (IJ.,k)I

=

0 (IJ.-II3) (IJ.-.+oo) (IeIN0). dk

Now we apply the Euler-Maclaurin sumformula: For every fixed reINwe have

q q (57)

'LI

(k)=

I

1

(x)dx

+

1/21

(q)

+

1/21

(P) k=p p

+

i:.

Bu

[fU-l)(q)_fU-l)(P~

/=1 (21)!

1

+o[

/1J"')(%111

(tf'p1 .

Taking/(k)= Im(IJ.,k)(I-1{IJ.,k»/I,

p=f

11;1213

1,

q

=ll1+

I1213

J '

we find, for every reIN,using (54), (56) and (57)

1l+tL213

(58)

'L

rQ.t,k) (1-/(1J., k»/I

=

J

r(IJ.,k)(I-1(IJ.,k»/Idie

Ik-jL151l213 1l-jL213

Simularly

1l+tL:n

(59)

'L

(u-k)/m(IJ.,k)(l-/{IJ.,k»/I

=

J

(IJ.-k)r(IJ.,k) (l-/(IJ., k»/I dk Ik-jLISIl:n 1l-jL:n

7. The asymptotic behaviour of Sand T

According to section 6 the sums

'L

canbe replaced by integrals with small errors which

IIl-k 1511213

are, for every r>O , 0(IJ.-r) (IJ.-.+oo). Then these integrals are transformed into integrals

B{jl.) dk

I ...

dx dxwhereA(11) andB(IJ.)are asymptotically equivalent with 1/2+2 11116 (IJ.-.+oo)

11

-By means of (38) and (47) we get complete asymptotic expansions for the two integrands.

(60)

(61)

[m{JJ.,k)(1-[(JJ.,k»": :::

:i:

SI(X)~-l/2 (JJ.~oo)

,

'~l

bothuniformly inXElR ,x

=

O{JJ.l/6)(JJ.~00). The functionsSI(X) andtl(X)are absolutely integr-able over(-00,00). Now we proceed as follows. LetNEIN.Then

JI.+J1213

t

[m{JJ.,k)(l-/{JJ.,k»" : dx JI.--JI. N B{JI.) [ N

1 lJ

=

L

~-l12

J

sl(x)dx+O ~--21"7 '~l -A{JI.)

It is easily seen that the functions SI(X) are of the form

2 2 2

p(x,e-X

_,1

(x» e-x +q(x,e-X _{,/(x» I (x)(l-I (x»,}

where p and q are polynomials. Hence

J

Is,(x)ldxand

J

Is,(x)ldxare

o

(e-eJl.II3)

(JJ.~00).

X<-A(JI.) x>B{JI.)

It follows that S (J..l,m,n)has a complete asymptotic expansion

00

(62) S{JJ.,m,n):::

i

~-l12

J

sl(x)dx

(JJ.~00)

/=-1 . - 0

A simular argument holds forT(JJ.,m,n).

00

(63) T{JJ.,m,n):::

:i:

J..l-I12

J

tl(x)dx

(JJ.~oo).

1~2 We calculate (64) S-l (x)=.J2!"'(x)!"(-x) (65) SO(X)=2/31t-I12 [

nr(X)f"-I(-x)-mr-l(X)f"(-X~

e-x2-2/3xf"'(x)f"(-x) (66) L2(X) =2x !"'(x)!"(-x) (67) Ll(x)

=

23123-11t-112

[n!"'(x)f"-I(-x)-mr-l(X)f"(-X~

xex2-21I2X2r(x)!"(-x) 2

The term with factor

e-

x in the right hand of (65) gives 0 upon integration.

2

Integration by parts of the term with factor xe-x in the right hand of (67) gives

00

12 -8. Computation of some coefficients

~ ~

A(1, 1)="2

J

f (x)(I-f (x»dx=-"21t- 1I2

J

x (l-2f (x»e-X2dx

--~ ~ =2"21t-1/2

J

f (x)xe-X2dx="21t-1

J

e-2x2dx=1t-l12

--~ A (l,2)=A(2, 1)= 112"2

J

(f(x)f2(-x)+f(-x)f2(x»dx

--~ =112"2

J

f(x)f(-x)dx=lI2A(l,I)

--B(1,1)=-2/3

J

x! (x)f (-x)dx=O . B (1,2)=-B(2, 1)=-2/3

J

x!(x)f2(-x)(f(x)+f(-x»dx

--~ =-2/3

J

x!(x)f3(-x)dx=B(1,3)=-B(3,1) B(1,2)=-2/3

J

x(f (x}-2f2(x)+f3 (x»dx ~

= 1I31t-1I2

J

x 2e-x\l-4f (x)+3f2(x»dx

~ = 1I61t-lJ2

J

_e-x2_{(1-4f (x)+3f2(x)+1t-1I2xe-x2 (-4+6f (x»)dx} ~ ~

=

1/6

J

(1-4f (x)+3f2(x»!, (x)dx+1t-1

J

xe-2x2f (x)dx

--00 ..J?: 1/4 -3/2

J

_3x2dx 3-1 = 1t

e

=

12 1t .

--~ D(1,1)=2/3A(I,I}-"2

J

x 2f(x)f(-x)dx

fi

~ =2/3A(1,I)-2

~

1t-1/2

J

x 3e-X2f(x)dx

--- -- 13-=2/3A(1 1)-

_,

2..-

₁₂

x-l12 = 1I4x-112

_.

00 D(I,2)=D(2,1)=213A(I,2)-1I2.J2

I

x2(f(x)f2(-x)+f2(x)f(-x»dx= 00 =2/3A(l,2)-1I2.J2

I

x2f(x)f(-x)dx - 0 0