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JORGE DOMINGUEZ Y DOMINGUEZ

S & T Performance Assesment of Mexican Research Units

Limiting properties of the empirical probability generating
function ¢" (t) are investigated. It is shown that ¢ (t) converges
uniformly almost surely to ¢(t), the probability generating
function of the underlying distribution. The weak convergence of
to a Gaussian process is discussed. Extensions
to multivariate discrete distributions are presente

VICTOR MANUEL PEREZ ABREU CARRION

i. . ••• ~ • 
COMUNICACIONES DEL CIMAT 
,. . .. ·. ···: .· ·· . 
. . ON THE LIMITING BEHAVIOR 
OF THE EMPIRICAL 
* Universidade Estadual de Campinas 
IMECC, Caixa Postal 1170 
13081 Campinas, SP - Brasil 
**cENT R 0 DE 
1-N V E S T I G A C I 0 N E N 
MATEMATICAS 
Apartado Postal 402 
Guanajuato, Gto. 
Mexico 
Tels. (473) 2-25-50 
2-02-58 
6 
A NOTE ON THE LIMITING BEHAVIOR 
OF THE EMPIRICAL PROBABILITY GENERATING FUNCTION 
by 
Mauro S. Marques* 
Universidade Estadual de Campinas · 
IMECC, Caixa Postal 1170 
13081 Campinas, SP - Brasil 
and 
Victor Perez-Abreu 
Centro de Investigaci6n en Matematicas, A. C. 
A. P. 402, Guanajuato, Gto., 36000- Mexico 
ABSTRACT 
Limiting properties of the empirical probability generating 
" function ¢ (t) are investigated. It is shown that ¢ (t) converges 
n n 
uniformly almost surely to ¢(t), the probability generating 
function of the underlying distribution. The weak convergence of 
n1/ 2 ( ¢ (t) - ¢(t)) to a Gaussian process is discussed. Extensions 
n 
to multivariate discrete distributions are presented 
Key words and phrases: 
function, weak convergence, 
process. 
Empirical probability generating 
discrete distribuion, Gaussian 
* This work was done while the author was visiting the Centro de 
Investigaci6n en Matematicas at G~anajuato, Gto., Mexico. 
1. INTRODUCTION 
Let X
1
,x
2
,... be nonnegative integer valued independent 
random vari'ables with the same discrete distribution {Pk}ke::O and 
(X) k 
probability generating function (pgf) ¢(t) = L t Pk ltl ~ 1. Let 
k=O 
~ 
¢ (t) be the empirical probability generating function (epgf) of 
n . 
the first n variables, i. e. 
~ 1 n X. ¢ (t) = \ t 1 
n nih ( 1. 1) It I ~ 1 
Kocher lakota and Kocherlakota ( 1986) have suggested the use 
of the epgf for procedures in testing hypotheses of goodness of 
fit for discrete distributions. Press (1972), Heathcote (1972) and 
Feuerverger and Mureika (1977) considered, among others, the use 
of the empirical characteristic function for . problems ·in 
statistical inference for stable laws. Epps et al (1982) have 
~ proposed methods based in the empirical moment generating function 
to .test hypotheses ~f goodness of fit in the presence of nuisance 
parameters. The study of the limiting behaviour of the empirical 
characteristic function was initiated by Feuerverger and Mureika 
" (1977) and Csorgo (1981). Recently, Malboroda (1986) has proved 
the law of large numbers and the central 1 imi t theorem for the 
empirical moment generating function in certain function spaces. 
In the present paper, we study the limiting behavior of the· 
empirical probabi I ity generating function. We show a uniformly 
strong law of large numbers (Theorem 1) and the weak convergence 
1/2 ~ 
of the processes Y (t)=n (¢ (t)-¢(t)) to a Gaussian process Y(t) 
n n 
in the space C([O, 1]) (Theorem 2). This last result is proved 
under the assumption that the second moment of {Pk}ke::Q is finite. 
2 
--------------~---~ -- -------------·--~---------------· 
These results are stated in Section 2 and their proofs are given 
in Sect ion 3. Extensions to multi variate discrete distributions 
are presented in Section 4. 
Although characteristic functions characterize completely 
distribution functions, we think that the use of probabi 1 ity 
generating functions, which also characterize discrete 
distributions, could be more appropriate and easier to handle in 
these situations, specially having in mind statistical 
applications for discrete data. On the other hand, the obvious 
relation M(t) = between the moment and probability 
generating functions for discrete distributions makes results on 
probability generating functions special cases of results for 
moment generating functions. However, as pointed out in Feller 
(1968), results for the lat ters are usually stated and proved 
under unnecessary r~strictions, as for example the existence of 
J 
all moments, which is not~ assumed in the present work and is a 
basic assumption in Malborada (1986). 
2. MAIN RESULTS 
Suppose (Q,~,P) is a probability space where there is defined 
a sequence {X }· >1 n n-
of nonnegative integer valued independent 
random variables with common distribution {Pk}k~O and probability 
generating function ¢{t). We recall that (see Feller (1968)) ¢(t) 
is an analytic function in (-1, 1), Pk = 1/k! ¢(k)(O) k~O and the 
factorial moments of {Pk}k~O are given in terms of the behavior of 
the derivatives of ¢(t) at t = '1. Therefore, since most of the 
information about {Pk}k~O is contained on the behavior of ¢(t) in 
[0,1] we will consider the epgf ¢ (t) as an stochastic process in 
n 
3 
.. 
... 
[0,1], although all our results in this section hold if we 
consider the interval [-1,1]. We shall denote by C[0,1] the space 
of real valued continuous functions on the interval [0,1] with the 
usual norm llfll = sup lf(t) 1. Clearly ¢ and ¢ belong to C[O, 1]. 
o~t~1 n 
A 
Using the fact that E(¢ (t)) = ¢(t) 
n t e [0,1] we can easily 
show the following relations: 
EC¢ (tJ¢ (s)) = ¢(ts) + n-1¢Ct)¢(s) s,t e [0,1], n n n n 
V AR ( ¢ ( t ) ) = ! { ¢ ( t 2 ) - ( ¢ ( t ) ) 2} t E [ 0 , 1 ], n n 
COY(¢ (t)¢ (s)) =n -
1
{¢(ts)- ¢(t)¢(s)} s,t E (0,1] . n n 
Define for n::::l 
1 n . 
p = \' 1 
k,n n i~l {Xi=k} k::::O 
(2. 1) 
(2.2) 
(2.3) 
(2.4) 
Then Pk is the frequency of the value k in the first n ,n 
observations, E( P k, n) = I:k and the . epgf can be expressed as 
follows: 
. CXl . 
¢ ( t ) = L t kp k t E [ 0' 1 ]. 
n k=O ,n (2.5) 
The first limit theorem corresponds to the strong law of 
large numbers which gives 
estimator of ¢. 
Theorem 1.· · 
II¢ - ¢11 ---7 0 a. s. n 
n-700 
that is 
A 
sup 
o~t~l 
1¢ (t)- ¢(t)j ---7 
n 
n-700 
the 11•11-consistency 
a. s. 
4 
of ¢ 
n 
as ·an 
(2.6) 
(2.7) 
The p~oof of this theo~em is given in Section 3. 
The next step is the study of the central 1 im.i t theo~em o~ 
weak convergence of the p~ocesses 
y (t) = n1/ 2 c¢ (t) - ¢(t)) 
n n 
t E [0,1) (2.8) 
in the space C[O, 1]. The one dimensional central limit theopem 
gives, fop each t, the convergence in law of Y (t) to a Gaussian 
n 
Pandom vaPiable with zero mean and variance 
(2.9) 
Moreover, for any k2::1 and t 1, ... tk e [0,1] f~om the 
multidimensional central limit theo~em we obtain the convergence 
in law of (Yn(t
1
J, ... ,Yn(tk)) to a k-dimensional Gaussian random 
vector with zero mean and covariance {K( t., t.)}. ._1 k' where 1 J l,J- , ... , . 
K(s,t) = ¢(st) - ¢(s)¢(t) s,t e (0,1]. (2.10) 
The second limit theo~em of this paper is a centr'al iimit theorem 
giving the weak convergence of {Y } 1 as p~ocesses with sample n n2:: 
paths in C[ 0, 1]. As usual we denote weak convergence by 9. The 
p~oof of this result is given in Section 3. 
Theorem 2. Let the second moment of {Pk}k:!::O be finite. Then 
Y 9¥ in C[O, 1J where Y(t) is a zero mean continuous Gaussian 
n 
process with covariance K(s,t) given by (2.10). 
Remarks. 
a) Theorems 1 and 2 hold if we consider Y and Y as elements in 
n 
C[-1,1]. 
b) ··The p~ocess Y(t) can be represented by the stochastic· 
integral 
CXI 
Y(t) = J txdB(F(x)) 
-CXI 
5 
(', 
· .. ·;· 
where B is a continuous version of the Brownian bridge and F 
is the distribution function of {Pk}k~l· 
3. PROOFS 
The proofs of Theorems 1 and 2 may be obtained from results 
on Law of Large Numbers and Central Limit Theorems for Banach 
space valued random variables. However we shall present here 
easier proofs using classical results as the one dimensional 
strong law of large numbers and theorems on weak convergence in 
C[0,1]. An alternative proof, using results from Araujo and Gine 
(1980) will be given in Section 4 where we present extensions to 
multivariate discrete distributions. 
Proof of Theorem 1. From (2.5) we have that for any t e [0,1] 
"' 
I ¢ C t J -¢ C t J I 
n 
. (3. 1) 
The usual strong law of large numbers gives that for any k ~ 0 
a.s. (3.2) 
co 
Then applying the dominated convergence theorem to r IP -P I we 
k~O k,n k 
obtain from (3.1) that 
sup 1¢ (t)-¢(t)l ~ 0 n . 
o~t~1 n7ro 
a.s. • 
Proof of Theorem 2 To show that Yn =9 Y ~n C(O, 1] we will apply 
Theorem 8.1 of Billingsley (1968). The weak convergence of the 
finite dimensional distributions has been previously observed 
in Section 2. Then we only have to prove tightness of {Yn(t)}n~l 
in C(O, 1] .. In order to do this we shall use the following result. 
6 
.. -------- ---------· - -----~--~ 
--------~--~----- ----------------------
Lemma 1. Assume the second moment of {Pk}k~ is finite. Then 
there exists a constant 6 depending only on ¢ such that for any 
s,t e[0,1] 
2 IK(t,t) + K(s,s) - 2K(s,t)l s 6(t-s) 
Proof. From (2.10) we have that for s,t e [0,1] 
K(t,t)+K(s,s)-2K(s,t) = ¢(t2)-(¢(t)) 2+¢(s2)-(¢(s)) 2 
+2¢(t)¢(s)-2¢(st) 
Then taking 
we obtain the lemma . 
• 
=¢(t
2
)+¢(s2)-2¢(st)-(¢(t)- ¢(s))2 
~ ¢(t2)+¢(s2)-2¢(st) 
co 
2\ k-1 k-2 k-1 2 =(t-s) L (t +t s+ ... +s ) Pk 
k=O 
(3.3) 
(3.4) 
(3.5) 
We now go back to the proof of Theorem 2. For any s,t e [0,1] 
using Lemma 1 we have that 
E(Y (t)-Y (s)) 2 = K(t,t)+K(s,s)-2K(s,t) ~ 
n n 
7 
2 
9 ( t-s) . 
Next, by the one dimensional central limit theorem Y (0) = 
n 
v'n(P
0 
-P
0
) converges to a normal random variable and therefore 
. ,n 
{Y (0)} 
1 
is tight. Then from Theorem 12 .. 3 in Billingsley (1968) 
n n::: 
we have that {Y (t)} >
1 
is tight in C[O, 1]. as we wanted to 
n n-
prove. 
• 
4. EXTENSIONS TO MULTIVARIATE DISCRETE DISTRIBUTIONS 
In this section we present extensions of Theorems 1 and 2 to 
multivariate discrete distributions. Let X = CX
1
, ... X ) be an - r 
r-dimensional discrete random vector with probability generating 
function 
(4. 1) 
and for n:::1 
i i . i . . 
let X: = (X
1
, ... ,Xr) 1=1, ... ,n be an independent 
random sample of X: defined on a probability space (n, ~. P). Define 
- J 
the empirical probability generating f'unction of the first n 
observations 
i i 
A 1 n x1 . xr 
""'(t) =-'t1 ... t t = (t1, ... ,t) 
'~"n n L r - r It I ::::1. (4.2) 
i =1 
Let C([-1,1]r) be the metric space of continuous functions from 
[~1,1]r to~ with the metric 
I I I f I I I = sup I f C.tJ I 
. r 
_t.e[-1,1] 
We shall introduce the following notation: for 
t•s = Ct
1
s
1
, ... ,t s ). -- r r 
(4.3) 
r _t., §. e [ -1 , 1 ] 
The following relations are easily shown: let s, t e [ -1, f]r, 
then 
"' 
E(¢n(_t.)) = ¢(1.) .. . ... (4. 4) 
8 
(4.5) 
(4.6) 
A 
Theorem 3. I I l¢n-¢1 I I~ as n~ a.s. 
Proo:f. Let Pk k = PCX1=k1, ... ,X =k) and 1" .. r r r 
~~1· · · k = ~ t 1{xi=k x1=k }. r 1-1 , .•. , 
1 1 r r 
(4.7) 
Then 
l.t..l::$1 (4.8) 
and 
A Loo k1 kr"n 
¢ (.t_)= t1 ... t pk k 
n k k =0 r 1· · · 1· · · r r 
(4.9) 
~ From the classical strong law o:f large numbers, :for each positive 
"n 
pk k ~ pk k a. s · 
1. . . 1" .. r n~ r 
(4.10) 
Then the proo:f o:f the theorem ends quite analogous to the proo:f o:f 
Theorem 1. 
Ill 
Theorem 4. Let Y (t) = nl/2 (¢ (t)-¢(t)) and assume that n- n- -
E(X.) 2 <co :for i = 1, ... ,r. Then Y 9Y in C([-1,1]r) where Y is a 
1 n 
Gaussian random :field in C([-1,1]r) with zero mean and covariance 
:function 
9 
r 
~ • .t_ E ( -1 , 1 ] . (4.11) 
(. 
Proo:f. Since for each n ~ 1 Y (•) is an element of the Banach n 
space C([-1,1{), in order to prove the theorem we shall verify 
the assumptions in Corollary 7.17 of Araujo and Gine (1980): using 
(4.5) we obtain 
for each r i E [ -1, 1] . 
Next for each n ~ 1 and ~.i e [-1,1]r 
where 
00 k kA 00 k k .... 
[ 
1 r n [ 1 r n t ... t pk k s 1 ... s pk k 1 r . . . r ... 
k ••• k =O 1 r k ••• k =O 1 r 
1 r 1 r 
00 k k k kA 
[ 
1 1 2 r n 
(t -s )t ... t Pk k + 
1 1 2 r ... 
k ••• k =O 1 r 
1 r 
00 k k k k k 
r s 1(t 2_s 2)t 3 ... t r 
L 1 2 2 3 r 
k ••• k =0 
1 r 
An 
p 
k ... k 
1 r 
00 k k 
s ... s (t L 1 r-1 
k k 
r r) p"n -s 
r r k: .. k 1 r-1 
k ••• k =0 
1 r 
A · CQ A 
Pn(j) = ~ Pn 
k ... k 
k k 1 r 
i J 
i=1, .... ,r 
1 r 
+ ... + 
r 
Then fori.~ E [-1,1] 
¢ncu-¢n(§_) :s mru: lti-sil ( [
00
kP~(1)+ ... + Lk. 
00
=kOP~(r)J. 
1:SI:Sr k=O 
In a similar way we can show that fori,§.. e [-1,1]r 
00 
where Pk(j) = ~ P · k ... k . 
k k 1 r 
i J . 
·i=l, ..• ,.r _____ _ 
10 
------------- -----
(4. 12) 
= 
. (4.13) 
(4. 14) 
(X) 
By assumption we have that fori= 1, ... ,r L k2Pk(i) < oo and 
k=O 
(X) " 
therefore L kP~(i) e L2 (Q) for i = 1, ... ,r. Then if 
k=O 
OJA OJ,.. OJ (X) 
M = L kP~(i)+ ... + [ kP~(r)+ [ kPk(l)+ [ kPk(r) , 
k=O k=O k=O k=O 
(4.15) 
(4.13) and (4.14) we have that for 
§.,t_ E (-1, 1) r 
IY (t)-Y (s)f ~ Me(_s,t_) 
1 - 1 -
(4. 16) 
where e is the continuous metric 
e(s,t) =max lt.-s. 1. 
- - . l l 
1~1~r 
(4.17) 
Conditions (4.12) and (4. 16) imply that the assumptions in 
Corollary 7.17 of Araujo and Gine (1980) are satisfied and 
therefore there exists a process Y in C( [ -_1, 1 ]_I"j __ wLth _coYariance 
(4. 11) such that Y ~yin C([-1,1]r). 
n 
Acknowlegements 
• 
The authors thank Professor Federico O'Reilly for valuable 
discussions and Professor Luis Briceno for having suggested the 
proof of Lemma 1. , 
11 
,, 
I 
REFERENCES 
[1] Araujo, A. and E. Gine (1980), <Jhe ~e.ni.rt.af. 2-i.mU <Jheoltem f-OIL 
'Rea£ and :Banach ~ 'Rartdo.m V~. (Wiley, New 
York.) 
[2] Billingsley, P. (1968), ~~ a& 7l~ .#.~. 
(Wiley, New York.) 
" 
[3] Csi:irgo, S. (1981), Limit behaviour of the empirical 
characteristic function. Ann. 7l~. 9, 130-144.144. 
[4] Epps, T. W., K. J. Singleton and L. B. Pulley 
of separated families of distributions 
empirical moment generating function. 
391-399. 
(1982), A Test 
·based on the 
:B.Lo.met:!Uka 69 ' 
[5] Feller, W. (1968), .i1.n J~ ta :P~ <Jhea!u.j. a.n.d. 
U4 ~· Vol. I. Third Edition. (Wiley, New 
York.) 
[6] Feuerverger, A. and R. A. Mureika 
characteristic function and 
~~. 5,88-97. 
( 1977), The empirical 
its applications. Ann. 
[7] Heathcote, C. 
symmetric 
172-181. 
R. (1972), A test of . goodness of fit for 
random variables. ~. J. ~t.a;t;M. 14, 
- J 
[8] Kocherlakota, S. and K. Kocherlakota (1986), Goodness of fit 
tests for discret.e distributions. ~amnw,n. ~tal.,Lo;t. <Jheolt. 
[ 9] 
JH.eth. 15, 815-$29. 
Malboroda, R. E. (1986), 
generating function of 
results of observations. 
~~. 32, 141-151. 
An estimator of the moment 
a random variable. from the 
<J heo!L. :P!LO.&.a.&. a.n.d. .Math. 
[10] Press, S. J. (1972), 
multivariate stable 
~. 67, 842-846. 
Estimation in univariate and 
distributions. J. . .i1mert. ~taiJ.,(:,l. 
12 



English

A Note on the Limiting Behavior of the Empirical Probability Genereting Function

Se hace una revision amplia del tema de la Colinealidad en la
Regresi6n Lineal. Aunque este es. un problema netamente numerico
debido a la "mala calidad" de los datos, los estadisticos estan
involucrados, pues la calidad de los estimadores se ve afectada. Se
muestran los efectos que provoca la presencia de la coiinealidad y
sus causas mas frecuentes. Se pre~entan varies procedimientos
utilizados Prara su diagn6stico, algunos de ellos apoyados en
herramientas desarrolladas en el terrene del Analisis Numerico como
puede ser la Descomposici6n en Valores Singulares. Asi tambien se"'
dice c6mo poder remediar el problema o en su defecto se recomienda
como alternativa el usa de estimadores sesgados o del tipo ~ayesiano.
Finalmente, se propane una estrategia para atacar el problema

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A Note on the Limiting Behavior of the Empirical Probability Genereting Function

El Problema de la Colienidad en la Regresión Lineal

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