1
The Probabilities of Large Deviations for the Chi-square and Log-likelihood Ratio
Statistics
Sherzod Mirakhmedov
Institute of Mathematics. National University of Uzbekistan
100125, Tashkent, Durmon yuli st., 29
e-mail: shmirakhmedov@yahoo.com
Abstract. A new large deviation results for the Pearson’s chi-square and Log-likelihood ratio
statistics are obtained. Here attention is focused on the case when the number of groups increases to
infinity and the probabilities of groups’ decreases to zero, as the sample size tends to infinity.
1. Introduction. Let   (1 ,...,N ) be a random vector having a multinomial distribution
M  n, p1 ,..., pN  , where p1  ...  pN  1 , pm  0 , m  1,..., N and 1  ...   N  n . Many tests of
goodness-of-fit can be reduced to testing a hypothesis about the parameters p1 ,..., pN . Most notably
test statistics for this kind of testing problem are Pearson’s chi-square (  2 ) statistic
(m  npm )2
,
npm
m 1
N
 N2  
and the log-likelihood ratio (LR) statistic:
N
m
m 1
npm
 N  2m ln
.
In the present paper we study P TN  xN  , where TN is standardized version of the  N2 or  N
statistics and xN   , in the situation when max pm  0 , N   as n   .
Similar problems have been considered in the literature. We refer to Kallenberg (1985), in
particularly his results for the Cramer’s zone, viz. xN  o( N ) , proposes that N min pm  c  0 and
additionally N  o( n ) ,
nN 3/2 xN   for the  N2 case. We refer also to Sirajdinov et al (1989)
and Ivchenko and Mirakhmedov (1995). In these papers the large deviation problem in Cramér’s
zone has been studied for the class of statistics h1 (1 )  ...  hN (N ) , where hm (u ) is a Borel function
defined on the non-negative axis and satisfying the Cramer condition: max E exp H hk ( )    , 
1 k  N
H  0 , here 
Poi( ) , the Poisson distribution with expectation   lim(n / N )  (0, ) . The
n 
Cramér’s condition is fulfilled for the  N , but for the  N2 does not
-------------MSC (2000): 60F10, 62G10, 62G20
Key words and phrases. Chi-square statistic, Log-likelihood ration statistic, multinomial distribution, large
deviations, Poisson distribution.
2
The assertions of the present paper corresponds to the case xN  o( N 1/6 ) , nevertheless they
covers the situations which does not covered by aforementioned papers. Our results, together with
corresponding results of Kallenberg (1985), allowed to explore the intermediate asymptotic relative
efficiency due to Inglot (1999) of the  2 and LR goodness of fit tests; the results are object of
separate paper Mirakhmedov (2016). Also our proofs are based on the technique different from
those aforementioned authors. Core part of our technique is that we reduce the cumulants of
statistics of type RN to the cumulants of a sum of independent random variables (r.v.), next we use
a modified variant of Lemma 2.3 of Saulis and Statulevicius (1991).
The rest of the paper is organized as follows. The main results are presented in Section 2; in
Section 3 the proofs are given; for the reader’s convenience, the auxiliary Assertions are collected
in Appendix. In what follows Ck is an universal positive constant; all asymptotic statements are
considered as n   ; 
F stands for “ r.v  has the distribution F ”.
2. Results
Let pmax  max pk , pmin  min pk , n  n / N and (u ) be the standard normal distribution
1 k  N
1 k  N
function.
2.1. Chi-square statistic. Set n  max 1, (npmin )1  ,
N
1
 2N ,
m 1 npm
 N2  
 N2 
1
n
N
 1
  Np
m 1

m

 1  2 N   N2  N n1 .

 
Theorem 2.1. Let pmax  o(1) . Then for all xN such that 0  xN  o min  N3 /  N2 n
(2.1)

1/3
1/4
, n1/6 , pmax
it holds
P  N2  xN N  N  1  ( xN ) 1  o(1) 
(2.2)
and
P  N2   xN N  N   ( xN ) 1  o(1)  .
(2.3)
Corollary 2.1. Let n  C0  1/ 2 and
C1  Npmin  Npmax  C2 N 1/3 ,
then for all xN such that 0  xN  o( N 1/6 ) the relations (2.2) and (2.3) are valid.
Corollary 2.2. Let pm  N 1 1   (n)dm,n  , m  1,..., N , where  (n)  0 and
N
d
m 1
m,n
0,
1 N 2
d m,n  d 2   .

N m1
(2.4)

3
Then for arbitrary n and xN such that 0  xN  o  N 1/6 min(1, n2/3 )  the relations (2.2) and (2.3) are
valid.
Remarks 2.1. For xN   the relation (2.2) can be written in the form log P  N2  xN N  N 
  xN2 / 2(1  o(1)) . From Kallenberg (1985, Eq. (2.17)) such kind relation follows if log N  o( xN2 ) ,
xN  o( N 1/6 ) and N  o(n3/8 ) . Last condition excludes the case recommended by Mann and Wald
(1942) who obtained the relation N
cn2/ 5 concerning the optimal choice of the number of groups
in chi-square goodness of fit test.
2.2. Likelihood ratio statistic  N . Here we restrict our attention to the case n   ; the
corresponding results for the case n    (0, ) follows from Sirajdinov et al (1987).
Theorem 2.2. Let n   and
C3  Npmin  Npmax  C4n3 ,
(2.5)
1/4
Then for all xN such that 0  xN  o  min( N 1/6 , pmax
)  it holds




P  N  xN 2 N  N  1  ( xN ) 1  o(1)  ,
(2.6)
and
P  N   xN 2 N  N  ( xN ) 1  o(1)  .
(2.7)
Corollary 2.3. Let n   and pm  N 1 1  o(1)  , m  1,..., N , then for all xN such that
0  xN  o( N 1/6 ) the relations (2.6) and (2.7) are valid.
Remarks 2.2. Let n   , Npmax  O(min( N 1/3 , n3 )) , xN   and xN  o( N 1/6 ) . Then from (2.6) it


follows that log P  N  xN 2 N  N   xN2 / 2 1  o(1)  . From Kallenberg (1985, Eq. (2.13)) such
relation follows under additionally conditions N  o  n3/7  and log N  o( xn2 ) .
3. Proofs
In what follows 1 ,...,  N are independent r.v.s with m
Poi(npm ) , 
Poi(n ) and
k
( ) is a
cumulant of the k th order of the r.v.  ,
N
RN   hm m  and
m 1
N
TN   hm  m  ,
(3.1)
m 1
where (1 ,..., N ) is a multinomial r.vec. as in introduction, hm ( x) , m  1,..., N are Borel functions
defined on the non-negative axis.
Main Lemma. Let


E hm (m )  O Ehms (m ) ,
s
(3.2)
4


E hm (m ) (m  npm )2  O s a (npm )b Ehms (m ) ,
s
(3.3)
for all m  1, 2,..., N , some non-negative real a and b , and each integer s : 3  s  k , where an
integer k such that
k
p
i 1
ji
 o(1) ,
(3.4)
for any k-subset ( j1 ,..., jk ) of the set (1, 2,..., N ) . Then for all k satisfying (3.4) and
3  k  o  Kn (a, b)  ,
b
where Kn (a, b)   n1b pmax

1/( a 1)
(3.5)
, a  max(1, a) , it holds
( RN ) 
k
k
(TN ) 1  o 1  .
(3.6)
Proof. We start from the following
Lemma 3.1. For any non-negative k it holds
 n
ER  n
k
N


 S 
ETNk exp i N  d ,
n

n
where
def


n  2 nP{S N  0}
1
n !e n
1

2 nn n
2

((1,...,N )) 
Proof. It is well known that

 1 
1  o    .
 n 

((1,...,  N ) / SN  0) , where
(3.7)
( X ) stands for the
distribution of a random vector X , S N    m  npm  . Hence ERNk  E TNk S N  0  . On the other
N
m 1


hand E TNk ei SN   E ei SN E TNk S N  . Now Lemma 3.1 follows by Fourier inversion. Notice that
Eq. (3.7) follows due to fact that S N
Poi(n) and Stirling’s formula.
n
Let k be as in (3.4) and (3.5). Use Lemma 3.1 to write
k
ERNk  n  l ,k l
l 1
 n



E  h j1 ( j1 )

n


k1

 ...  h jl ( jl )

kl
 S 
exp i N   d ,
n 

(3.8)
where l ,k is the summation over all l-tuples (k1 ,..., kl ) with positive integer components such that
k1  ...  kl  k ; l is the summation over all l-tuples ( j1 ,..., jl ) such that ji  jr for i  r and
jm  1,2,..., N ; m  1,2,..., l .
l
l
i 1
i 1
Set Sl , N   ( ji  np ji ) , dl , N   p ji , l
 n



E  h j1 ( j1 )

n


k1

 ...  h jl ( jl )

kl
1,..., k and write
 S 
exp i N   d
n 

5
 n




E  h j1 ( j1 )

n
 n


  E  h j1 ( j1 )

 n


k1

k1

 ...  h jl ( jl )

 ...  h jl ( jl )

kl
 S 
 S  Sl , N 
exp i l , N   E exp i N
 d
n 
n 


 2

 Sl , N   
 S N  Sl , N 
exp i
   E exp i
  exp  1  dl , N   d
n  
n 


 2

 2
 
exp

1

d
E  h j1 ( j1 )




l,N

 2
 
 n
 n

kl



 E  h j1 ( j1 )


k1

 ...  h jl ( jl )

kl

k1

 ...  h jl ( jl )

kl

 Sl , N   
  1  d
 exp i
n   


 n
 2
 def

exp

1

d



d  J1  J 2  J 3 .
l
,
N
 
 2

 n
We have

 i ( m  npm ) 
 it /
E exp 
  exp npm  e
n




n
1 
(3.9)
 2

it  
 3

exp

p

pm  ,

m

n 
6 n 
 2
(3.10)
and
 i (m  npm ) 


 2 pm 2 
2 
E exp 
  exp 2npm sin
  exp  2   ,
n
2 n
 




2
2
2
because sin u / 2  u /  , u   .
(3.11)
In order to get an upper bound for the J1 we first write the integral J1 as the sum of two
n / 2 and
integrals, J1 and J1 say, over intervals
n/2
n respectively. Next,
noting that S N  Sl , N is the sum of independent r.v.s m  npm over the set of indexes (1,..., N )


except ( j1 ,..., jl ) we use to E exp i (S N  Sl , N ) / n in the J1 and J1 the equation (3.10) and
inequality (3.11) respectively. Then quite clear algebra gives
  n /2
 3
kl 
k1
 2


 

J1  E h j1 ( j1 )  ...  E h jl ( jl )   exp  1  dl , N  exp 
(1  dl , N )   1 d
 2
 


 
6 n
 n /2


 2
 
 S N  Sl , N 

  E exp i n   exp  2 1  dl , N  d 
 n

 
   n
2

3
  n /2
 6 
2 
 E h j1 ( j1 )  ...  E h jl ( jl )   exp 
1  dl , N   6 n (1  dl ,N )d
 12

  n /2
kl
k1
 n

 C5 exp  1  dl , N  .
 4

Hence

 1 
J1  O 
 E h j1 ( j1 )
 n 

k1

 ...  h jl ( jl )

kl
.
(3.12)
since (3.2) and (3.4). Next apply inequality eit  1  it  t 2 / 2 to get: for all k satisfying (3.5)
6
2
kl  l
k1
C6 
 
J2 
E  h j ( j )  ...  h jl ( jl )    ji  np ji  
n  1 1
 i 1
 


ki
C7 l
l  E h ji ( ji )  ji  np ji
n i 1
 C8 E h j1 ( j1 )




 C9 E h j1 ( j1 )
h jm ( jm )




k1
 ...  h jl ( jl )

 
k1
l
m  i , m 1
 ...  h jl ( jl )


2
k1
 o 1 E h j1 ( j1 )

km
kl

k l a
ki (np ji )b

n i 1
kl

(npmax )b 1 max(1,a )
k
n

 ...  h jl ( jl )

kl
,
(3.13)
because (3.3) and the fact that k1a  ...  kla  k max(1,a ) . By a simple algebra we obtain

J 3  2 E h j1 ( j1 )

k1

 ...  h jl ( jl )
  1  o 1 .
kl
(3.14)
since the condition (3.4). Now apply (3.12), (3.13) and (3.14) in the (3.9) to get


E
  h j1 ( j1 )



k1

 ...  h jl ( jl )

kl
 S 
exp i N   d
n 


 2 1  o 1  E h j1 ( j1 )

k1

 ...  h jl ( jl )

kl
.
(3.15)
Remark that in (3.13) and (3.14), and hence in (3.15), the o(1) is uniform in all l-tuples (k j1 ,..., k jl )
and k satisfying the condition (3.5). The relations (3.7), (3.8) and (3.15) imply
ERNk  ETNk 1  o 1  ,
(3.16)
for every integer k satisfying the condition (3.5). Main Lemma follows from (3.16) and Assertion 3.
Remark 3.1. The condition (3.3) is used to estimate of the integral J 2 only. Let the condition
(3.3) is fulfilled with a1 , b1 for all m 
 (1, 2,..., N ) and with a2 , b2 for all m  (1, 2,..., N ) \
,
then Eq. (3.6) is valid for every integer k such that 3  k  o  min( Kn (a1 , b1 ), Kn (a2 , b2 )  instead of
(3.5).
Let  x  stand for the integer part of x , 
Poi( ) and v ( )  E     .
v
Lemma 3.2. For any integer v  2 one has
v /2
v ( )  v !  cl ,v  l ,
(3.17)
l 1
where for all of l  1, 2,..., v / 2 ,
0  cl ,v  1/ l ! ,
also
(3.18)
7
(v  1)cl ,v 1  lcl ,v  cl 1,v1 ,
(3.19)
where l  1, 2,..., (v  1) / 2 if v is an even, and l  1, 2,..., (v  1) / 2 1 ,
(v  1)c( v1)/2,v1  c( v1)/2,v1 if v is an odd; here we put c0,v 1  0 .


Proof. Apply the Bruno’s formula to Eei (  )  exp   ei  1  i  to get
 
1

v
!
cl ,v  l ,

km
l 1
m  2 km !( m !)
v2
v
v ( )  v !  k ... k 
2
where

is summation over all non-negative k2 ,..., kv such that 2k2  ...  vkv  v and
v
1
, and hence 0  cl ,v  1/ l ! . On the other hand Kenney and
km
m 2 km !( m !)
l  k2  ...  kv , cl ,v  
Keeping (1953) give
v 1 ( )  vv 1 ( )  
Just to keep notation simple we put
Let
is even, then (v 1) / 2
d
 v ( ) .
d
v/2
(3.20)
and cˆi ,v  v !cl ,v . Note that cˆ1,v
and (v 1) / 2
(v
1.
2) / 2

 1


l 1
l 1
l 1
l 2
1 for any integer
0.
Therefore (3.17) and (3.20) gives
 cˆl ,v1 l   vcˆl ,v1 l 1   lcˆl ,v l    vcˆl 1,v1  lcˆl ,v   l   .
For the even v the property (3.19) follows. Let now v is odd, then (v 1) / 2
1.
(v 1) / 2
and
From (3.17) and (3.20) obtain
 1


 1

l 1
l 1
l 1
l 2
l 1
 cˆl ,v1 l   vcˆl ,v1 l 1   lcˆl ,v l   vcˆl 1,v1 l   lcˆl ,v  l

   vcˆl 1,v 1  lcˆl ,v   l  vcˆ ,v 1  1   .
l 2
This proves (3.19) for the odd v . Lemma 3.2 is proved completely.
Using (3.15) we obtain v ( )  
d
d
v ( )  v / 2 v ( ) . This together with (3.20) imply
v ( )  v1 ( )    v / 2  v ( ) .
(3.21)
Use formula (3.20) for v  2 ( ) , next again apply (3.20) to the derivative of v 1 ( ) , after this use
the inequalities
2
d2
d 2
v
( )
v/2
d
d
v 1
2
v
( )
( ).
v
( ) (because lcˆl ,v
1
cˆl ,v ( see (3.19)), 
d
d
v ( )  v / 2 v ( ) and
Then one can observe that for any integer   0
(  1)v ( )  v2 ( )  2v(  v)v ( ) .
(3.22)
In the sequel some notations regarding to the general statistic RN ( see (3.1)) are useful :
8
N
N
m 1
m 1
AN   Ef m ( m ) ,  n  n1  cov  f m ( m ),  m  ,
N
 N2  Varf m ( m ) ,  N2   N2  n n2 .
(3.23)
m 1
Note that under the conditions (3.2) and (3.3)
ERN  AN  o( N ) and VarRN   N2 (1  o(1)) .
2
N
We notice that formula for the
and
2
N
(3.24)
given in (2.1) are the special case of (3.23)
corresponding to the chi-square statistic; that is why the same symbols are used.
Proof of Theorem 2.1. The chi-square statistic is (3.1) kind with f m ( x)   x  npm  / npm , hence
2
AN  N ,  N2 and  N2 as in (2.1). Set m  m  npm  / npm . We have Em2 s (m  npm )2
 4s(npm  2s) Em2 s , since (3.22), hence the conditions (3.2) and (3.3) are fulfilled with b  1 and
a  1 if npm  2s , and with b  0 and a  2 if npm  2s . It is evident that the condition (3.4) is
1
fulfilled for the k  o( pmax
) . Therefore, from Main Lemma and Remark 3.1 it follows that
 N 2
 1  o 1  ,
k   m 
 m1 
(  N2 ) 
k

(3.25)

1/2
for every integer k such that 3  k  o min  pmax
, n1/3  .
We would remind the notation
see that (2k )! 22k e /
2k (k !)2 .
n
1
max 1,(npmin )
Due to this, putting
m
. Since 2 m mme
(
2
m
E
2
m

k 1
l 0
k 1
 (np
m
2
m
) / Var
obtain E mk  2k (2k )!(Varm2 )  k /2  (npm ) l  (k !) 2 27 n / Varm2

m
k 2
m!
e m mm e
m
one can
and using Lemma 3.2 we
for k  3 , since
2
1
)  l  k  nk  2 and Varm  (npm )  2 . Hence, by Assertion1
l 0
k  m    k !
2

28 n / Varm2

k 2
. That is
k
    k !  2  
2
2
m
n
 N 2
2
28 n
k    m    k !
 m1 

8

k 2
k 2
Varm2 , for k  3 . Therefore
 N2 ,
(3.26)
N
because  m , m=1,2,… are independent r.v.s, and  N2  Var m2 , see notation (2.1). Now (3.25),
m 1
(3.26) and the fact that  N2  Var  N2 1  o(1)  , see (3.23), imply
k

2
N
/ Var 
2
N

 N
 k    m2 / Var  N2
 m1
13
2

2  2  n N 

k
!





3

 N 
k 2
,
9


1/2
for all k satisfying 3  k  o min  pmax
, n1/3  . Theorem 2.1 follows from this and Assertion 2 with
1/2
  213 n N2 /  N3 and k  min  pmax
, n1/3  .
Proof of Corollary 2.1. Under conditions of Corollary 2.1 we have  N2 1  (2c0 )1    N2   N2
1/ 4
and n  max(1, 2 / c1 ) , also pmax
 c21/ 4 N 1/ 6 . Corollary 2.1 follows.
Proof of Corollary 2.2 is straightforward since in this case n  max(1, n1 ) ,
 N2  N  2  d 2 2 (n)n1 (1  o 1  and  N2  2 N 1  (2n )1 (1  o(1))   2 N n .
Proof of Theorem 2.2. Observe that the  N is RN kind statistic with f m ( x)  x ln  x / npm  (we
assume that 0  ln 0  0 ). Write

 

f m (m )  npm 1  ˆm ln 1  ˆm ,
(3.27)
N
where ˆm  m  npm  / npm  1 . Put gm ( x)  f m ( x)   n ( x  npm ) . It is evident that RN   g m (m ) ,
m 1
hence it is sufficient to check the conditions of Main Lemma for gm (m ) . We shall use the
inequality

1
1
x 2l 1 
x 2l  2

0  (1  x) ln(1  x)   x  x 2  x3  ... 
,

2
6
2l (2l  1)  (2l  1)

(3.28)
x  1 and integer l  0 , to find, see denotes (3.23),
AN 
 1  N 
 1 
N 1 N 1 
 
1  O 
   1  O    ,
2 12 m1 npm 
 npm   2 
 n  
 n  1
(3.29)
 1 
1 N 1 
2
1  O 
   1  O  n  ,

12n m1 npm 
 npm  
N
 N2  Var g m (m ) 
m 1
(3.30)
 1  N 
 1 
N 1 N 1 
 
1  O 
   1  O    .
2 6 m1 npm 
 npm   2 
 n  
(3.31)
From (3.28) we have x  (1  x) ln(1  x)  x  x 2 / 2 , hence

c
n


 npm (1   n )ˆm  gm (m )  npm (1   n )ˆm  ˆm2 ,
(3.32)
since ˆm  1 , npm  c1n and 1   n  (c0n2 )1 by (3.30). From Lemma 3.2 we obtain
Eˆm  O  !(npm )
/2
 for any integer
3.2, whereas for an odd
 1 (to see this for an even

 1 we can directly use Lemma
 1 we first use the inequality E ˆm  Eˆm1

/( 1)
and then apply Lemma
10
3.2). Therefore, it is not hard to understand that the conditions (3.2) and (3.3) with a  1 and b  1
1/ 2
are satisfied. Thus, Main Lemma gives: for all integer k such that 3  k  o  pmax

 N

g (m )  1  o 1  .
k m
 m1

( N ) 
k
On the other hand using (3.32), inequality (2k )! 22k e /

(3.33)
2k (k !)2 and
the condition (2.4) we obtain

k

k !(npm )k / 2 
k
k
2
11 k  2
E gm (m )  2k  npm  (1   n )k E ˆm  Eˆm2 k  2k 1  (2k )!
  (k !)  2  .
2k
c0n


Therefore, Assertion 1 imply
k
 gm (m )   (k !)2  212 
k 2
, because Var gm (m )  21 (1  o(1)) .
Hence
 1 N


g (m )   (k !) 2 211 N 1/ 2
k N  m
m 1




k 2
,
since (3.29) and (3.31). Thus from (3.33) taking into account that
k

( N1 N )  (k !)2 212 N 1/ 2

k 2
N
2
N
Vargm ( m ) we obtain
m 1
,
1/2
for all integer k such that 3  k  o  pmax
 . Noting that Var N   N2 1  o(1)  , see (3.24), Theorem
1/2
2.2 follows from Assertion 2 with k  o  pmax
 and   212 N 1/2 .
Proof of Corollary 2.3 is straightforward.
Appendix. Below  is a r.v. with E  0 , Var  1 ,
k
( ) and  k ( ) cumulant and moment
of k th order respectively of the r.v.  . The following conditions and assertions are taken from the
book by Saulis and Statulevicius (1991).
Bernstein condition ( B ) : there exists the constants   0 and B  0 such that
 k ( )  (k !) 1 Bk 2 , for all k =3,4,….
Statulevicius condition ( S ) : there exists the constants   0 and   0 such that
k
( )  (k !) 1 ( k 2) , for all k =3,4,….
Assertion 1. If  satisfy condition ( B ) then it also satisfy condition ( S ) with    2B  .
1
Assertion 2. Let a r.v.  depending on a parameter  satisfy condition ( S1 ) (i.e. v  1 ) for all k
such that 3  k  k . Then for all x such that 0  x  121 min


k , 1/ 3 there exist a constant c such
that P   x   1  ( x)  1  C ( x  1) 1/3  and P    x   ( x) 1  C ( x  1) 1/3  .
In the book of Saulis and Statulevicius (1991) no assertion in the form of Assertion 2 is given.
However, Assertion 2 can be proved by reasoning alike to those presented by Saulis and
11
Statulevicius (1991) at page 35, in course of proof of Lemma 2.3, see relations (2.62), (2.65) and
(2.66) and Lemma 2.2 of the book.
Assertion 3. One has
( )  k !   1
k
here

m1  m2 ... mk 1
k
 m1  m2  ...  mk  1!
l 1
1  l ( ) 


ml !  l ! 
ml
is summation over all non-negative integer m1 , m2 ,..., mk such that m1  2m2  ...  kmk  k
.
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3. Ivchenko G.I. and Mirakhmedov S.,(1995). Large deviations and intermediate efficiency of the
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4. Mann H.B. and Wald A. (1942). On the choice of the number of intervals in the application of
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