Nonparametric Statistics, 2003, Vol. 15(1), pp. 1–10

DISPARITY BASED GOODNESS-OF-FIT TESTS FOR
AND AGAINST ORDER RESTRICTIONS FOR
MULTINOMIAL MODELS
B. BHATTACHARYAa,* and A. BASUba

Department of Mathematics, Southern Illinois University, Carbondale, IL, USA;b
Applied Statistics Unit, Indian Statistical Institute, Calcutta, India(Received January 2001; Revised June 2002; In final form July 2002)

We consider disparity based test statistics to test the equality of a multinomial probability vector to a given
probability vector against an isotonic=antitonic order restriction. The problem of testing the isotonic restriction asa null hypothesis against unrestricted alternatives is also considered. In both cases the asymptotic distributions ofthese test statistics are found to be of the chi-bar squared type similar to the popular likelihood ratio test statistics
for these cases. However our numerical studies demonstrate that depending on the situation, several of thedisparity test statistics are more powerful than the latter.
Keywords: Blended weight chi-square distance; Blended weight Hellinger distance; Hellinger distance;
Isotonic=antitonic order; Likelihood ratio test; Minimum disparity estimation; Power divergence

1

INTRODUCTION

Test statistics such as Pearson’s chi-square and likelihood ratio are two of the most popularmeans of testing
P equality of a multinomial probability vector (PV) ð p ¼ ð p1 ; p2 ; . . . ; pk Þwith pi  0; ki¼1 pi ¼ 1) to a given PV. There are, however, less known test statistics
available, such as Neyman modified chi-square, Freeman-Tukey, and modified log likelihood
ratio for the same testing scenario. In an attempt to unify these statistics, Cressie and Read
(1984) (also see Read and Cressie, 1988) introduced the family of power divergence test
statistics. For two PV’s p and q ¼ ðq1 ; q2 ; . . . ; qk Þ, the power divergence family of test
statistics is denoted by f2nI l ð p; qÞ; l 2 Rg, where n is the sample size andI l ð p; qÞ ¼

"
k
X
i¼1

pi
lðl þ 1Þ

( 
)
#
pi l
qi  pi
1 þ
qi
lþ1

ð1Þ

with the cases of l ¼ 0; 1 defined as the continuous limits at those values of l. It can beeasily seen that the statistics Neyman modified chi-square, discriminant information,
Freeman-Tukey, log likelihood ratio and the Pearson’s chi-square are special cases of (1)* Corresponding author.

ISSN 1048-5252 print; ISSN 1029-0311 online # 2003 Taylor & Francis Ltd
DOI: 10.1080=1048525021000049658

2

B. BHATTACHARYA AND A. BASU

with l ¼ 2; 1; 1=2; 0; 1 values respectively. Cressie and Read studied the differences inbehavior of (1) asymptotically and for finite sample sizes for different l values. They alsosuggested using the statistic based on l ¼ 2=3 as a competitor to the Pearson’s chi-squareand log likelihood ratio statistics.
For a finite set X ¼ fx1 ; . . . ; xk g, a binary relation  on X is a simple order if it is reflexiveðx  x; 8x 2 X Þ, transitive (x  y; y  z imply x  z; 8x; y; z 2 X Þ, antisymmetric(x  y; y  x imply x ¼ y; 8x; y 2 X ) and every two elements of X are comparable(8x; y 2 X either x  y or y  x). A binary relation  on X is a partial order if it is reflexive,transitive and antisymmetric. A quasi order is reflexive and transitive only. Thus every simple
order is a partial order and every partial order is a quasi order.
A real-valued function f on X is isotonic with respect to the quasi-ordering  on X if forx1 ; x2 2 X with x1  x2 imply f ðx1 Þ  f ðx2 Þ. Instead, if the function f satisfies f ðx1 Þ  f ðx2 Þ,then f is said to be antitonic on X . For example, f is isotonic on X with respect to the simple
order (nondecreasing) x1  x2      xk if f ðx1 Þ  f ðx2 Þ      f ðxk Þ. Consider the partial
order restriction  on X defined by x1  xi ; 8i ¼ 2; . . . ; k, known as simple tree. A function
f is isotonic on X with respect to the simple tree restrictions if f ðx1 Þ  f ðxi Þ; 8i ¼ 2; . . . ; k.
Suppose g is a given function on X and w is a given positive function on X . A function g
on X is anPisotonic regression of g with weights w if and only if g  is isotonic and g
minimizes x2X ½gðxÞ  f ðxÞ2 wðxÞ in the class of all isotonic functions f on X .
In this paper we assume that a random sample is available from a multinomial distribution
with PV p ¼ ðp1 ; p2 ; . . . ; pk Þ and consider the hypotheses H0 : p ¼ p0 for a specified
p0 ¼ ðp01 ; p02 ; . . . ; p0k Þ; H1 : p is isotonic and H2 : p is unrestricted. Statistical inference
under isotonic cone restrictions and related topics are discussed in Robertson, Wright and
Dykstra (1988). Assuming p0 is isotonic, we consider testing H0 vs. H1  H0 and also
H1 vs. H2  H1 using disparity based test statistics, and study the asymptotic distributionsof the test statistics when the null hypotheses are true. Robertson (1978) considered thelikelihood ratio tests for the same hypotheses.
Dykstra and Lee (1991) found a general solution to minimizing (1) subject to q belonging
to an isotonic cone and p being the vector of observed relative frequencies of the multinomial. The solution, based on l, is expressed in terms of convex projections onto the isotonic
cone. Bhattacharya (1997) considered a more general form of the isotonic cone restrictions,obtained the estimates and performed hypothesis tests under these restrictions. In this paper,
based on the disparity approach of Lindsay (1994), we consider general classes of test
statistics such as those based on the power divergence, blended weight Hellinger distance,and blended weight chi-square families, all of which are subclasses of disparity tests. Basuand Sarkar (1994) considered testing H0 vs. H2  H0 using such disparity test statistics.
Although in existing order-restricted testing literature the likelihood ratio test is the mostcommon choice as a test statistic, many of the other disparity test statistics used in thispaper will be shown to have better power under certain alternatives. Cressie and Read
(1984), Read and Cressie (1988), and Basu and Sarkar (1994) demonstrate similar cases
when the alternative is unrestricted.
In Section 2, we describe the test statistics used in this paper. In Section 3, we obtain theasymptotic distributions of the test statistics for the two testing situations considered. In
Section 4, we present several numerical investigations, and show that with appropriately
defined simple tree order restrictions as an alternative, the test statistics developed here produce higher power at the ‘dip’ and ‘bump’ alternatives compared to the statistics developedfor unrestricted alternatives by Cressie and Read (1984), Read and Cressie (1988) and Basuand Sarkar (1994). Since under the isotonic restrictions, theoretical asymptotic analysis of
the moments of the test statistics appear to be intractable, we compare the moments of thestatistics through simulations for the simple order and the simple tree order to examine

DISPARITY TESTING

3

their convergence to the asymptotic limit. Such moment comparisons under the orderrestrictions does not exist in the current literature.

2

DISPARITY TEST STATISTICS

Suppose n observations are available from a multinomial distribution with PV p; and let
x ¼ ðx1 ; . . . ; xk Þ denote the vector of observed frequencies for the k categories. Let G be astrictly convex thrice differentiable function on ½1; 1Þ with Gð0Þ ¼ 0, Gð1Þ ð0Þ ¼ 0 andGð2Þ ð0Þ ¼ 1, where GðiÞ represents the ith derivative of G. For the rest of this paper wewill assume that Gð3Þ is bounded and continuous at 0.
A disparity between two PVs p and q generated by G is defined by


k
X
pi
rG ð p; qÞ ¼
G
 1 qi :ð2Þ
qi
i¼1
Let p ¼ ð p1 ; p2 ; . . . ; pk Þ where each pi ¼ xi =n for 1  i  k. The disparity test statistic for
testing H0 against H1  H0 generated by G is given by 2nrG ð p ; p0 Þ, wherep ¼ ð p1 ; p2 ; . . . ; pk Þ is the isotonic regression of p with equal weights (Robertson et al.,1988) under H1. Appropriate algorithms are available to compute the isotonic regression
estimates for well known restrictions; for example, when H1 is the simple order, the pool adjacent violators algorithm ( page 10 of Robertson et al., 1988) may be used, and when H1
denotes the simple tree order, the algorithm given on page 19 of Robertson et al. (1988)
may be used. These algorithms are used in Section 4 for computing appropriate estimates.
1 2
Letting di ¼ ðp1
0i pi  1Þ, the Pearson chi-square statistic is generated by GðdÞ ¼ 2 d .
The log likelihood ratio chi-square and the power divergence family are generated byGðdÞ ¼ ðd þ 1Þ logðd þ 1Þ  dandGl ðdÞ ¼

ðd þ 1Þðlþ1Þ  ðd þ 1Þd;lðl þ 1Þlþ1respectively. The (twice) squared Hellinger distance (the Freeman-Tukey divergence)corresponds to GðdÞ ¼ 2½ðd þ 1Þ1=2  1. Lindsay (1994) introduced the blended weight
Hellinger and the blended weight chi-square distance families. Here we modify those to
the restricted alternative case. The blended weight Hellinger distance family
fBWHDa ; 0  a  1g defined by()2kX
pi  p0i1ð3ÞBWHDa ð p ; p0 Þ ¼ 2
 1=2 þ ð1  aÞðp Þ1=2
0i
i¼1 að pi Þcorresponds toGa ðdÞ ¼ 2

1



d
½aðd þ 1Þ1=2 þ ð1  aÞ2:Note that the (twice) squared Hellinger distance is a member of fBWHDa g with a ¼ 1=2.
Another family of disparities is the blended weight chi-square fBWCSa ; 0  a  1g obtainedby taking a weighted average of the denominators of the Pearson’s and Neyman’s chi-squares.
Its form is
BWCSa ð p ; p0 Þ ¼ 21kX

ð pi  p0i Þ2
api þ ð1  aÞp0i
i¼1

ð4Þ

4

B. BHATTACHARYA AND A. BASU

and for this family of disparity test statistics
Ga ðdÞ ¼

21 d2:ðad þ 1ÞIn the unrestricted case, Lindsay (1994) used the families (3) and (4) for estimation purposes
and Basu and Sarkar (1994) used those for goodness-of-fit tests.
Here the disparities rG are presented in a standardized form so that the corresponding GðÞ
functions have the appropriate properties without changing the disparity statistics themselves.
Notice that we have represented the I l measure in a slightly different but equivalent way
ð1Þ
compared to Cressie and Read (1984). Our standardizations guarantee that GP
ð0Þ ¼ 0k2
ð2Þ
1
and G ð0Þ ¼ 1, and thus the leading term of any disparity rG ð p; pÞ equals 2
i¼1 di pi ,1where di ¼ pi pi  1, when expanded in a Taylor series in di around 0. As a result, theleading term of any disparity test statistic equals the Pearson’s chi-square statistic. As wewill see, this helps to establish the asymptotic distribution of the disparity test statistics inthe next section.

3

HYPOTHESIS TESTSWhen testing the hypothesis
H0 against the alternative H1  H0 , the test statistic to be used isP
2nrG ð p ; p0 Þ ¼ 2n ki¼1 Gð pi =p0i  1Þp0i . Using a Taylor series expansion (as a function ofpi around p0i ), it follows that
 

k
X
p
rG ð p  ; p 0 Þ ¼
G i  1 p0i
p0i
i¼1
¼

k
X

Gð0Þp0i þ

i¼1

þ

k
X

k
k
X
X
ð pi  p0i ÞGð1Þ ð0Þ þ
21 ð pi  p0i Þ2 Gð2Þ ð0Þp1
0i
i¼1

i¼1

2
61 ð pi  p0i Þ3 Gð3Þ ðp1
0i xi  1Þp0i

i¼1¼ S1 þ S 2 þ S3 þ S4
say, where pi  xi  p0i . Since Gð0Þ ¼ 0 and as both pi and p0i sum to 1 over i, the first two
terms S1 and S2 are equal to 0. Also
6nS4 ¼kX

2
nð pi  p0i Þ3 ½Gð3Þ ðp1
0i xi  1Þp0i 

i¼1

(


k
X
i¼1

nð pi  p0i Þ2)

sup j pi  p0i jisup p2
0i
i




x

1Þ
;
sup Gð3Þ ðp1
0i i
i


where fsupi p2
p is consistent for p0, and p, sup
0i g is bounded, supi j pi 
Ppk 0i j ¼ op ð1Þ (since2
are continuous
i¼1 nð pi  p0i Þ ¼ Op ð1Þ (since it is a continuous
Pk functions of 2p) andfunction of
nð
p

p
Þ
which
is
bounded) under H0. Since ðxi  p0i Þ ¼ op ð1Þ fori
0i
i¼1
every i, it follows that Gð3Þ ðp1x
1Þ¼
Op ð1Þ by the assumptions on Gð3Þ . Therefore,i0i
ð2Þ
6nS4 ¼ op ð1Þ. Since G ð0Þ ¼ 1 the result follows by noting that

2nS3 ¼ n

k
X
i¼1

2

p1
0i ð pi  p0i Þ

DISPARITY TESTING

5

is the Pearson chi-square statistic whose asymptotic chi-bar square distribution under thesimple null hypothesis is well known (Robertson et al., 1988).
When testing H1 vs. H2  H1 , the test statistic is given by


k
X
pi

2nrG ð p; p Þ ¼ 2n
G   1 pi :pi
i¼1
In this case, it may be shown that H0 : p0i ¼ 1=k, 8i is least favorable within H1 (Robertson,1978). By a Taylor series expansion P
similar to above, its asymptotic distribution may beshown to be the same as that of n ki¼1 ð pi  pi Þ2 =pi whose asymptotic chi-bar square
distribution is also well known (Robertson et al., 1988).
Consider the quasi-order restriction, p , induced by p and  on f1; 2; . . . ; kg which
requires that i p j only when i  j and pi ¼ pj . Let I p be the set of functions on
f1; 2; . . . ; kg isotonic with respect to p . Let Z ¼ ðZ1 ; . . . ; Zk Þ where Z1 ; . . . ; Zk are independent standard normal variables, and for i ¼ 1; . . . ; k, let Pp ði; kÞ denote the probability
that PðZ j I p Þ, the equal weights least square projection of Z onto I p , takes on exactly i
distinct values. This probability is well known as the equal weight level probabilities forthe given isotonic order. For p ¼ ð1=k; . . . ; 1=kÞ, this probability is denoted by Pði; kÞ.
The following theorem is obtained from above discussions.
THEOREM 1 For a constant c1 ; when testing H0 against H1  H0 the asymptotic distribution of the test statistic 2nrð p ; p0 Þ under H0 is given bylim Pð2nrð p ; p0 Þ  c1 Þ ¼

n!1

k
X

Pði; kÞPðw2i1  c1 Þ

i¼1

where w2i is a chi-square random variable with i degrees of freedom with w20  0.
When testing H1 as a null hypothesis against the alternative H2  H1 , for a constant c2
and p 2 H1,kXlim Pp ð2nrð p; p Þ  c2 Þ ¼
Pp ði; kÞPðw2ki  c2 Þ
n!1

i¼1

where the subscript p denotes that the probability is computed with p as the parameter value.
Moreover, in this case H0 : p0i ¼ 1=k, 8i is asymptotically least favorable within H1 , that is,lim Pp ð2nrð p; p Þ  c2 Þ 

n!1

k
X

Pði; kÞPðw2ki  c2 Þ:

i¼1

Clearly, the same test statistics are used in presence of antitonic order restrictions withappropriately modified estimates. In the next section we consider the simple order and thesimple tree order for which the level probabilities are available from Robertson et al.
(1988) upto k  20.

4

NUMERICAL RESULTS

We begin by describing the performance of several disparity tests with some exact computations. To keep a clear focus we concentrate mostly on the multinomial case with n ¼ 20 and
k ¼ 4 and a test size of 0.05 for these exact computations; however, to demonstrate greater

6

B. BHATTACHARYA AND A. BASU

applicability of the proposed methods, we also consider a few other combinations of n and k
in the power divergence case. Several values of l are chosen for the power divergence case,and several choices of a are used for the BWHDa and BWCSa families. For a given value of k,
we consider the symmetric null hypothesis H0 : p0i ¼ 1=k for all i against appropriate
alternative hypotheses of simple tree order H1 : p1  pi, 8i ¼ 2; . . . ; k or the antitonic
order H10 : p1  pi, 8i ¼ 2; . . . ; k. Consider the PV p with p1 ¼ ð1 þ gÞ=k,pi ¼ ðð1  g=ðk  1ÞÞ=k, i ¼ 2; . . . ; k indexed by a single parameter g, 1  g  k  1,
and notice that negative values of g lead to ‘dip’ alternatives belonging to the situation
described by H1, and positive values of g produce ‘bump’ alternatives in H10 . We have computed exact powers for g ¼ 1:5 and 0:9 as in Cressie and Read (1984) and Basu and Sarkar(1994). For a given disparity r, we first consider three test statistics T1 ¼ 2nrð p; p0 Þ,
T2 ¼ 2nrð p ; p0 Þ, T3 ¼ 2nrð p ; p0 Þ where T1 is the conventional test statistic of Cressie
and Read, and Basu and Sarkar, originally developed to test H0 against H2  H0 , while T2
and T3 are our proposed statistics for isotonic and antitonic tree orders respectively (e.g.
for testing H0 against H1  H0 and H10  H0 respectively), and p and p are the isotonic
and antitonic regression of p with equal weights. We have computed exact powers forg ¼ 1:5 and 0:9 as in Cressie and Read (1984) and Basu and Sarkar (1994). The powersof the test statistics are presented in Tables I–V. Tables I, IV, V deal with the k ¼ 4,n ¼ 20 case and present the results for the power divergence, the BWHD and the BWCS
families respectively under this scenario. Tables II and III represent two different combinations of k and n under which the power divergence statistic is also studied; they help
demonstrate the uniformity in the behavior of the test statistics in these scenarios as wellcompared to the k ¼ 4, n ¼ 20 case. In general, Tables I–V illustrate the following:
For each given disparity, the power values of T2 are higher than those of T1 for all thedisparity tests when g ¼ 0:9 (i.e. for the dip alternative); this is expected, since T2
specifically utilizes the information that the alternative belongs to H1 , while T1 simply
states that the null is false.
For the bump alternative ðg ¼ 1:5Þ, T3 has higher power than T1 for all the disparity tests,
which is again expected.
The increase in power in T2 over T1 for the dip alternative is generally higher compared tothe increase in power in T3 over T1 for the bump alternative. This is because we used an
TABLE I Power of the T1, T2 and the T3 Test Statistics for the Cressie–Read Family for then ¼ 20, k ¼ 4 Case, Obtained via Exact Computations (Rounded to Four Places of Decimals). The
Size of the Test is 0.05.
T2

T1

T3

l

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

5.00
2.00
1.00
0.50
0.30
0.00
0.30
0.50
2=3
0.70
1.00
2.00
5.00

0.63160.65000.79600.80090.85250.86400.86400.86400.86400.86470.87450.89620.90250.74340.74340.73420.72630.71080.70450.70450.70450.70450.63630.51500.32900.24220.00000.00000.00000.00000.00000.00000.00000.00000.00010.00010.00010.00020.00020.93570.93260.92480.92480.90950.89050.80170.80170.76200.76200.74340.4791
0.46300.63660.65500.80720.87490.89850.91320.93570.93750.93750.93750.93930.95100.96710.00220.00220.00210.00180.00170.00160.00150.00150.00150.00150.00150.00030.0001DISPARITY TESTING

7

TABLE II Power of the T1, T2 and the T3 Test Statistics for the Cressie–Read Family for then ¼ 25, k ¼ 4 Case, Obtained via Exact Computations (Rounded to Four Places of Decimals). The
Size of the Test is 0.05.
T2

T1

T3

l

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

5.00
2.00
1.00
0.50
0.30
0.00
0.30
0.50
2=3
0.70
1.00
2.00
5.00

0.78180.83280.88890.89150.91440.92150.93050.93100.93430.93430.94070.95620.9566

0.89630.89610.89430.89430.89130.87910.82230.8016
0.77620.77620.73320.42780.38230.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.97980.97930.96350.9524
0.94340.93040.91510.91510.89950.87350.84130.70320.48090.79880.87510.93190.94980.94980.95700.96890.96890.96990.96990.9701
0.9826
0.98460.00230.00230.00230.00230.00230.00220.00220.00220.00220.00220.00200.00030.0001

extreme dip alternative (g ¼ 0:9 is near the end of the range) but a moderate bumpalternative. More extreme bump alternatives like those generated by g ¼ 2 or 2.5 would
lead to greater increases in power for T3 over T1 for such alternatives.
The powers are decreasing functions of l within the power divergence family, andincreasing functions of a for the BWCS and BWHD families for the dip alternatives, but thereverse happens for bump alternatives.
For bump alternatives, the powers of T2 are lower than those of T1 (in fact lower than thesize of the test), since here the truth is further away from the alternative compared to thenull; similarly, powers of T3 are smaller than those of T1 under dip alternatives.
Thus in the scenarios considered in this limited study it is observed that it may be preferable to use disparity tests of the form T2 with small values of l (in the Cressie–Read family)or large values of a (within the BWHD or BWCS family) against suspected isotonic alternatives. Similarly disparity tests of the form T3 may be preferable with large values of l (in the
Cressie–Read family) or small values of a (within the BWHD or BWCS family) against
TABLE III Power of the T1, T2 and the T3 Test Statistics for the Cressie–Read Family for then ¼ 20, k ¼ 5 Case, Obtained via Exact Computations (Rounded to Four Places of Decimals). The
Size of the Test is 0.05.
T2

T1

T3

l

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

5.00
2.00
1.00
0.50
0.30
0
0.30
0.50
2=3
0.70
1.00
2.00
5.00

0.22530.22530.22530.33610.44680.61000.66050.68060.69070.69070.66970.73060.74980.58800.58800.58800.58750.56930.4466
0.37600.32160.28510.2851
0.27200.18960.14640.00000.00000.00010.00010.00030.00030.00030.00030.00030.00040.00040.00140.00140.83600.80690.77660.75300.75020.7502
0.66330.53600.4954
0.4936
0.40890.27720.24520.40740.41880.48770.56430.61570.69560.75020.78510.81260.81260.81730.85300.88030.01450.01450.01450.01450.01450.00510.00390.00220.00200.00200.00200.00110.0004

8

B. BHATTACHARYA AND A. BASUTABLE IV Power of the T1, T2 and the T3 Test Statistics for the BWHD Family for the n ¼ 20,
k ¼ 4 Case, Obtained via Exact Computations (Rounded to Four Places of Decimals). The Size of
the Test is 0.05.
T2

T1

T3

a

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00

0.87450.86400.86400.86400.85250.80090.79600.73530.70170.65000.65000.51500.70450.70450.70450.70480.72630.73410.74100.74280.74340.74340.00010.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.74340.76200.80170.89050.90940.92480.92480.92480.92880.92880.9326

0.93930.93750.93750.91320.89850.87490.86290.80130.74030.70670.65500.00150.00150.00150.00160.00170.00180.00180.00210.00210.00220.0022suspected antitonic alternatives. A general recommendation, of course, will require more
extensive investigation and we do not wish to claim the results to be any more generalthan can be suggested by our limited numerical findings. However, since the test statistic
T2 ðT3 Þ explicity utilizes the information about isotonic (antitonic) ordering, we expect
T2 ðT3 Þ to perform better than T1 when the isotonic (antitonic) ordering is indeed true.
We next perform a simple comparison of the convergence of the statistics to their asymptotic chi-bar square limits through the speed of convergence of the moments of the disparity
tests under the null hypothesis H0 : p ¼ p0 as k goes to infinity. For simplicity we restrict
ourselves to the Cressie–Read family in this case,palthoughsimilar analysis can be easily
ﬃﬃﬃ
done with the BWHD and BWCS. Defining wi ¼ nð pi  p0i Þ we have the Taylor series
expansion
2nI l ð p ; p0 Þ ¼

k
k
l 1X
w3i ðl  1Þðl  2Þ X
w4i
 pﬃﬃﬃþþ Op ðn3=2 Þ
2
3
3
n
12n
p
p
p
0i
i¼1
i¼1 0ii¼1 0i

k
X
w2
i

ð5Þ

under H0. Since it appears to be intractable to obtain the moments of 2nI l ð p ; p0 Þ using (5),we compare the simulated moments of 2nI l ð p ; p0 Þ for different values of l with the
TABLE V Power of the T1, T2 and the T3 Test Statistics for the BWCS Family for the n ¼ 20,
k ¼ 4 Case, Obtained via Exact Computations (Rounded to Four Places of Decimals). The Size of
the Test is 0.05.
T2

T1

T3

a

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

g ¼ 1.5g ¼ 0.9

0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00

0.87450.86470.86400.86400.85260.85260.80090.79600.73530.70170.65000.51500.63630.70450.70450.70490.70490.72630.73410.74100.74280.74340.00010.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.74340.76200.80170.85720.89050.92480.92480.92480.92480.92880.9326

0.93930.93750.93750.9276
0.91320.89850.86290.80920.80130.70670.65500.00150.00150.00150.00150.00160.00170.00180.00190.00210.00220.0022

2.8331.07(0.0464)15.5000.95(0.4505)

2

3

0.833
1.22
(0.0051)

k¼3

1

Moments

6.804
0.79(0.0569)
44.1050.67(1.0449)

1.593
1.00
(0.0032)

k¼7

8.958
0.73
(0.1094)61.8630.63(1.8042)

1.929
0.93
(0.0052)

k ¼ 10

Simple order

13.9450.68(0.1511)107.6190.59(2.6966)

2.598
0.87
(0.0079)

k ¼ 204.167
6.37(0.0013)23.5005.01(0.0062)

1.167
8.80
(0.0002)

k¼3

29.0490.74(0.0040)255.1130.64(0.0248)

4.376
0.83
(0.0001)k¼764.6420.46(0.0049)734.991
0.39(0.1168)

7.008
0.56
(0.0001)

k ¼ 10

Simple tree order298.4850.41(0.0452)
6120.870
0.39(0.6352)

16.2360.44(0.0012)k ¼ 20TABLE VI First Row is the Exact Moment of the Chi-bar Squared Distribution, Second Row is the Value of Lambda at Which the Corresponding
Simulated Moment is Closest to the Exact Moment and the Third Row is |Exact Moment 7 Simulated Moment| in Parentheses, for the First Three
Moments for the Simple Order and the Simple Tree Order Alternatives.

DISPARITY TESTING
910

B. BHATTACHARYA AND A. BASU

moments of (its limiting asymptotic) chi-bar square distribution. We consider the first threemoments for the restrictions of the simple order and the simple tree order with 100,000 replications and a sample size of 500. We consider k ¼ 3, 7, 10, 20. The results are expected to bebetter for larger k. The rth moments of the chi-bar square distribution for the simple order andthe simple tree order are given bykXPði; kÞði  1Þði þ 1Þ    ði þ 2r  3Þi¼1

where the level probabilities Pði; kÞ are appropriately defined (Robertson et al., 1988).
In Table VI, we provide the exact moments from chi-bar squared distribution using theabove formula. The l values, correct up to second decimal places, for which the simulated
moments of the Cressie–Read family statistics is closest to the corresponding moment isprovided along with the final absolute difference in parenthesis. For each k, the minimizing
value of l seems to decrease for higher moments for both orders. Also the magnitude of theabsolute difference increases slightly with higher moments. It is seen that we do not get clear
choices of l ¼ 1 or l ¼ 2=3 as in Cressie and Read (1984), but they do converge tosomewhere in or around the interval ð1=3; 2=3Þ. In the case of tree order, the case of
k ¼ 3 produces values of l much larger than 1, which is unexpected. On the whole, however,
we expect the convergence of l for the Cressie–Read family statistics to be reasonable in theinterval ð1=3; 2=3Þ for large k.
References
Basu, A. and Sarkar, S. (1994). On disparity based goodness of fit tests for multinomial models. Statistics andProbability Letters, 19, 307–312.
Bhattacharya, B. (1997). Testing multinomial parameters under order restrictions. Communications in Statistics–
Theory and Methods, 26, 1839–1865.
Cressie, N. and Read, T. R. C. (1984). Multinomial goodness-of-fit tests. J. Roy. Statist. Soc. Ser. B, 46, 440–464.
Dykstra, R. L. and Lee, C. I. C. (1991). Multinomial estimation procedures for isotonic cones. Statistics andProbability Letters, 11, 155–160.
Lindsay, B. G. (1994). Efficiency versus robustness: the case for minimum Hellinger distance and related methods.
Annals of Statistics, 22, 1081–1114.
Read, T. R. C. and Cressie, N. (1988). Goodness-of-fit Statistics for Discrete Multivariate Data. Springer, New York.
Robertson, T. (1978). Testing for and against an order restriction on multinomial parameters. J. Amer. Statist. Assoc.,
73, 197–202.
Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. John Wiley & Sons,
New York.