Bivariate symmetry and a simple generalization of the sign test for testing spherical symmetry of a bivariate distribution

J King Saud Univ., Vol. 21, Science (Special Issue), pp. 227-232, Riyadh (2009i1430H.)

Bivariate Symmetry and a Simple Generalization of Sign Test for Testing Spherical Symmetry of a Bivariate Distribution

A. A. Al-Shiha

Department o/Statistics & Research College o/Science. University P. 0. Box 2455, Riyadh 1451, Saudi Arabia

E-mail: aalshiha@ksu.edu.sa

Key words: Sign test, Circular symmetry, Spherical symmetry, Bivariate symmetry.

Abstract. Ahmad and Cerrito have proposed two definitions of bivariate symmetry, however; neither of these satisfies the property of two random variables considered by many authors in the literature. This has led us to propose another that satisl1es the desirable properties. We also propose an extension of univariate test to test the symmetry of a bivariate distribution. The sampling distribution of the proposed test follows approximately a chi-square distnbution. We conduct a simulation study to evaluate the performance of the proposed test for testing circular symmetry of a continuous bivariate distribution.

1. Introduction

Several tests for testing symmetry of a univariate distribution are proposed in the literature, see for Gastwirth (J 97 Hill and Rao (1977), and Randles et. al (1980). McWilliams (1990) showed the use of a runs statistic for symmetry of a distribution. Modarres and Gastwirth (1996) a modified runs test for symmetry.

Tajuddin (l994) proposed a test for based on Wilcoxon test. KhaJique and Tajuddin (I have studied the use of a test statistic in

asymmetry of a distribution.

The problem of symmetry in a bivariate or a multivariate distribution is tackled in different ways. Several papers such as Bell and Haller (1969) and Hollander (J 971) have considered the of testing the bivariate

H 0: F(x,y) V (x,y) E problem is as a test of interchangeability by Ernst and Schucany (1999) among others. Beran (1979), Alzaid et. al (1990) and others have studied the elliptical symmetry of multivariate distributions, which also

the of random variables.

Heathcote et. al (1995) have talked about the

of a distribution, whereas Small has discussed various kinds of

227

including the concept of an angular

Ahmad and Cenito (1991) have discussed the requirements of a bivariate

proposed two different definitions of the definitions do not regard the

property of random variables under bivariate symmetry. We this situation in Section 2, and this lead us to another definition of bivariate Oja and Nyblom (J 989) have several bivariate of the sign test for a univariate distribution. These tests include the bivariate tests proposed by Blueman (I Brown et. a) ( 1992) and (I all the above-mentioned authors and others such as Chaudhuri and Sengupta (1993) have the extensions of the test for the location Koltchinskii and Li (1998) have a test for spherical of a distribution in ,p 2, with unknown location Zhu and Neuhaus (2003) conditional test procedures for testing symmetry of multivariate distributions. Earlier, Romano (1989),

(1991) and Smith (1977) have considered spherical symmetry for a distribution with known centre.

We present a bivariate

228 A.A, AI-Shiha: Bivariate Symmetry and a Simple Generalization of the Sign Test for,,, which can be extended to a multivariate case, of the

simple test for circular of a

bivariate continuous distribution with known location We describe the test statistic in Section 3, We conduct a simulation study similar to the one carried on Koltchinskii and Li (1 to examine the of our proposed test statistic, We the simulation results in Section 4. Finally, we end up with the conclusion of this paper in Section 5.

2. Concepts of Bivariate Symmetry Ahmad and Cerrito (1991) discuss that there was

no result from the

made to define a form of

random variables. They have as

desirable properties for any definitions of bivariate symmetry:

(I) The densities are

(2) The conditional densities are symmetric,

The bivariate normal distribution is ~\!tTIm,ptr

with to the definition, and

Statistical tests can be developed to test for symmetry.

Based on above they define two kinds of symmetry for the bivariate distributions, which are:

(J) (2)

Weakly The random variable J)

with joint

there exist constants

-a) c-x)

is weakly symmetry if

E 91 such that:

-b) h(b ,and a,y-b) = -x.b y).

with joint density there exist constants

The random variable Y) is strongly symmetry if

E

m

such that:

for some in 9t, and for some in ~H,

Ahmad and Cerrito (\991) mentioned that Hollander (197]) defines the of a random variable as the bivariate symmetry; however, neither of the definitions of weakly or

the considered by

Hollander (1971). Ahmad and Cerrito (1991), in Section 3 of their paper, have considered all the examples of bivariate symmetry for the densities that satisfy the property of of random variables. Thus, we feel that this property of

of random variables is also

to understand the of of a bivariate distribution, We can call a bivariate distribution with probability function is

about (WLOG) if it satisfies the conditions:

(I) jj(x) - x), V X E ~H ,

(2) y), V y ~,

(3) E

, and

of distributions the bivariate symmetry considered by Ahmad and Cerrito (1991).

(I) y) ==

---'-'-=====,

^{for E , where}

Q=

(2) (3)

(4)

y) c/

y) = k /

+ ] , for E

E ,and

1

x+ y

I),

for

It is clear that all the above densities follow the conditions mentioned for the definition of a properly bivariate We give a counter example in Table I where the distribution is strongly symmetric it lacks the property of of random variables and hence is not bivariate symmetric. We note from probability distribution in Table I that both densities as well as the conditional densities are

as an

Table L A distribution that is strongly symmetric but not I roperly symmetric

X\Y -2 -1 0 I 2 Total

-2 ,03 .01 .02 .01 .03 0,1

-I ,06 ,02 ,04 .02 ,06 0.2

0 .12 ^,04 .08 ,04 .12 0,4

I ,06 .02 .04 .02 06 0.2

I

2 ,03 ,01 ,02 ,01 .03 0,1 I

Total 0.3 O.! 0.2 0,1 0.3 1.00

1. King SOlid Univ., Vol. 21, Science (Special Issue), Riyadh (20091l430H.) 229

Now, we give some more examples below to understand the concept of bivariate symmetry:

(a) Table 2 gives an example of a joint distribution with identical marginal densities not independently distributed that do not satisfY the conditions of weakly symmetry.

(b) Table 3 gives an example of a density with marginal densities symmetric and independently distributed where fi(x) and

12(Y)

^{are symmetric}

and independent. Here .f(x,y) IS strongly symmetric but not properly symmetric as long as a ^:t b. When a = b, j(x,y) is also properly symmetric.

(c) Finally, we present Table 4 where the marginal densities are not independent and (X, Y) IS

properly symmetric.

Table 2

X\Y -I 0 I Total

-I 0.10 0.10 0.05 0.25

0 0.05 0.35 0.10 0.50

I 0.10 0.05 0.10 0.25

Total 0.25 0.50 0.25 1.00

T bl 3 a e

XfY -I 0 I Total

-I ab a(I-2b) ab a

0 b( 1-2a) (1-2a)(1-2b) b(l-2a) 1-2a

I ab a(I-2b) ab a

Total b 1-2b b I

T bl 4 a e

X/V -I 0 I Total

-I a b a 2a+b

0 b 1-4a-4b b l-4a-2b

I a b a 2a+b

Total 2a ·-b J -4a-2b 2a+b I

3. The Proposed Test Statistic for Testing Bivariate Spherical Symmetry

Let (xi,Yi),i = 1,2, ... ,11, be a random sample of

11 observations from a bivariate distribution Fx,y(.x,y) with known mean vector 11 =

C!J

x,f.-1_{y )'.}

With no loss of generality, we can assume f.-1_x = f.-1_y = O.

Let the number of observations falling in the quadrants I, II, lI£, and IV be 111,112,113, and 114

respectively, and PI, P2, P3, P4 be the proportions of observations falling in the corresponding quadrants.

Then, it is well known that the distribution of

(111,112,113,114) follows a multinomial distribution. If we assume that the given bivariate distribution

F X ,Y(x,y) is spherically symmetric about the origin, then the distribution of (111,112,113,114) will be multinomial distribution with the parameters

PI = P2 = P3 = P4 = 1/4.

Our goal is to test H a : the bivariate distribution

IS spherically symmetric, against HI : the bivariate distribution is not spherically symmetric. Under H a'

the expected number of observations in each quadrant is 11 / 4 . The test statistic is given by:

4 4 11 2

S=-L)

^l1i--) . 11 i=J 4

Under H a, and for large value of 11, the sampling distribution of S follows approximately a chi-square distribution with 3 degrees of freedom. The test rejects H a (the hypothesis of spherically symmetry) if the value of S exceeds x~(3), the (I-a)IOOlh percentile of a chi-square distribution with 3 degrees of freedom, at a specified value of a .

4. Simulation Study Empirical Size of the Test

We compare the empirical a with the nominal a values of the proposed test statistic for the choices of a = O. 1,0.05 and 0.01 and various sample sizes from different underlying distributions. We consider all distributions considered by Koltchinskii and Li (1998). They have considered the following distributions under the null hypothesis that the distribution is spherically symmetric.

(I) H~I) = the standard bivariate normal distribution, i.e.,

I 2 2

j(x,y)=-exp{-(x + y )/2} .

2J[

(2) H~2) ⁼ the uniform distribution on the unit disk, i.e.,

I 2 2

j(x,y)=-;O<x +y <1.

J[

(3) H~3) = the uniform distribution on the unit circle, i.e.,

230 A.A. AI-Shiha: Bivariate Symmetry and a Simple Generalization of the Sign Test for ...

I 2 2

j(x,y)=-; X + y = l.

2;r

We remark here as mentioned by Koltchinskii and Li (J 998) that the distributions considered under H ⁰ are spherically symmetric.

We draw 10000 random samples of various sizes from each of the above-mentioned distributions and we observe the number of times the observed test statistic exceeds the critical value for different choices of level of significance. The results obtained are presented in Table 5 and Figure 1. Table 5 and Figure 1 compare the empirical a values with the nominal a values (a = 0.0 I, 0.05 and 0.1 ) and different choices of sample size, n = 10(10)100,200 .

Table 5. Empirical a at different nominal values of a for different underlying distributions under H ⁰

Sample Size 10 20 30 40 50 60 70 80 90 100 200

0.12 0.11 0.10 0.09 O.OB 0.07 0.06 0.05 O.M 0.03 0.02 0.01 0.00

•

•

•

. •

•

0 0

Underlying Distributions H(I)

0

H(2) 0

H(3) 0

Nominal Alpha ( a )

.01 .05 .10 .01 .05 .10 .01 .05 .10 .0111 .0363 .1065 .0113 .0354 .1079 .0092 .0339 .1021 .0066 .0416 .09J4 .0076 .0438 .0890 . 0068 .0424 .0911 .0073 .0512 .0909 .0085 .0542 .0956 .0068 .0490 .0917 .0088 .0452 .0964 .0106 .0480 .0962 .0076 .0407 .0927 .0101 .0510 .0942 .0080 .0494 .0963 .0094 .0517 .0975 .0076 .0483 .1051 .0098 .0483 .1024 .0100 .0465 .1006 .0115 .0515 .0949 .0088 .0471 .0878 .0100 .0506 .0946 .0109 .050J .0985 .0112 .0499 .1018 .0100 .0532 .1060 .0110 .0461 .0992 .0092 .0495 .1032 .0108 .0523 .1025 .0105 .0459 .0918 .0096 .0473 .0949 .0088 .0493 .0989 .0111 .0513 .J052 .0087 .0487 .0968 .0120 .0478 .0977

En..,;rical Alpha (al for Ho(ll

• •

0.1

•• • •

• • .

•

^•

• •

^0.05

0 e- 0.01

50 100 150 200

Fig. I (a). Empirical a for H~I) H~I) = the standard bivariate normal distribution

0.12 0.11 0.10 '0.09 0.08 0.07 0.06 0.05 O.M 0.03 0.02 0.01 0.00

•

·

• • •

•

• .

·

e -

.

50

En..,;rical Alpha (al for Ho(2l

•

0.1

• •

•

• • __ a •

-

^0.05

0.01

100 150 200

Fig. I (b). Empirical a for H~2) H~2) ⁼ the uniform distribution on the unit disk

0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 O.M 0.03 0.02 0.01 0.00

-"-

·

• •••

•

. .

o o • 50

En..,;rical Alpha (al for Ho(3l

• •

^0.1

•

~ 0.05

.

0.01

100 150 200

Fig. I (c). Empirical a for H~3) H~3) = the uniform distribution on the unit circle

From Table 5 and Figure I, we observe that the nominal values of a ^lies within two standard error of the empirical a. Thus, we see that the proposed test is a valid test.

The Power of the Test

To consider the power of the test statiStIC, the following distributions are considered under the alternative hypothesis that the distribution is not spherically symmetric:

(I)

Hr

^{l) =} the distribution of random vector with two independent exponential variates with parameters

A.I

=

I and ~

=

2, respectively; i.e.,

j(x,y) = 2exp{-x - 2y}; for x > 0, y > O.

(2) HF) = the distribution of the random vector with two independent components, exponential with parameter A. = 1 and standard normal; i.e.,

J(x,y)=

~exp{-x-y2 / 2};forx > 0,YE~

^.

...;2;r

1. King Saud Univ., Vol. 21, Science (Special Issue), Riyadh (2009/1 430H.) 231

(3) HF) = the mixture (with parameter 112) of two bivariate normal distributions in 9,2 with unit covariances and with means (0,0) and (3,0); i.e.,

/(",y) =

~

^{[exp{-x2 12}} + exp{-(x - 3)2 12}]

2v 21l

x exp{-y2/2};jor (x,Y)E912.

(4) H_I(4) = the uniform distribution in equilateral triangle with center point (0,0); i.e.,

j(x,y)

=

I;

-.fi

~2y-I~lxl~.fi.

Table 6. Power of the test against different alternatives and different choices of a and sample sizes

Sample size

10 20

30 40

50 60 70 80 90 100 200

1.0

0.8

0.6

0.4

0.2

H(I) I

.01 .05

0513 .1200 .0836 .2325 .1653 .3874 .2716 .4821 .3521 .6097 .4335 .6802 .5342 .7583 .6168 .8191 .6885 .8653 .7562 .9023 .9855 .9972

OiSlribution under Alternalivcs H(2)

I

H() I Nominal Alph~ (a )

.10 .01 .05 .10 .01 .05

.2583 .0232 .0739 . 1849 .0102 .0334 .3579 .0316 .1204 .2171 .0074 .0423 .4944 .0584 .1993 .2956 .0083 .0509 .6176 .0929 .2439 .3626 .0088 .0417 .7135 .1277 .3269 .4432 .0090 .0501 .7909 .1607 .3731 .5097 .0079 .0488 .8409 .2018 .4318 .5482 .0096 .0449 .8904 .2567 .4840 .6192 .0100 .0514 .9210 3066 .5358 .6746 .0101 .0499 .9489 .3513 .6018 .7166 .0093 .0473 .9989 .7439 .8942 .9431 .0099 .0500

Power of the Testfor Hl(l)

.~ .... :.:=.-::.:..;.. :..-;,. ;..-~----- ,

..

,"

...

"

^/

.10

.1013 .0955 .0908 .0914 .0975 .1041

.0884 .1101 .0992 .0967 .0979

, ~

/ /

4 . : I

:" /;

•• I / I

.

^I

^"

"

0.0'-,-_ _ ---, _ _ _ -,-_ _ - , _ _ _ -,-1

50 100 150 200

Fig. 2 (a). Power of the Test for

f(x,)~=2exp{-x-2y}; x>O, y>O

H(4) I

.01 .05 .10

.0101 .0357 .1093 .0073 .0403 .0906 .0070 .0505 .0916 .0090 .0444 .0894 .0083 .0504 .0974 .0094 .0488 .1042 .0086 .0506 .0954

.0094 .0477 .1002

.0093 .0474 .0976

.0124 .0517 .1005 .0085 .0451 .0966

0.9 0.8 0.7 0.6 0.5 OA 0.3 0.2 0.1

Power of the Test for Hl(2)

0.0l..,-_ _ _ .----_ _ --,-_ _ _ r -_ _ ---r...J

50 100 150 200

Fig. 2 (b). Power of the Test for

I 2

j(x,y) = r.:;-exp{-x -)' /2); x> 0, Y E <J"l

\l2;r . .

Power of the Test for Hl(3)

0.12. . -- - - - - -- -- - - , ,---:::,.-:::::--,

0.10 . -------

..

0.08

0.06

0.04

~.,.)I-...

... --... __ _ - -- - - ----

...

0.02

... ..

0.00"'r-_ _ ----. _ _ _ , - -_ _ ~---..,...

50 100 150 200

Fig. 2 (e). Power of the Test for H_I(3) H?):

f(x,y)=

I , 2 2 2

~[exp(-x-/2) +exp{-(x-3) 12))exp{-y 12); (x,y)E9~

2-J2;r

We remark here as mentioned by Koltchinskii and Li (1998) that the distributions considered under HI are not spherically symmetric although the last three are symmetric in some forms.

We draw 10000 random samples of various sizes from each of the above-mentioned distributions and we observe the number of times the observed test statistic exceeds the critical value for different choices of level of significance. The results obtained are presented in Table 6 and Figure 2. Table 6 and Figure 2 present the power of the test against various alternatives for nominal a = 0.0 I, 0.05 and 0.1 and different choices of sample size, n = 10(10)100,200.

From Table 6 and Figure 2, we observe that the test is quite powerful against the alternatives H}I) and H](2); however, it fails to reject Ho (the

232 A.A. AI-Shiha: Bivariate Symmetry and a Simple Generalization of the Sign Test for ...

hypothesis of spherical symmetry) for the alternatives H~3) and H

I(4)

5. Conclusion

In Section 4, we noticed that the test statistic S is a valid test for testing spherical (circular in this case) symmetry of a bivariate distribution. It also has a high power for rejecting false H ⁰ for two cases of the four cases considered. However, like many other test statistics it commits an error of type II, i.e., it does not reject H 0 whenever H 0 is false for some cases.

The test fails to reject when the distribution has an equal probability in each of the four quadrants i.e.

when the following condition is fulfilled:

P(X > flx'Y > fly)

=

^P(X < flx'Y > fly)

=

P(X > flx,Y < fly)

=

P(X < flx'Y < fly).

The proposed test is very simple and its sampling distribution is known. The proposed test can be easily extended to test the spherical symmetry of a p- variate distribution.

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