One of the common problems of practical importance is that of determining whether there is independence 

between a pair of random variables. In this paper, the problem of testing independence of bivariate random variables against 

a weighted alternative model with possible missing values on both responses is considered. The model considered here is due 

to Shei, Bai and Tsai [9] which is the generalization of Hajek and Sidak [12] model with weighted contamination. A new rank 

test based on ranks is proposed and its asymptotic normality is established. Locally most powerful tests for the model is 

derived. The asymptotic null distributions of the test statistics are also provided for the purpose of practical us

Kumari, Savitha, .

Parameshwar V. Pandit, .

ePrints@Bangalore University

American Journal of Mathematics and Statistics 2015, 5(4): 196-199 DOI: 10.5923/j.ajms.20150504.05  Rank Tests for Independence Against Weighted Alternative with Missing Values Parameshwar V. Pandit1,*, Savitha Kumari2 1Department of Statistics, Bangalore University, Bangalore, India 2Department of Statistics, SDM College, Ujire, India  Abstract  One of the common problems of practical importance is that of determining whether there is independence between a pair of random variables. In this paper, the problem of testing independence of bivariate random variables against a weighted alternative model with possible missing values on both responses is considered. The model considered here is due to Shei, Bai and Tsai [9] which is the generalization of Hajek and Sidak [12] model with weighted contamination. A new rank test based on ranks is proposed and its asymptotic normality is established. Locally most powerful tests for the model is derived. The asymptotic null distributions of the test statistics are also provided for the purpose of practical use. Keywords  Independence tests, Locally most powerful rank test, Weighted dependence alternative, Missing values  1. Introduction The problem of interest for the statisticians is that of determining whether there exists a relationship between two characteristics in a population. In the literature several authors attempted the quantification of the concept of stochastic dependence for bivariate distributions. Rank tests for independence based on complete data can be found in Spearman [1], Kendal [2], Bhuchongkul [3], Puri and Sen [4], Shirahata [5] among others. Iman and Conover [6] proposed a measure of dependence, which is the Pearson correlation coefficient computed on Savage [7] scores, reflects the importance on the top ranks. Shieh [8], proposed a weighted Kendal’s tau statistic. Shieh, Bai and Tsai [9] proposed some rank tests and derived locally most powerful rank test for testing independence against a weighted contaminated alternative. Pandit [10] considered this problem with a different weighted alternative and derived locally most power rank test. However, in practical situations, some observations on either of the variables may be missing. In such a situation the tests mentioned above cannot be applied. Wei [11] derived a locally most powerful rank test for independence against the alternative given by Hajek and Sidak [12] in presence of missing values. In this paper, we propose rank tests and derive locally most powerful rank test for independence against weighted alternatives in presence of missing values.  * Corresponding author: panditpv12@gmail.com (Parameshwar V. Pandit) Published online at http://journal.sapub.org/ajms Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved   Let (X1, Y1), (X2, Y2), … , (Xn, Yn), (Xn+1, . ),…, (Xn+m), (. , Yn+1), …, (. , Yn+k) be a random sample from a bivariate distribution function F(x,y). The problem is to test ),(,)().(),(: 210 yxallforyFxFyxFH  . Here, The alternative considered is as below: ZYYZXuXX  *** ,)(    (1) where X*, Y* and Z* are mutually independent and u(x) is monotone in x. (Shieh, Bai and Tsai [9]). The alternative due to Hajek and Sidak [12] is a particular case of (1). Under (1), it is clear that if 0 , X and Y are independent and larger the   is, the more dependent are X and Y. Thus the constant   may be regarded as a dependence or mixing coefficient. The alternatives stated in (1) indicate the positive dependence of the random variables X and Y. If we assume negative dependence between X and Y, then the model (1) is ZYYZXuXX  *** ,)(     (1*) where X*, Y* and Z* are mutually independent and u(x) is monotone in x. Let R1, R2,…, Rn+m be the ranks of the first coordinates X1, X2, …, Xn+m of the sample and Q1, Q2,…, Qn+k be the ranks of the second coordinates Y1, Y2, …, Yn+k  of the sample. Rank tests developed for testing independence against the alternatives considered are the functions of Ri’s and Qj’s, i=1,2,…, n+m, j=1,2,…, n+k. In section 2, we propose a new test for testing independence and the LMPR test for the alternative (1) is considered in section 2. The LMPR test for the alternative (1) is derived in section 3. In section 4, we give some remarks.  American Journal of Mathematics and Statistics 2015, 5(4): 196-199  197   2. New Rank Test for Testing Independence The model of dependence considered is that considered in Shei, Bai and Tsai [9]. The model for a random sample with missing values is as specified below. Let    * * ** * *( ) , , 1,2,...,( ) , , 1,2,..., ; 1,2,..., .i i i i i i ij j j j l l m lX X u X Z Y Y Z i nX X u X Z Y Y Z j m l k          The variables X*, Y* and Z are independent and   is a real nonnegative parameter.  First, we propose a test statistic for testing bivariate independence against model (1), which is the modified Spearman’s coefficient to include missing observations. The statistic is defined by    niiiisknQmnRuW1*2121. Here, *)( niIui  , where pnn )1(*  and 10  p  is roughly the proportion of the observed items. The choice of p  is to have less loss in significance level.  The asymptotic distribution of the statistic *sW  is given in the following theorem. Let I(f) denote the Fisher information,   dxxfxfxffI )()()()(2. Theorem 1: Assume that H0 holds, I(f10)< and I(f20)< . Then, 12)1( 2*npnWs  converges in distribution to standard normal. Proof: Consider,    niiiisknQmnRuVarWVar1*2121)(    and under H0 , 2*11 1( )2 2ns i i iin m n kVar W E u R Q                 2*1 2121   niiiknQmnRE  1 11 2 1 2* ( ) ( )*( * 1) ( , ) ( , )n Var R Var Qn n Cov R R Cov Q Q .     (2) Further, we have    ,121)()(,2)1( 211mnRVarmnRE     121)()(,2)1( 211knQVarknQE ,  1 21 2( 1)(3( ) 2)( ) ,12( 1)( , )12n m n mE R Rn mCov R R     .  1 21 2( 1)(3( ) 2)( ) ,12( 1)( , )12n k n kE Q Qn kCov Q Q      Substituting these in equation (2) and as n, we have   .12/)1( 2*pnnWnVar s  Assuming that I(f10)< and I(f20)<, and applying Theorem V.1.6a in Hajek and Sidak [12], we have the required result. 3. Locally Most Powerful Rank Test for Weighted Alternative Here, we consider the model of dependence which is the generalization of the model considered by Bhuchongkul [3], and, Hajek and Sidak [12]. The model for a random sample with missing values is as specified below. Let    * * ** * *( ) , , 1,2,...,( ) , , 1,2,..., ; 1,2,..., .i i i i i i ij j j j l l m lX X u X Z Y Y Z i nX X u X Z Y Y Z j m l k              The random variables X*, Y* and Z are independent;  is a real nonnegative parameter and w(x) monotone in x. Under the above model it is clear that if  =0, X and Y are independent.  Now, let X* and Y* have the p.d.f.s f10(x) and f20(y) respectively and the distribution of Z is arbitrary. Let x*=t(x,  z) be the unique solution of the equation x=x*+w(x*) z , for given x and z . The joint p.d.f., q , of (X1, Y1), (X2, Y2), … , (Xn, Yn), (Xn+1, . ),…, (Xn+m), (. , Yn+1), …, (. , Yn+k) is given by   mnnjknnlljniii zyfxfyxhq1 120*101)().(.),( , where   )().().(),( 20*10 zdMzyfxfyxh and )(zM  is the distribution function of Z with mean z  and finite variance 2z . Let X(i) and Y(i) be the i-th order statistic of {X1,X2,…, Xn+m} and {Y1,Y2,…,Yn+k} respectively. Further, let 198 Parameshwar V. Pandit et al.:  Rank Tests for Independence Against Weighted Alternative with Missing Values      )()(),( )(101010 iimn XfufEfra   and   )(),( )(202020 iikn YffEfqa  denote the score functions corresponding to 10f  and 20f  respectively. In order to obtain LMPR test we assume the following conditions: (i) The derivatives )( 10wf  and 20f  are continuous, (ii)  dxxuf )()( 10 and 20 ( )f x dx  , (iii) The missing observations on either the first coordinate or the second coordinate occur at random. The following theorem states the LMPR test. Theorem 2: Under the conditions (i) to (iii) and for the model (1), the test statistic niiknimn fqafraV120101 ),().,( with critical region cV 1 , where c is a constant, is locally most powerful rank test for testing 0:0 H  against 0:1 H at corresponding level of significance. Corollary 1: If f10 and f20 are from Logistic family, then the test based on *sW  with critical region cWs *, where c is a constant, is asymptotic LMPR test for testing 0:0 H  against 0:1 H  for model (1). 4. Power Comparisons In this section, we compare the powers of the proposed test *sW  with those of top-down statistic rT and Kendall’s τ. The alternative used is as in (1). For power comparisons, the alternatives considered are 1. Xi, Yi and Zi follow normal with mean zero and variance one. 2. Xi, Yi and Zi follow logistic(0,-1). The correlation coefficient between X and Y denoted by  in terms of  is ))(( 22222222zyzxzpp . Here it is to be noted that =0 implies the independence. For simulation selected values of  are considered. The results are presented in table 1. In the table 1, m is the number of missing observations corresponding to x-values and k is the number of missing observations corresponding to y-values. From the above table it is easily seen that the test V is more powerful than the top-down statistic, rT due to Iman and Conover [6] and Kendall’s test, τ. Similar results are obtained for n=30, 50.  Table 1.  Empirical Powers of new test , top-down statistic rT and Kendall’s τ for n=20 and p=0.9 m k  N(0,1) Logistic(0,1) *sW  rT τ  rT τ 2 2 0.1 0.1325 0.0994 0.0812 0.1578 0.0872 0.0905   0.4 0.3242 0.1089 0.1582 0.3160 0.0951 0.1567   0.7 0.5122 0.1279 0.2665 0.5283 0.1056 0.2381 2 3 0.1 0.2624 0.1552 0.1652 0.2764 0.2311 0.2153   0.4 0.6234 0.5570 0.5284 0.7762 0.6732 0.5932   0.7 0.9132 0.9981 0.8973 0.9372 0.8973 0.8693 3 3 0.1 0.2925 0.1072 0.0929 0.3078 0.1991 0.2125   0.4 0.6356 0.1187 0.1696 0.6961 0.5862 0.5685   0.7 0.9212 0.1286 0.2765 0.9383 0.8984 0.8791 3 4 0.1 0.3514 0.1645 0.1673 0.3454 0.2732 0.2053   0.4 0.6531 0.5483 0.5096 0.7785 0.6578 0.5937   0.7 0.9437 0.9104 0.8973 0.9478 0.9073 0.8774  *sW*sW American Journal of Mathematics and Statistics 2015, 5(4): 196-199  199   5. Some Remarks and Conclusions 1.  The paper considers the problem of independence against a weighted alternative when the data have missing values. The alternative considered is the generalization of Hajek and Sidak [12], accommodating weighted contamination. 2.  The alternative considered here is the model used in Shei, Bai and Tsai [9]. 3.  A new rank test is developed and the asymptotic normality of the test statistic is established. 4.  Locally most powerful rank(LMPR) test for this problem is derived for the alternative used in Shei, Bai and Tsai [9] and the LMPR derived in Wei [11] for the alternative due to Hajek and Sidak [12] can be obtained as a particular case. 5.  The new test proposed here is shown to be LMPR when the marginal distributions of X* and Y* belong to logistic family.  REFERENCES  [1] Spearman, C. (1904). The proof and measurement of association between two things. Am. J.Psych. 15, 72-101. [2] Kendall, M.G. (1962). Rank Correlation Methods. Griffin: London. [3] Bhuchongkul, S. (1964). A class of nonparametric tests for independence in bivariate populations. Ann. Math. Statist. 35, 138-149. [4] Puri, M.L. and Sen, P.K. (1971). Nonparametric Methods in Multivariate Analysis. Wiley: New York. [5] Shirahata, S. (1974). Locally most powerful rank tests for independence. Bull Math. Stat. 16, 11-21. [6] Iman, R.L. and Conover, W.J. (1987). A measure of top-down correlation. Technometrics, 29, 351-357. [7] Savage, I.R (1956). Contributions to the theory of rank order statistics- two-sample case. Ann.Math.Statist.27,590-615. [8] Shieh, G.S. (1998). A weighted Kendal’s statistic. Stat. Prob. Letters, 39; 17-24. [9] Shei, G.S., Bai, Z. and Tsai, W.Y. (2000). Rank tests for independence – with a weighted contamination alternative. Statistica Sinica, 10, 577-593. [10] Pandit, P.V. (2006). Locally most powerful and other rank tests for independence-with a contaminated weighted alternative. Metrika, 64, 379-387. [11] Wei, L.J. (1983). Tests for independence in the presence of missing values. Austral. J. Statist., 25, 85-90. [12] Hajek, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press; New York.  

Rank Tests for Independence Against Weighted 

Alternative with Missing Values

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Rank Tests for Independence Against Weighted Alternative with Missing Values

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