Nonparametric estimation of P(X < Y) using kernel methods

METRON - International Journal of Statistics 2006, vol. LXIV, n. 1, pp. 47-60

AYMAN BAKLIZI – OMAR EIDOUS

Nonparametric estimation of P(X < Y) using kernel methods Summary - In this paper we consider point and interval estimation of P(X < Y ). We used the kernel density estimators of the densities of the random variables X and Y , assumed independent. The resulting statistic is found to be similar in many ways to the Mann-Whitney statistic, however it assigns smooth continuous scores to the pairs of the observations rather than the zero or one scores of the Mann-Whitney statistic. A simulation study is conducted to investigate and compare the performance of the estimator based on this statistic and the Mann-Whitney estimator. Bootstrap confidence intervals based on the two statistics are investigated and compared. Key Words - Bootstrap intervals; Kernel density estimator; Mann-Whitney statistic; Stress-strength.

1. Introduction A wide range of problems in engineering and medicine, as well as some other fields involve inference about the quantity p = P(X < Y ) where X and Y may denote the stress and the strength variables in the context of mechanical reliability of a system. The system fails any time its strength is exceeded by the stress applied to it. In a medical setting, P(X < Y ) measures the effect of the treatment when X is the response for a control group and Y refers to the treatment group. Several authors have considered statistical inference about P(X < Y ) under a parametric setup, see for example Reiser and Guttman (1986), Johnson et al. (1994) and the references therein. When there is not enough evidence for a specific parametric model for the distribution generating the data, one may consider nonparametric methods. There are several authors who discuss point and interval estimation in the nonparametric case (Sen, 1967; Govindarajulu, 1968 and 1976; Halperin et al., 1987 and Hamdy, 1995), see Kotz et al. (2003) as a general reference. Let X 1 , . . . , X n 1 and Y1 , . . . , Yn 1 be two independent Received December 2004 and revised September 2005.

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AYMAN BAKLIZI – OMAR EIDOUS

random samples from the distributions of X and Y respectively. n 1 n 2The MannWhitney statistic (Lehmann, 1975) is defined as W = i=1 j=1 φ(X i , Y j ) where φ(X i , Y j ) = 1 if X i < Y j and 0 otherwise. A version corrected for ties is also given in Lehmann (1975). It is readily seen that E(W ) = n 1 n 2

E

i=1

j=1

φ(X i , Y j )

= n 1 n 2 P(X < Y ), therefore an unbiased estimator n

n

1 2 ˆ 1 is the uniformly for p is given by pˆ 1 = n 1n i=1 j=1 φ(X i , Y j ). In fact, p 1 2 minimum variance unbiased estimator for p. The variance of pˆ 1 is given by

var( pˆ 1 ) = 

(n 1 −1)



1 n1n2 F12 d F2 +(n 2 −1)





F22 d F1 −(n 2 −1)+(2n 2 −1) p−(n 1 +n 2 −1) p 2

where F1 and F2 are the distribution functions of X and Y respectively. Several authors have considered the problem of constructing distribution-free confidence intervals for p using this unbiased estimator. Sen (1967) showed that √ n2 n 0 ( pˆ 1 − p) is asymptotically normally distributed where n 0 = nn 1+n . He also 1√ 2 provided a consistent estimator for the asymptotic variance of n 0 ( pˆ 1 − p) which can be used to construct approximate confidence intervals. Govindara√ julu (1968) showed that min(n 1 , n 2 )( pˆ 1 − p)/SG is asymptotically normal where SG2√is an unbiased, distribution-free and consistent estimator of the variance of min(n 1 , n 2 )( pˆ 1 − p)/SG . Halperin et al. (1987) have shown that √ min(n 1 , n 2 )( pˆ 1 − p)/ν H is asymptotically normal where ν H = p(1 − p)    n1  



Fn21 dG n 2 +n 2

G 2n 2 d Fn 1 −n 2 +(2n 2 −1) pˆ 1 −(n 1 +n 2 −2) pˆ 12 pˆ 1 (1 − pˆ 1 )



 

+ 1 .

They use this result to construct distribution free intervals for p. Hamdy (1995), using the approach of Halperin et al. (1987), has constructed the pivotal quantity with an asymptotic chi-squared distribution with one degree of freedom, which can be used to construct confidence intervals for p. This pivotal quantity is given by Z 2 ( p) = ( pˆ 1 − p)2 /(θˆ (A L − AU ) + AU )

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Nonparametric estimation of P(X