Estimation of the Parameters of Bivariate Normal Distribution with Equal Coe ffi cient of Variation Using Concomitants of Order Statistics

To provide an optimum estimator for the parameters, the use of priori information has a crucial role in univariate as well as bivariate distributions. One such prior information is to utilize the knowledge on coefficient of variation in the inference problems. In the past plenty of work was carried out regarding the estimation of the mean μ of the normal distribution with known coefficient of variation. Also inference about the parameters of bivariate normal distribution in which X and Y have the equal (known) coefficient of variation c, are extensively discussed in the available literature. Such studies arise in clinical chemistry and pharmaceutical sciences. It is interesting to note that concomitants of order statistics are applied successfully to deal with statistical inference problems associated with several real life situations. A problem of interest considered here is the estimation of parameters of bivariate normal distribution in which X and Y have the same coefficient of variation c using concomitants of order statistics. For that consider a sample of n pairs of observations from a bivariate normal distribution in which X and Y have the same coefficient of variation c, we derive the best linear unbiased estimator (BLUE) of θ2 and derive some estimators of θ. Efficiency comparisons are also made on the proposed estimators with some of the usual estimators, finally we conclude that efficiency of our best linear unbiased estimator (BLUE) θ̃2 is much better than that of the estimators θ̂2 and θ∗ 2.


Introduction
The use of prior information in inference is well established in the Bayesian arena of statistical methodology.Such information is usually incorporated into a model by choosing an appropriate prior distribution.In some instances prior information can be incorporated in classical model as well.Searls and Intarapanich (1990) considered an estimator of variance when the kurtosis of the sampled population is known.But in some situations of biological and physical sciences, where the scale parameter is proportional to the location parameter, knowing the proportionality constant is equivalent to knowing the population coefficient of variation (Gleser & Healy 1976), however, inferential testing procedures become more complex, and the property of completeness is no longer held true, and, thus the standard theory of uniformly minimum variance unbiased estimation (UMVUE) is not applicable.Several authors, have studied these problems; Searls (1964) used the coefficient of variation to improve the precision of sample mean as an estimator of the population mean.If the parent distribution is normal with mean θ and standard deviation cθ thus, distributed as N(θ, c 2 θ 2 ), the problem of estimating θ has been studied (Kunte, 2000;Guo & Pal, 2003); the best linear unbiased estimator (BLUE) of θ for N(θ, c 2 θ 2 ) distribution for different values of c using order statistics are discussed in Thomas and Sajeevkumar (2003); and estimating the location parameter θ of the exponential distribution with known coefficient of variation by ranked set sampling are discussed by Irshad and Sajeevkumar (2011); and Sajeevkumar and Irshad (2011) have also dicussed estimation of the location parameter θ of the exponential distribution using censored samples.
Estimation of means of two normal populations is frequently undertaken, assuming that the variances of the populations are known.Sen and Gerig (1975) developed an estimator under the equality of the two coefficients of variation assumption; Azen and Reed (1973) consider the problem of estimation of parameters of bivariate normal distribution with equal coefficient of variation, however, they fail to give explicit expression for the estimates of the parameters.Sajeevkumar and Irshad (2012) have recently studied the estimation parameters of bivariate normal distribution with equal coefficient of variation using concomitants of record values.This paper discuss the estimation of parameters of bivariate normal distribution with equal coefficient of variation say BV ND(θ 1 , θ 2 , cθ 1 , cθ 2 , ϑ) with means θ 1 > 0 and θ 2 > 0, standard deviations cθ 1 and cθ 2 , and correlation coefficient ϑ using concomitants of order statistics.
Let (X i , Y i ), i = 1, 2, • • • , n be a random sample from an arbitrary bivariate distribution with probability density function (pdf) f (x, y).If the sample values on the variate X are ordered as X 1:n , X 2:n , • • • , X n:n , then the accompanying Y-observation in an ordered pair with X-observation equal to X r:n is called the concomitant of the r th order statistic X r:n and is denoted by Y [r:n] .There is extensive literature available on the application of concomitants of order statistics such as in: biological selection problems (Yoe & David, 1984), ocean engineering (Castillo, 1988), development of structural designs (Coles & Tawn, 1994).Harrel and Sen (1979) used concomitants of order statistics to estimate the correlation coefficient ϑ of a bivariate normal distribution.An excellent review of work on concomitants of order statistics is available in David and Nagaraja (2003).David (1973) considered the bivariate normal model say BV ND(θ 1 , θ 2 , φ 1 , φ 2 , ϑ) in which the variable Y is linked with the variable X through the regression model: where Z [r] denotes the particular error Z r associated with X r:n .Due to the independents of X r and Z r , we have, set of X r:n is independent of Z [r] .Yang (1977) showed that: and, where m(X r:n ) = E(Y/X r:n = x) and σ 2 (X r:n ) = Var(Y/X r:n = x).
2. Inference on θ 2 When c and ϑ are Known Using Concomitants of Order Statistics Y [n:n] be the concomitants of order statistics arising out of ordering ) be the vector of concomitants of order statistics.Using (1), ( 2) and (3), we obtain means and vaiances of , and are given by: ) and, where, Then by consider θ 2 as the location parameter of Y variable, a linear unbiased estimator of θ 2 based on concomitants of order statistics is given by (same argument that of Balakrishnan & Rao, 1998, p. 13): and, Since the X variable is symmetric about θ 1 , then by the same argument that of (David & Nagaraja, 2003, p. 188), Using ( 7), ( 8) and ( 9) reduces to: and, Introducing cθ 2 as the scale parameter of Y variable, a linear unbiased estimator of cθ 2 based on concomitants of order statistics is given by (same argument that of Balakrishnan & Rao, 1998, p. 13): and, Using ( 9), ( 12) and ( 13) reduces to: and, From ( 12) we can obtain another linear unbiased estimator θ * 2 of θ 2 based on concomitants of order statistics and is given by: and, using ( 9), ( 16) and ( 17) reduces to: and, where 1 is an n × 1 matrix.The BLUE θ2 of θ 2 is given in the following theorem.
U is an identity matrix of dimension n and V = ((V i, j )) then the BLUE of θ 2 is given by: and, ) be the vector of concomitants of order statistics.From (4) we have: where 1 is an n × 1 matrix and δ = ϑ(δ 1:n , δ 2:n , • • • , δ n:n ) .From ( 5) and ( 6) the dispersion matrix of Y [n] is obtained as: where , where ϑ and c are known, then ( 22) and ( 23) defines a generalised Gauss-Markov setup, the BLUE θ2 of θ 2 is given by: θ2 and, Using ( 9), ( 20) and ( 21) reduces to: and, This proves the theorem.
Remark 1 The estimator θ2 defined in ( 24) is a convex combination of the estimators θ2 and θ * 2 defined respectively in (10) and (18).

Proof.
Let ).Now we should find a such that Var(R) is minimum.That is we have to find a such that ∂Var(R) ∂a = 0. which implies: .
For this values of a, we have R = θ2 .Hence the proof.
We have calculated the coefficients of Y [i] in the BLUE θ2 , for different values of n, c and ϑ.Also we have computed the numerical values of Var( θ2 ) given in (11), Var(θ * 2 ) given in (19), Var( θ2 ) given in (25), the efficiency Var( θ2 ) of θ2 compared to θ2 , the efficiency Var( θ2 ) of θ2 compared to θ * 2 , for different values of n, c and ϑ, and are given in the following tables.It may be noted that in all the cases efficiency of our estimator θ2 is much better than that of the estimators θ2 and θ * 2 . Table and the relative efficiencies E 1 and E and the relative efficiencies E 1 and E and the relative efficiencies E 1 and E  4), ( 5) and ( 6) we have: ) )).From ( 27) and ( 28) we noted that Gauss-markov theorem does not applicable to derive the BLUE of ϑ.Hence we derive two linear unbiased estimators of ϑ, and are given below.
Theorem 2 Suppose W be an n dimensional coloumn vector and W Y * Then an estimator θ1 obtained by minimising W W subject to W Y * [n] is unbiased for ϑ is obtained as: and an estimator θ2 obtained by minimising W VW subject to W Y * [n] is unbiased for ϑ is obtained as: Also: and, The proof of the theorem is omitted, since it is similar to the proof of Chacko and Thomas (2008).
The estimators θ1 and θ2 given in ( 30) and (31) can also written as θ1 are two unbiased estimators of ϑ with variances given by ( 32) and (33) then θ1 is more better than θ2 if |ϑ| < Z, where Z = The proof of the theorem is omitted, since it is similar to the proof of Chacko and Thomas (2008).
We have calculated the values d i and e i contained in θ1 and θ2 , their variances and the interval (−Z, Z) in which θ1 is more better than θ2 for different values of n and are given in Table 4.
and the value of Z such that when ϑ ∈ (−Z,

Table 2 .
Coefficients of

Table 3 .
Coefficients of When c and θ 2 are Known Using Concomitants of Order StatisticsIn this part, we derive certain estimators for ϑ of a BV ND(θ 1 , θ 2 , cθ 1 , cθ 2 , ϑ) with known coefficient of variation by concomitants of order statistics.Take