a

Introduction

This section includes some fundamental findings that are pertinent to the current investigation. A quick review of literature on record values, concomitants of record values are recited. An introduction to the Morgenstern family of distributions is also displayed in this section. The study of record values in many ways parallels to the study of order statistics, indeed they are inextricably related. In any physical phenomena, occurrence of extreme situations attracts the attention of the experimenter. Record value theory developed in statistics; enter forcibly into the study of such situations. In these situations record values and record breaking data are also found useful to draw inferences about the underlying population distribution. Record values and associated statistics are of great importance in several real life problems involving weather, economics and sports data. The statistical study of record values started with Chandler (1952) [8] and now it has spread in different directions. Resnick (1973) [14] and Shorock (1973) [16] documented the asymptotic theory of records. Glick (1978) [9] provides a survey of literature of records. For a detailed discussion on the developments in the theory and application of record values, see Arnold et al. (1998) [4] , Nagaraja (1988) [12] Nevzorov and Balakrishnan (1998) and Ahsanullah (1995) [13] . In the bivariate set up, concomitants of order statistics and concomitants of record values assume their importance in place of the order statistics and the record values arising from a univariate population. Development of theory and applications of concomitants of record values also provide a very rich source of information on bivariate distributions which further helps to generate methods for the analysis of data arising from them. For a detailed description on the theory of concomitants of record values, see Ahsanullah (1994) [1] . For some latest developments in the theory and application on concomitants of record values, see Ahsanullah and Nevzorov (2000) [3] and Chacko and Thomas (2006) [5] . Let (X, Y) be a bivariate random variables with cumulative distribution function (cdf) F(x, y), which is absolutely continuous and let the corresponding joint pdf be denoted by f(x, y). Let ) , (

… be a sequence of independent observations arising from the distribution f(x, y). Suppose the upper (or lower) record values corresponding to the first component variate X in the sequence ) , (

alone are available. Then in the available literature, methods are developed to utilize the information contained in these record values to deal with the inference problems associated with random variable X (see Gulati and Padgett, 2003) [10] . But there are situation such as in animals or human beings where measurements on the units of a study variate Y is expensive or painful, but the measurement on an auxiliary variable X which is correlated with Y is very easy and the main interest lies on the study of those units having probably large (or Small) values on Y. In this case the study can be conducted by generating a sequence of independent X observations on the units, constructing the associated sequence of record values and then making the measurement of Y on those units which have record values on X. Then the resulting observations made on Y are known as concomitants of record values. A detailed account of applications of concomitants of record value in inference problems are available in Chacko and Thomas (2008) [6] .

Record value

Let

,... , 2 1 X X be a sequence of independent and identically distributed (iid) random variables having a common distribution function F(x) which is absolutely continuous and probability density function(pdf) f(x).An observation j X will be called an upper record value (or simply a record) if its value exceeds that of all previous observations. Thus

The first observation 1 X is taken as the initial record 1 R . The next record 2 R is the observation following 1 R which is greater than 1

R and so on. The records ,... , 2 1 R R as defined above are sometimes referred to as the sequence of upper records. Similarly, an observation j X will be called a lower record value if its value is less than that of all previous observations. If we write n R to denote the th n upper record value then its pdf is given by

If we write n L to denote the th n lower record value, then its pdf is given by

and the joint pdf of the th m and th n lower record values for m < n is given by, ) Y is given by (see Arnold et al, 1998, p.272) [4] .

where ) , ( 2 [11] . For a detailed discussion on the distribution theory and properties of concomitants of record values see, Ahsanullah (1994) [1] , Arnold et al (1998) [4] and Ahsanullah and Nevzorov (2000) [3] .

Morgenstern Family of Distributions

Morgenstern family M of bivariate distribution functions F(x, y) possessing a representation of the form,

The family M of bivariate distributions with distribution function F x,y (x, y) as given in (1.9) is also called in the literature as Farlie Gumbel Morgenstern (FGM) family of bivariate distributions. Several applications of concomitants of order statistics in estimating the parameters of Morgenstern Family of Distributions (MFD) are elucidated in Chacko and Thomas (2006) [5] . The system provides a very general expression of a bivariate distribution from which members can be derived by substituting expression of any desired set of marginal distributions. Since both the joint cdf and joint pdf are given in terms of marginal cdf and pdf, it is very easy to generate a random sample from a distribution belonging to MFD. Thus the members of this family can be used in simulation studies, especially when weak dependence between variates is of interest. Important Morgenstern family of distributions is Morgenstern type bivariate logistic distribution and Morgenstern type bivariate exponential distribution. Chacko and Thomas (2006) [5] estimated two parameter of a five parameter Morgenstern type bivariate logistic distribution using concomitants of record values. Chacko and Thomas (2011) [7] estimated one parameter of a three parameter Morgenstern type bivariate exponential distribution using concomitants of record values. Sajeev Kumar and Sumi (2022) [15] estimated two parameters of a five parameter Morgenstern type bivariate exponential distribution using concomitants of order statistics. Hence in this work our aim is to estimate two parameters of a five parameter Morgenstern type bivariate exponential distribution using concomitants of record values. Throughout this paper we assume concomitants of record values mean concomitants of upper record values.

Estimation of parameters 2  and 2  of Morgenstern type bivariate exponential distribution by concomitants of record values

The bivariate random variable (X, Y) is said to have a Morgenstern type bivariate exponential distribution (MTBED) its pdf is given by a

The pdf defined in (2.1) is an extension of the pdf of MTBED given in Chacko and Thomas (2011) [7] . Here we have introduced new location parameters 1  and 2  . Clearly the marginal distributions of X and Y variables are univariate exponential distributions. The joint pdf of standard MTBED is obtained by making the substitutions

Clearly the marginal distributions of each of V and W of (V,W) with pdf defined by (2.2) is standard exponential distributions given by the pdf's

be a sequence of independent observations drawn from (2.

2). Let

Since the marginal distribution of W of (V, W) is standard exponential distribution with pdf defined by (2.4), we have the following W is the largest order statistics of a random sample of size 2 drawn from the distribution of marginal random variable X. On substituting (2.7) and (2.8) in (2.9) and (2.10) we get, for

Thus the variances and covariances of concomitants of record values of a random sample of size n arising from (2.2 are given by for (

be a sequence of independent observations drawn from a population with pdf defined in (2.1). Clearly we

for i=1, 2,… Then by using (2.19), (2.20) and (2.21). We have for  involved in (2.1) and is given in the following theorem.

X i i be a sequence of independent observations drawn from the population with pdf defined by (2.1).

be the vector of concomitants of record values arising from (2.1). Let '

 respectively be the mean vector and dispersion matrix of vector of concomitants of record values arising from (2.2).

Then the BLUE's of the parameters 2

 and 2

Corresponding variances are given by Var

then by using (2.22) we can write

Also to compare the efficiency of our estimator 2  defined in (2.26), we have take the unbiased estimator

U are respectively the concomitants of first and th n upper record values arising from (2.1). We have evaluated

References

1. Ahsanullah M. Record values, random record models and concomitants, Journal of Statistical Research. 1994;28:89-109.
2. Ahsanullah M. Record Statistics. Nova Science Publishers, New York, 1995.
3. Ahsanullah M, Nevzorov VB. Some distributions of induced record values. Biometrical journal. 2000;42:1069-1081.
4. Arnold BC, Balakrishnan N, Nagaraja HN. Records. Wiley, New York, 1998.
5. Chacko M, Thomas PY. Concomitants of Record values arising from Morgenstern type bivariate logistic distribution and some of their applications in parameter estimation. Metrica. 2006;64:317-331.
6. Chacko M, Thomas PY. Estimation of parameters of bivariate normal distribution using concomitants of record values. Statistical papers. 2008;49:263-275.
7. Chacko M, Thomas PY. Estimation of parameter of Morgenstern Type Bivariate Exponential Distribution using concomitants of order statistics. Statistical methodology. 2011;8(4):363-376.
8. Chandler. The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B. 1952;14:220228.
9. Glick N. Breaking records and breaking boards. Amer. Math. Monthly. 1978;85:2-26.
10. Gulati S, Padgett WJ. Parametric and Nonparametric Inference from Record breaking data. Lecture notes in statistics, Springer, New York, 2003.
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12. Nagaraja HN. Record values and related statistics-A review. Communications in Statistics-Theory and Methods. 1988;17:2223-2238.
13. Nevzorov VB, Balakrishnan N. A record of records. In Handbook of statistics (Eds. N.Balakrishnan and C.R.Rao) 16, 515570, North-Holland, Amsterdam. The Netherlands, 1998.
14. Resnick SI. Limit laws for record values. Stochastic Processes and Their Applications, 1973, 67-82.
15. Sajeevkumar NK, Sumi AR. Use of concomitants of order statistics in the estimation of parameters 2  and 2  of Morgenstern type bivariate exponential distribution. The Indian society for probability and statistics published online, 2022. https://doi.org/10.1007/s41096-022-00132-7.
16. Shorrock RW. Record values and inter-record times. Journal of Applied Probability. 1973;10:543-555.