Estimating the dependence parameter of Farlie-Gumbel-Morgenstern type bivariate gamma distribution using ranked set sampling

Yusuf Can Sevil; Tuğba Yıldız

doi:10.3390/IOCMA2023-14419

Abstract:

The Farlie-Gumbel-Morgenstern (FGM) is a flexible family that has been widely used to model weak dependence between the random variables. The family was introduced by Morgenstern (1956), Gumbel (1960) and Farlie (1960). Characteristic properties of FGM family, see Nelsen (1999). In the literature, we can see some studies that investigate the FGM family when marginals have uniform, normal, exponential, logistic or gamma distributions, see Bekci and Bairamov (1999), Scaria and Nair (1999), Smith and Moffatt (1999), Abo-Eleneen and Nagaraja (2002), Ucer and Yildiz (2012), Yildiz and Ucer (2017). Balakrishnan and Lai (2009) provide different bivariate distributions in the FGM family.

Ranked set sampling (RSS) was introduced by McIntyre (1952) as cost effective sampling method. One of its useful properties is that it can be used to collect data when exact measurements of the sampling units are either difficult and/or expensive. In the literature, it can be seen many applications of RSS in areas such as environmental, ecological, agricultural, biological and medical. Using RSS and its modifications, some authors dealt with the estimation of the dependence parameter. Stokes (1980) investigated maximum likelihood (ML) estimator of the association parameter for bivariate normal distribution. Modarres and Zheng (2004) suggested ML estimator of dependence parameter of bivariate normal distribution and bivariate extreme value distribution using RSS. Estimation of the association parameter based on bivariate RSS proposed by Al-Saleh and Samawi (2005) when the ranked set samples are obtained from bivariate normal distribution. Recently, Sevil and Yildiz (2022) considered the estimation problem of the association parameter for Gumbel (Type Ⅱ) bivariate exponential distribution.

In the present work, we consider FGM type bivariate gamma distribution that is developed by D’Este (1981) and Gupta and Wong (1984). We investigate maximum likelihood (ML) estimators based on simple random sampling (SRS) and RSS. We provide algorithms to generate sampling data from FGM type bivariate gamma distribution. Finally, we present biases and efficiency of the ML estimator based on RSS with respect to the ML estimator based on SRS.

Keywords: Ranked set sampling; algorithm for sampling data; order statistics; dependence parameter; FGM family; maximum likelihood estimation

References

Abo-Eleneen Z, Nagaraja H (2002) Fisher information in an order statistic and its concomitant. Annals of the Institute of Statistical Mathematics 54:667-680.

Al-Saleh MF, Samawi HM (2005) Estimation of the correlation coefficient using bivariate ranked set sampling with application to the bivariate normal distribution. Communication in Statistics-Theory and Methods 34:875-889.

Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer-Verlag.

Bekci M, Bairamov I (1999) Concomitants of order statistics in FGM type bivariate uniform distributions. ISTATISTIK, Journal of Turkish Statistical Association 2:135-144.

D’Este GA (1981) A Morgenstern-type bivariate gamma distribution. Biometrika 68:339-340.

Farlie DJ (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47:307-323.

Gumbel EJ (1960) Bivariate exponential distributions. Journal of the American Statistical Association 55:698-707.

Gupta AK, Wong C (1984) On a Morgenstern-type bivariate gamma distribution. Metrika 31:327-332.

McIntyre (1952) A method for unbiased selective sampling, using ranked set sampling. Australian Journal of Agricultural Research 3:385-390.

Modarres R, Zheng G (2004) Maximum likelihood estimation of dependence parameter using ranked set sampling. Statistics & Probability Letters 68:315-323.

Morgenstern D (1956) Einfache Beispiele zweidimensionaler Verteilung. Mitteislingsblatt für Mathematische Statistik 8:234-235.

Nelsen RB (2007) An introduction to copulas. Springer Science & Business Media.

Scaria J, Nair U (1999) On concomitants of order statistics from Morgenstern family. Biometrical Journal 41:483-489.

Smith MD, Moffatt PG (1999) Fisher’s information on the correlation coefficient in bivariate logistic models. Australian & New Zealand Journal of Statistics 41:315-330.

Sevil YC, Yildiz TO (2022) Gumbel’s bivariate exponential distribution: estimation of the association parameter using ranked set sampling. Computational Statistics 37:1695-1726.

Stokes SL (1980) Inference on the correlation coefficient in bivariate normal populations from ranked set samples. Journal of the American Statistical Association 75:989-995.

Ucer BH, Yildiz TO (2012) Estimation and goodness-of-fit procedures for Farlie-Gumbel-Morgenstern bivariate copula of order statistics. Journal of Statistical Computation and Simulation 82:137-147.

Yildiz TO, Ucer BH (2017) Fisher information of dependence in progressive type Ⅱ censored order statistics and their concomitants. International Journal of Applied Mathematics & Statistics 56:1-10.