Parameter Estimation in Multivariate Gamma Distribution
Abstract
Multivariate gamma distribution finds abundant applications in stochastic modelling, hydrology and reliability. Parameter estimation in this distribution is a challenging one as it involves many parameters to be estimated simultaneously. In this paper, the form of multivariate gamma distribution proposed by Mathai and Moschopoulos [10] is considered. This form has nice properties in terms of marginal and conditional densities. A new method of estimation based on optimal search is proposed for estimating the parameters using the marginal distributions and the concepts of maximum likelihood, spacings and least squares. The proposed methodology is easy to implement and is free from calculus. It optimizes the objective function by searching over a wide range of values and determines the estimate of the parameters. The consistency of the estimates is demonstrated in terms of mean, standard deviation and mean square error through simulation studies for different choices of parameters.References
. Balakrishnan N & Chin-Diew Lai (2009) Continuous Bivariate Distributions. Second edition. Springer Publications.
. Bo Ranneby (1984) Maximum Spacing Method. An Estimation Method Related to the Maximum Likelihood Method. Scandinavian Journal of Statistics 11(2): 93-112.
. Cheng RCH, Amin NAK (1983) Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society (B) 45(3):394-403.
. Cheriyan KC (1941) A bivariate correlated gamma-type distribution function. Journal of the Indian Mathematical Society 5:133-144.
. Clarke RT (1980) Bivariate Gamma distributions for extending annual streamflow records from precipitation: some large-sample results. Water Resources Research 16(5):863-870.
. Fisher RA (1922) On the Mathematical Foundations of Theoretical Statistics. Philosophical Transactions of Royal Society London (A) 222:309-368.
. Jensen DR (1970) The joint distribution of quadratic forms and related distributions. Australian Journal of Statistics 12:13–22.
. Johnson NL, Samuel Kotz, Balakrishnan N (1994) Continuous Univariate Distributions, Volume 1. Wiley Publications.
. Kibble WF (1941) A two-variate gamma type distribution. Sankhya 5: 137–150.
. Mathai AM & Moschopoulos PG (1992) A form of Multivariate Gamma Distribution. Annals of the Institute of Statistical Mathematics, 44 (1): 97-106.
. Mathai AM, Moschopoulos PG (1991) On a multivariate gamma. Journal of Multivariate Analysis 39:135–153.
. McKay AT (1934) Sampling from batches. Journal of the Royal Statistical Society (B) (1): 207– 216.
. T. Princy, An extended compound gamma model and application to composite fading channels, Statistics, Optimization and Information Computing, vol. 3, pp. 42–53, 2015.
. Royen T (1991) Expansions for the multivariate chi-square distribution. Journal of Multivariate Analysis 38:213–232.
. Samuel Kotz, Balakrishnan N & Johnson NL (2000) Continuous Multivariate Distributions, Volume 1: Models and Applications, Second edition. Wiley-Interscience Publications.
. Sarmanov IO (1970a) Gamma correlation process and its properties. Doklady Akademii Nauk, SSSR 191:30–32.
. Sarmanov IO (1970b) An approximate calculation of correlation coefficients between functions of dependent random variables. Mathematical Notes, Academy of Sciences of USSR 7:373–377.
. Yue S, Oarda TBMJ, Bobee B (2001) A review of bivariate gamma distributions with hydrological application. Journal of Hydrology 246:1-18.
. Yue S (2001) A bivariate gamma distribution for use in multivariate flood frequency analysis. Hydrological Processes 15: 1033-1045.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
