Parameter and quantile estimation for the three-parameter gamma distribution based on statistics invariant to unknown location

Abstract The three-parameter gamma distribution is widely used as a model for distributions of life spans, reaction times, and for other types of skewed data. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter gamma distribution, which avoids the problem of unbounded likelihood, based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared to other prominent methods in terms of bias and mean squared error. Finally, we present two illustrative examples.

[1]  P. Billingsley,et al.  Probability and Measure , 1980 .

[2]  L. R. Shenton,et al.  Problems with Maximum Likelihood Estimation and the 3 Parameter Gamma Distribution , 2002 .

[3]  Masatake Mori,et al.  Double Exponential Formulas for Numerical Integration , 1973 .

[4]  Narayanaswamy Balakrishnan,et al.  Simple efficient estimation for the three-parameter gamma distribution , 2000 .

[5]  Anthony J. Jakeman,et al.  A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE‐PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS , 1991 .

[6]  Betty Jones Whitten,et al.  Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution , 1982 .

[7]  A. Clifford Cohen,et al.  Parameter estimation in reliability and life span models , 1988 .

[8]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[9]  W. Nelson.  Statistical Methods for Reliability Data , 1998 .

[10]  L. R. Shenton,et al.  Properties of Estimators for the Gamma Distribution , 1987 .

[11]  Wan-Kai Pang,et al.  A simulation based approach to the parameter estimation for the three-parameter gamma distribution , 2004, Eur. J. Oper. Res..

[12]  G. Tzavelas.  Sufficient Conditions for the Existence of a Solution for the Log-Likelihood Equations in the Three-Parameter Gamma Distribution , 2008 .

[13]  Richard L. Smith.  Maximum likelihood estimation in a class of nonregular cases , 1985 .

[14]  Wayne Nelson,et al.  Applied life data analysis , 1983 .

[15]  Betty Jones Whitten,et al.  Modified Moment Estimation for the Three-Parameter Weibull Distribution , 1984 .

[16]  P. Sen,et al.  Order statistics and inference : estimation methods , 1992 .

[17]  Russell C. H. Cheng,et al.  Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin , 1983 .

[18]  D. Griffiths.  Interval Estimation for the Three‐Parameter Lognormal Distribution Via the Likelihood Function , 1980 .

[19]  A. Cohen,et al.  Progressively Censored Sampling in the Three-Parameter Gamma Distribution , 1977 .

[20]  Hideo Hirose,et al.  Parameter estimation for the 3-parameter Gamma distribution using the continuation method , 1998 .

[21]  Charles E. Antle,et al.  Discrimination Between the Log-Normal and the Weibull Distributions , 1973 .

[22]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[23]  Hideo Hirose,et al.  Maximum likelihood parameter estimation in the three-parameter gamma distribution , 1995 .

[24]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[25]  Russell C. H. Cheng,et al.  Corrected Maximum Likelihood in Non‐Regular Problems , 1987 .

[26]  Peter Hall,et al.  Bayesian likelihood methods for estimating the end point of a distribution , 2005 .

[27]  J. Lawless.  Statistical Models and Methods for Lifetime Data , 2002 .

[28]  Albert H. Moore,et al.  Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples , 1965 .