New closed‐form efficient estimator for multivariate gamma distribution

Abstract

Maximum likelihood estimation is used widely in classical statistics. However, except in a few cases, it does not have a closed form. Furthermore, it takes time to derive the maximum likelihood estimator (MLE) owing to the use of iterative methods such as Newton–Raphson. Nonetheless, this estimation method has several advantages, chief among them being the invariance property and asymptotic normality. Based on the first approximation to the solution of the likelihood equation, we obtain an estimator that has the same asymptotic behavior as the MLE for multivariate gamma distribution. The newly proposed estimator, denoted as MLECE$$ {\mathrm{MLE}}_{\mathrm{CE}} $$ , is also in closed form as long as the n$$ \sqrt{n} $$ ‐consistent initial estimator is in the closed form. Hence, we develop some closed‐form n$$ \sqrt{n} $$ ‐consistent estimators for multivariate gamma distribution to improve the small‐sample property. MLECE$$ {\mathrm{MLE}}_{\mathrm{CE}} $$ is an alternative to MLE and performs better compared to MLE in terms of computation time, especially for large datasets, and stability. For the bivariate gamma distribution, the MLECE$$ {\mathrm{MLE}}_{\mathrm{CE}} $$ is over 130 times faster than the MLE, and as the sample size increasing, the MLECE$$ {\mathrm{MLE}}_{\mathrm{CE}} $$ is over 200 times faster than the MLE. Owing to the instant calculation of the proposed estimator, it can be used in state–space modeling or real‐time processing models.