Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in GaGLM

The Gamma distribution based generalized linear model (
$Ga$
GLM) is a kind of statistical model feasible for the positive value of a non-stationary stochastic system, in which the location and the scale are regressed by the corresponding explanatory variables. This paper theoretically investigates the asymptotic properties of maximum likelihood estimates (MLE) of
$Ga$
GLM, which can benefit the further interval estimates, hypothesis tests and stochastic control design. First, the score function and the Fisher information matrix for
$Ga$
GLM are derived. Then, the Lyapunov condition is derived to ensure the asymptotic normality of the score function normalized by the Fisher information matrix. Based on this condition, the asymptotic normality of the MLE of
$Ga$
GLM is proven. Finally, a numerical example is given to testify the asymptotic properties obtained in the research. The numerical results indicate that the MLE of
$Ga$
GLM converged to a normal distribution as the number of sample measurements increased.