The multi-parameter estimation of discrete distribution without closed-form solutions by the US algorithm

Strong and steady convergence characterizes the upper-crossing/solution (US) algorithm, which is an effective method for identifying roots of a complicated nonlinear equation $h(\theta )=0$. Only the case where one parameter of a distribution function can be directly specified by another parameter is taken into account by the research that is currently available. However, whether this approach can be applied in multi-parameter scenarios where one parameter cannot be clearly represented by the other is an issue deserving of more investigation. In order to extend the applicability of the US algorithm, this article used the Type Ⅰ discrete Weibull distribution with two parameters as an example. It then combined the US algorithm with the first-derivative lower bound (FLB) function method to estimate the complex situation where two parameters cannot be expressed as each other. Simulation studies and empirical analysis demonstrated that the US algorithm performs more accurately and steadily than the traditional Newton method.