General quadratic distance methods for discrete distributions definable recursively

Quadratic distance (QD) methods for inference and hypothesis testing are developed for discrete distributions definable recursively. The methods are general and applicable to many families of discrete distributions including those with complicated probability mass functions (pmfs). Even if no explicit expression for the pmf of some distributions exists, QD methods are relatively simple to implement: the QD estimator can be computed numerically using a non-linear least-squares method. The asymptotic properties of the QD estimator are studied. Test statistics for goodness-of-fit are formulated and shown to follow asymptotically a chi-square distribution under the null hypothesis. Estimation and model testing are treated in a unified way. Simulation results presented indicate that the QDE protects against a certain form of mis-specification of the distribution, which makes the maximum likelihood estimator (MLE) biased, while keeping the QDE unbiased.