Parameter estimation based on differential equations of empirical transforms

Abstract

A new class of robust parameter estimators is proposed based on integral transforms of probability distributions, such as the characteristic function or the Laplace transform. In the past, estimators of this type were mostly constructed as minimizers of squared distances between the model transform (of the assumed distribution) and the empirical transform (computed from the sample), e.g. [1], [2] or [3].

Here, instead of using the above distances, we construct estimators by minimizing the L2 norm of ’empirical’ differential equations satisfied by the transforms. Such functionals have been already used in literature but for goodness-of-fit testing, see e.g. [4], [5]. The motivation for applying the equations in the estimation context is that it often leads to computationally attractive and reliable estimators, having closed-form expressions and being robust to outliers.

We address issues of consistency and asymptotic normality and derive influence functions of the estimators. We show that a good trade-off between robustness and efficiency requires data-driven weight functions in the L2 statistical functionals. In simulations, we study finite sample performance in pure and contaminated models, making comparisons with other kinds of estimators.