Behaviour of the Monotone Single Index Model Under Repeated Measurements

Abstract

The generalized linear model is an important method in the statistical toolkit. The isotonic single index model can be thought of as a further generalization whereby the link function is assumed to be monotone non-decreasing as opposed to known and fixed. Such a shape constraint is quite natural in many statistical problems, and is fulfilled by the usual generalized linear models. In this paper we consider inference in this model in the setting where repeated measurements of predictor values and associated responses are observed. This setting is encountered in medical studies and is very different from the one considered in the classical monotone single index model studied in the literature. Here, we use nonparametric maximum likelihood estimation to infer the unknown regression vector and link function. We present a detailed study of finite and asymptotic properties of this estimator and propose goodness-of-fit tests for the model. Through an extended simulation study, we show that the model has competitive predictive performance. We illustrate our estimation approach using a Leukemia data set.

Existence, uniqueness and strong consistency of the maximum likelihood estimator for a model of accidents frequencies

“…where ℓ(θ) is the log-likelihood function. Although the maximum likelihood method is generally efficient in most cases, some statistical models may suffer from the non-existence of the MLE [4,8,9,18] or its non-uniqueness [1,15,16].…”