Conditional inference in parametric models

TL;DR

This paper introduces a novel conditional inference method using simulation conditioned on data statistics, providing explicit approximations for conditional likelihood and demonstrating invariance properties, with applications to exponential families and comparisons to bootstrap methods.

Contribution

It proposes a new simulation-based approach to conditional inference, including explicit likelihood approximations and invariance results, extending to exponential families with nuisance parameters.

Findings

The approximated conditional likelihood remains invariant when the conditioning statistic is sufficient.

The proposed Rao-Blackwellisation improves inference accuracy.

Monte Carlo tests validate the method's effectiveness.

Abstract

This paper presents a new approach to conditional inference, based on the simulation of samples conditioned by a statistics of the data. Also an explicit expression for the approximation of the conditional likelihood of long runs of the sample given the observed statistics is provided. It is shown that when the conditioning statistics is sufficient for a given parameter, the approximating density is still invariant with respect to the parameter. A new Rao-Blackwellisation procedure is proposed and simulation shows that Lehmann Scheff\'{e} Theorem is valid for this approximation. Conditional inference for exponential families with nuisance parameter is also studied, leading to Monte carlo tests. Finally the estimation of the parameter of interest through conditional likelihood is considered. Comparison with the parametric bootstrap method is discussed.