Conditional Inference in Parametric Models

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 Scheffé 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.

Long runs under a conditional limit distribution

“…Definition (29) shows that g u 1,n depends on the unknown parameter θ T . It can be seen that u 1,n is nearly sufficient for θ in g u 1,n in the sense that the value of g u 1,n (X k 1 ) does not vary when θ T is substituted by any other value θ of the parameter and the X i 's are generated under any density γ r T ,θ ′ (see [5]) this is indeed in agreement with the statement of Theorem 11. Hence on one hand , u 1,n can be used to obtain improved estimators of θ T and on the other hand, g u 1,n can be used to simulate samples distributed under a proxy of p u 1,n using any θ in lieu of θ T in (29), as is done in the following procedure:…”

“…Optimizing L η 0 (θ|X k 1 ) with respect to θ produces a consistent estimator of θ T . We refer to [5] for examples and discussion.…”

Statistics as a tool for decision making in agricultural and environmental experiments

“…During the last century, statistics revolutionized science by presenting useful models that modernized the research process in the direction of better research parameters, making it possible to guide decision making in a wide variety of areas. Statistical methods were developed as a mixture of science and logic for the solution and investigation of problems in various areas of human knowledge (Broniatowski & Caron, 2013;Garcia-Romero et al, 1995). With the advent of increasingly powerful personal computers, statistics has become more accessible to researchers in different fields.…”

Effect of pollution on physical and chemical water data: a multivariate statistical analysis

“…Statistics today has made a significant contribution to the decision-making process because much of what is produced is based on quantitative methods, and statistics is one of these areas. In the information and knowledge age, statistics uses mathematics to support professionals in business, government, and researchers (Broniatowski & Caron, 2014;Garcia-Romero et al, 1995). Today the use of statistics is widespread in private and public institutions.…”