## Inference Functions for Semiparametric Models

Publikation: Working paperForskning

### Standard

arXiv, 2020.

Publikation: Working paperForskning

### Bibtex

@techreport{939651d944e74de08ba6666b4923ab3c,
title = "Inference Functions for Semiparametric Models",
abstract = "The paper discusses inference techniques for semiparametric models based onsuitable versions of inference functions. The text contains two parts. In thefirst part, we review the optimality theory for non-parametric models based onthe notions of path differentiability and statistical functionaldifferentiability. Those notions are adapted to the context of semiparametricmodels by applying the inference theory of statistical functionals to thefunctional that associates the value of the interest parameter to thecorresponding probability measure. The second part of the paper discusses thetheory of inference functions for semiparametric models. We define a class ofregular inference functions, and provide two equivalent characterisations ofthose inference functions: One adapted from the classic theory of inferencefunctions for parametric models, and one motivated by differential geometricconsiderations concerning the statistical model. Those characterisations yieldan optimality theory for estimation under semiparametric models. We present anecessary and sufficient condition for the coincidence of the bound for theconcentration of estimators based on inference functions and the semiparametricCram{\e}r-Rao bound. Projecting the score function for the parameter of intereston specially designed spaces of functions, we obtain optimal inferencefunctions. Considering estimation when a sufficient statistic is present, weprovide an alternative justification for the conditioning principle in acontext of semiparametric models. The article closes with a characterisation ofwhen the semiparametric Cram{\e}r-Rao bound is attained by estimators derivedfrom regular inference functions.",
keywords = "estimating functions, quasi inference functions, statistical functional differentiability, statistical differential geometry, non-parametric models",
author = "Rodrigo Labouriau",
year = "2020",
month = nov,
day = "15",
language = "English",
volume = "2011.07275",
publisher = "arXiv",
type = "WorkingPaper",
institution = "arXiv",

}

### RIS

TY - UNPB

T1 - Inference Functions for Semiparametric Models

AU - Labouriau, Rodrigo

PY - 2020/11/15

Y1 - 2020/11/15

N2 - The paper discusses inference techniques for semiparametric models based onsuitable versions of inference functions. The text contains two parts. In thefirst part, we review the optimality theory for non-parametric models based onthe notions of path differentiability and statistical functionaldifferentiability. Those notions are adapted to the context of semiparametricmodels by applying the inference theory of statistical functionals to thefunctional that associates the value of the interest parameter to thecorresponding probability measure. The second part of the paper discusses thetheory of inference functions for semiparametric models. We define a class ofregular inference functions, and provide two equivalent characterisations ofthose inference functions: One adapted from the classic theory of inferencefunctions for parametric models, and one motivated by differential geometricconsiderations concerning the statistical model. Those characterisations yieldan optimality theory for estimation under semiparametric models. We present anecessary and sufficient condition for the coincidence of the bound for theconcentration of estimators based on inference functions and the semiparametricCramèr-Rao bound. Projecting the score function for the parameter of intereston specially designed spaces of functions, we obtain optimal inferencefunctions. Considering estimation when a sufficient statistic is present, weprovide an alternative justification for the conditioning principle in acontext of semiparametric models. The article closes with a characterisation ofwhen the semiparametric Cramèr-Rao bound is attained by estimators derivedfrom regular inference functions.

AB - The paper discusses inference techniques for semiparametric models based onsuitable versions of inference functions. The text contains two parts. In thefirst part, we review the optimality theory for non-parametric models based onthe notions of path differentiability and statistical functionaldifferentiability. Those notions are adapted to the context of semiparametricmodels by applying the inference theory of statistical functionals to thefunctional that associates the value of the interest parameter to thecorresponding probability measure. The second part of the paper discusses thetheory of inference functions for semiparametric models. We define a class ofregular inference functions, and provide two equivalent characterisations ofthose inference functions: One adapted from the classic theory of inferencefunctions for parametric models, and one motivated by differential geometricconsiderations concerning the statistical model. Those characterisations yieldan optimality theory for estimation under semiparametric models. We present anecessary and sufficient condition for the coincidence of the bound for theconcentration of estimators based on inference functions and the semiparametricCramèr-Rao bound. Projecting the score function for the parameter of intereston specially designed spaces of functions, we obtain optimal inferencefunctions. Considering estimation when a sufficient statistic is present, weprovide an alternative justification for the conditioning principle in acontext of semiparametric models. The article closes with a characterisation ofwhen the semiparametric Cramèr-Rao bound is attained by estimators derivedfrom regular inference functions.

KW - estimating functions

KW - quasi inference functions

KW - statistical functional differentiability

KW - statistical differential geometry

KW - non-parametric models

M3 - Working paper

VL - 2011.07275

BT - Inference Functions for Semiparametric Models

PB - arXiv

ER -