Maximum likelihood estimation for survey data with informative interval censoring

Angel G Angelov,Magnus Ekström

doi:10.1007/s10182-018-00329-x

Angel G Angelov, Magnus Ekström

Open Access

https://doi.org/10.1007/s10182-018-00329-x

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Journal: AStA Advances in Statistical Analysis	Publication Date: Jul 17, 2018
Citations: 5	License type: open-access

Affiliation: Umeå University

Abstract

Interval-censored data may arise in questionnaire surveys when, instead of being asked to provide an exact value, respondents are free to answer with any interval without having pre-specified ranges. In this context, the assumption of noninformative censoring is violated, and thus, the standard methods for interval-censored data are not appropriate. This paper explores two schemes for data collection and deals with the problem of estimation of the underlying distribution function, assuming that it belongs to a parametric family. The consistency and asymptotic normality of a proposed maximum likelihood estimator are proven. A bootstrap procedure that can be used for constructing confidence intervals is considered, and its asymptotic validity is shown. A simulation study investigates the performance of the suggested methods.

Highlights

In questionnaire surveys respondents are often allowed to give an answer in the form of an interval
We proved the existence, consistency, and asymptotic normality of a proposed parametric maximum likelihood estimator
In comparison with the scheme used in a previous paper (Angelov and Ekström 2017), the new schemes do not involve exclusion of respondents and this leads to a smaller bias of the estimator as indicated by our simulation study

Summary

Introduction

In questionnaire surveys respondents are often allowed to give an answer in the form of an interval. A format that does not involve any pre-specified values is the respondent-generated intervals approach, suggested by Press and Tanur (2004a, b), where the respondent is asked to provide both a point value (a best guess for the true value) and an interval They employed Bayesian methods for estimating the parameters of the underlying distribution. Estimating the underlying distribution using SSI data, requires some generally untestable assumptions related to how the respondent chooses the interval To avoid such assumptions, Belyaev and Kriström (2012, 2015) introduced a novel two-stage approach.

Scheme A

Scheme B

Model and estimation

Asymptotic results

Simulation study

Conclusion

The partial derivatives