Monotone local linear estimation of transducer functions

Hughes, D ORCID: 0000-0002-1287-9994 (2014) Monotone local linear estimation of transducer functions. PhD thesis, University of Liverpool.

Text HughesDav_Sep2013_2007866.pdf - Unspecified Available under License Creative Commons Attribution. Download (6MB)

Abstract

Local polynomial regression has received a great deal of attention in the past. It is a highly adaptable regression method when the true response model is not known. However, estimates obtained in this way are not guaranteed to be monotone. In some situations the response is known to depend monotonically upon some variables. Various methods have been suggested for constraining nonparametric local polynomial regression to be monotone. The earliest of these is known as the Pool Adjacent Violators algorithm (PAVA) and was first suggested by Brunk (1958). Kappenman (1987) suggested that a non-parametric estimate could be made monotone by simply increasing the bandwidth used until the estimate was monotone. Dette et al. (2006) have suggested a monotonicity constraint which they call the DNP method. Their method involves calculating a density estimate of the unconstrained regression estimate, and using this to calculate an estimate of the inverse of the regression function. Fan, Heckman and Wand (1995) generalized local polynomial regression to quasi-likelihood based settings. Obviously such estimates are not guaranteed to be monotone, whilst in many practical situations monotonicity of response is required. In this thesis I discuss how the above mentioned monotonicity constraint methods can be adapted to the quasi-likelihood setting. I am particularly interested in the estimation of monotone psychometric functions and, more generally, biological transducer functions, for which the response is often known to follow a distribution which belongs to the exponential family. I consider some of the key theoretical properties of the monotonised local linear estimators in the quasi-likelihood setting. I establish asymptotic expressions for the bias and variance for my adaptation of the DNP method (called the LDNP method) and show that this estimate is asymptotically normally distributed and first{-}order equivalent to competing methods. I demonstrate that this adaptation overcomes some of the problems with using the DNP method in likelihood based settings. I also investigate the choice of second bandwidth for use in the density estimation step. I compare the LDNP method, the PAVA method and the bandwidth method by means of a simulation study. I investigate a variety of response models, including binary, Poisson and exponential. In each study I calculate monotone estimates of the response curve using each method and compare their bias, variance, MSE and MISE. I also apply these methods to analysis of data from various hearing and vision studies. I show some of the deficiencies of using local polynomial estimates, as opposed to local likelihood estimates.

Item Type:	Thesis (PhD)
Additional Information:	Date: 2014-09 (completed)
Subjects:	?? HA ?? ?? QA ??
Depositing User:	Symplectic Admin
Date Deposited:	06 Aug 2015 12:50
Last Modified:	17 Dec 2022 00:47
DOI:	10.17638/02007866
Supervisors:	Zychaluk, K Clancy, D
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2007866