## Retrospective Theses and Dissertations

Dissertation

1987

#### Degree Name

Doctor of Philosophy

Statistics

#### Abstract

Let an observed random vector Z(,t) be represented as Z(,t) = z(,t)('0) + (epsilon)(,t), where z(,t)('0) is a fixed unobservable true value and (epsilon)(,t) is a stochastic measurement error. According to the nonlinear functional measure- ment error model, the true values z(,t)('0) are assumed to satisfy the nonlinear relation f(z(,t)('0); (beta)('0)) = 0, where (beta)('0) is a fixed but unknown vector of parameters. The measurement errors are assumed to be independently distributed with zero mean and covariance matrix (SIGMA)(,(epsilon)(epsilon)) = (sigma)('2)(UPSILON)(,(epsilon)(epsilon)), where (sigma)('2) is a positive scalar and (UPSILON)(,(epsilon)(epsilon)) is a fixed positive definite matrix. Under the model, maximum likelihood estimators of (beta)('0) and z(,t)('0) can be constructed for a given value of (UPSILON)(,(epsilon)(epsilon)). Properties of estimators of (beta)('0) and z(,t)('0) are presented for correctly specified (UPSILON)(,(epsilon)(epsilon)) and for (UPSILON)(,(epsilon)(epsilon)) misspecified;A weighted least squares estimator for (UPSILON)(,(epsilon)(epsilon)), which uses the model residuals is presented. The estimator of (UPSILON)(,(epsilon)(epsilon)) is shown to be consistent and asymptotically normally distributed for a sequence in which the sample size becomes larger and (sigma)('2) becomes smaller as the index of the sequence increases. A test of specification for (UPSILON)(,(epsilon)(epsilon)) is constructed using the least squares estimator of (UPSILON)(,(epsilon)(epsilon)). The limiting behavior of the test statistic is derived and the test is investigated in a Monte Carlo study;A second estimator of (UPSILON)(,(epsilon)(epsilon)) is derived by the method of maximum likelihood, assuming normally distributed measurement errors. The maximum likelihood estimator is shown to be consistent and its limiting distribution is derived. A likelihood ratio test for correct specification of (UPSILON)(,(epsilon)(epsilon)) is introduced and its limiting behavior is discussed.

#### DOI

https://doi.org/10.31274/rtd-180813-13063

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

Daniel J. Schnell

en

AAI8716815

application/pdf

199 pages

COinS