#### Date of Award

Spring 1985

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

Ram C. Dahiya

#### Committee Member

Michael J. Doviak

#### Committee Member

John P. Morgan

#### Abstract

A calibration method substitutes for measurements, X(,i), that are accurate but impractical or costly, a set of measurements, Y(,i), that are less accurate but simpler or less costly. There are two general types of calibration methods. The classical approach in which once the calibration sample is drawn, the estimates of the X values for a given unit is found without any consideration of the distribution of X values for the other units to be measured. This corresponds best to the literal meaning of the word "calibration". Maximum likelihood estimation is the statistical formulation of the classical approach.

The second approach to calibration takes into account the distribution of X values for the units to be measured and attempts to minimize the mean squared deviations between the predicted and true X values. This leads to inverse regression and Bayesian prediction.

Methods for both classical and inverse calibration are given for linear univariate and multivariate, and quadratic univariate and multivariate models. Asymptotic means and variances of the X estimators are derived. Estimators unbiased to O(1/N) are given for X values and important parameters. Confidence regions are constructed using the unbiased estimator of X.

The results for calibration models are applied to a related set of tare weight models to obtain estimators and confidence regions for tare weights.

Methods for numerical solution are given for the nonlinear estimation equations that occur with quadratic models.

Simulation experiments were conducted to check on the validity of estimators and confidence regions. Over a wide range of X values, the unbiased classical estimators provide good estimates of X as measured by unbiasedness, mean square error and Pittman closeness. Within two standard deviation of the mean of the calibration sample, the inverse estimator will generally show a smaller MSE than the unbiased classical estimator. Simulations indicate that values of X for quadratic models should be confined to within two standard deviations of the mean of the calibration sample.

#### DOI

10.25777/xb5b-p136

#### Recommended Citation

McKeon, James J..
"Statistical Calibration Theory"
(1985). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/xb5b-p136

https://digitalcommons.odu.edu/mathstat_etds/95