Ben Hur

Technische Universiteit Eindhoven Department of Mathematics and Computer Science

Master’s thesis

Confidence Intervals in Inverse Regression
by M. Thonnard

Supervisor: Dr. A. Di Bucchianico

Eindhoven, January 2006

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Preface
This Masters thesis is the result of 10 months of research at the Department of Statistics, Probability and Operations Research at the Eindhoven Universityof Technology (TU/e). With this thesis I will end my study of Industrial and Applied Mathematics, and I will receive the degree of Master of Science. The subject of my research, Confidence Intervals in Inverse Regression, was introduced to me by dr. A. Di Bucchianico, who became my supervisor on this project. I would like to take some time to thank everybody who, in one way or another, helped meduring my 10 months of research. First of all there is my supervisor. He helped me every time I had a question or problem. And always in the most patient and precise way, in spite of his very busy schedule. I would also like to thank I. Schreur-Piet from the Chemical Engeneering and Chemistry Department for the datasets that she gave me. I would love to thank my parents who made it possible for meto begin (and finish) my study. Not only financially, but also for their support and love. For this support and love I also want to thank my sister and all my friends, Erik and Wannes in particular, who helped me with some problems, and Barbara for her support.

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Summary
In most regression problems we have to determine the value of Y corresponding to a given value of x. We will consider theinverse problem, which is called inverse regression or calibration. We will only investigate the simple linear regression model, which is a model with one regressor x that has a relationship with a response Y , which is a straight line. It is not always easy to measure these variables, the regressor x or the response Y . Assume we have known values of x and their corresponding Y values, which bothform a simple linear regression model and we have also an unknown value of x, such as x0 , which can not be measured and we can observe the corresponding value of Y , say Y0 . Then, x0 can be estimated and a confidence interval for x0 can be obtained. There are many possibilities to solve this problem. In the introduction we give the two most used solutions to estimate the unknown x0 , theclassical method and the inverse method, and we will explain them briefly. Firstly, we will give some simple examples of the calibration problem. We will consider a physical and a chemical example, because inverse regression is frequently used in physical and chemical engineering. First, we will investigate the linearity between the two variables x and Y . Afterwards, we give the classical and inverseestimator of an unknown value x0 . Secondly, we will consider the history of inverse regression and we will present a systematic overview. Afterwards, we will apply Graybill’s method from Graybill (1976) on the centred simple linear regression model to obtain an estimator for the unknown value of x0 and a 100(1 − α)% confidence interval for x0 . For this model we will calculate the Least Squaresestimators for the intercept, the slope and the unknown value of x0 . Because this estimator of x0 is biased, we also present Nasz´di’s estimator, which is approximately corrected for bias. Then, we o calculate the Maximum Likelihood estimator of σ 2 with the likelihood function, in which we substitute the Least Squares estimators of the intercept, the slope and the unknown value of x0 . This method isnot the full Maximum Likelihood method, but a plug-in approach. We will also correct this estimator of σ 2 for bias. We will also obtain a confidence interval for x0 . Graybill claimed that the confidence coefficient of this interval is less than 100(1 − α)%. Then, we will consider a second method to obtain an estimator for the unknown value of x0 , Brown’s profile likelihood. The profile likelihood...