Fundamentals of Probability and Statistics for Engineers

Let us note in this example that, since x 2 x^21 , matrix C is constrained in that
its elements in the third column are the squared values of their corresponding
elements in the second column. It needs to be cautioned that, for high-order
polynomial regression models, constraints of this type may render matrix CTC
ill-conditioned and lead to matrix-inversion difficulties.

Reference

Rao, C.R., 1965, Linear Statistical Inference and Its (^) Applications, John Wiley & Sons
Inc., New York.

Further Reading

Some additional useful references on regression analysis are given below.

Anderson, R.L., and Bancroft, T.A., 1952, Statistical Theory in Research^ ,McGraw-Hill,
New York.
Bendat, J.S., and Piersol, A.G., 1966, Measurement and Analysis of RandomData , John
Wiley & Sons Inc., New York.
Draper, N., and Smith, H., 1966, Applied Regression Analysis John , Wiley & Sons Inc.,
New York.
Graybill, F.A., 1961, An Introduction to Linear Statistical Models ,Volume 1. McGraw-
H ill, N ew York.

Problems

11.1 A special case of simple linear regression is given by

Determine:

(a) The least-square estimator ;

(b) The mean and variance of

(c) An unbiased estimator for^2 , the variance of Y.

11.2 In simple linear regression, show that the maximum likelihood estimators for and
are identical to their least-square estimators when Y is normally distributed.

11.3 Determine the maximum likelihood estimator for variance^2 of Y in simple linear
regression assuming that Y is normally distributed. Is it a biased estimator?

11.4 Since data quality is generally not uniform among data points, it is sometimes
desirable to estimate the regression co efficients by minimizing the sum of weighted
squared residuals; that is, and in simple linear regression are found by minimizing

Linear Models and Linear Regression 359

ˆ

Yˆ x‡E:

B^

B^





^ ^

Xn

iˆ 1

wie^2 i;

for

;