Introduction to Probability and Statistics for Engineers and Scientists

9.8Weighted Least Squares 387

(b)The weighted sum of squares can also be seen as the relevant quantity to be minimized
by multiplying the regression equation

Y=α+βx+e

by

√

w. This results in the equation

Y

√

w=α

√

w+βx

√

w+e

√

w

Now, in this latter equation the error terme

√
whas mean 0 and constant variance.
Hence, the natural least squares estimators ofαandβwould be the values ofAandBthat
minimize

∑

i

(Yi

√

wi−A

√

wi−Bxi

√

wi)^2 =

∑

i

wi(Yi−A−Bxi)^2

(c)The weighted least squares approach puts the greatest emphasis on those data pairs
having the greatest weights (and thus the smallest variance in their error term). ■

At this point it might appear that the weighted least squares approach is not particularly
useful since it requires a knowledge, up to a constant, of the variance of a response at an
arbitrary input level. However, by analyzing the model that generates the data, it is often
possible to determine these values. This will be indicated by the following two examples.

EXAMPLE 9.8b The following data represent travel times in a downtown area of a certain
city. The independent, or input, variable is the distance to be traveled.

Distance (miles) .511.523456810

Travel time (minutes) 15.0 15.1 16.5 19.9 27.7 29.7 26.7 35.9 42 49.4

Assuming a linear relationship of the form

Y=α+βx+e

betweenY, the travel time, andx, the distance, how should we estimateαandβ? To utilize
the weighted least squares approach we need to know, up to a multiplicative constant, the
variance ofYas a function ofx. We will now present an argument that Var(Y) should be
proportional tox.

SOLUTION Letddenote the length of a city block. Thus a trip of distancexwill consist of
x/dblocks. If we letYi,i=1,...,x/d, denote the time it takes to traverse blocki, then
the total travel time can be expressed as

Y=Y 1 +Y 2 +···+Yx/d