Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.7 HYPOTHESIS TESTING

however, such problems are best solved using one of the many commercially

available software packages.

One begins by making a first guessa^0 for the values of the parameters. At this

point in parameter space, the components of the gradient∇χ^2 will not be equal

to zero, in general (unless one makes a very lucky guess!). Thus, for at least some

values ofi, we have

∂χ^2

∂ai

∣

∣

∣

∣

a=a^0

=0.

Our aim is to find a small incrementδain the values of the parameters, such that

∂χ^2

∂ai

∣

∣

∣

∣

a=a^0 +δa

= 0 for alli. (31.104)

If our first guessa^0 were sufficiently close to the true (local) minimum ofχ^2 ,

we could find the required incrementδaby expanding the LHS of (31.104) as a

Taylor series abouta=a^0 , keeping only the zeroth-order and first-order terms:

∂χ^2

∂ai

∣

∣

∣

∣

a=a^0 +δa

≈

∂χ^2

∂ai

∣

∣

∣

∣

a=a^0

+

∑M

j=1

∂^2 χ^2

∂ai∂aj

∣

∣

∣

∣

a=a^0

δaj. (31.105)

Setting this expression to zero, we find that the incrementsδajmay be found by

solving the set ofMlinear equations

∑M

j=1

∂^2 χ^2

∂ai∂aj

∣

∣

∣

∣

a=a^0

δaj=−

∂χ^2

∂ai

∣

∣

∣

∣

a=a^0

.

It most cases, however, our first guessa^0 will not be sufficiently close to the true

minimum for (31.105) to be an accurate approximation, and consequently (31.104)

will not be satisfied. In this case,a^1 =a^0 +δais (hopefully) an improved guess

at the parameter values; the whole process is then repeated until convergence is

achieved.

It is worth noting that, when one is estimating several parametersa,the

functionχ^2 (a)maybeverycomplicated. In particular, it may possess numerous

local extrema. The procedure outlined above will converge to the local extremum

‘nearest’ to the first guessa^0. Since, in fact, we are interested only in the local

minimum that has the absolute lowest value ofχ^2 (a), it is clear that a large part

of solving the problem is to make a ‘good’ first guess.

31.7 Hypothesis testing

So far we have concentrated on using a data sample to obtain a number or a set

of numbers. These numbers may be estimated values for the moments or central

moments of the population from which the sample was drawn or, more generally,

the values of some parametersain an assumed model for the data. Sometimes,