Conformal Geometric Algebra in Stochastic Optimization Problems ...

146 CHAPTER 5. PARAMETER ESTIMATION
it is advisable to employ the non-parametric estimation. No assumption about the
distribution of the population is made. As a benefit, no wrong assumptions can
be made. The outcome is, comparable to interpolation, an approximation of the
pdf. The simplest example would be a histogram. Further to mention is the kernel
density estimation.
Subsequently, the key ideas concerning parametric estimation are to be conveyed.
Besides, the five quality criteria unbiasedness, minimal variance, consistency, efficiency,
sufficiency and robustness for estimators are briefly explained. After that a
small example on parameter estimation is given.
Note that random variables are labeled with an underset tilde, e.g. y ∽ . However,
most of the time it is dealt with realizations y of random variables y ∽ .
5.1.1 Point Estimation: a Motivation
A case in point is the estimation of the mean. Let X ∽ 1,...,X ∽ k be independent
and identically distributed (i.i.d.) random variables. Assume a normal distribution
with mean µ and variance σ 2 , denoted by X ∽ i ∼ N(µ; σ 2 ). Let x1 ...xk be a sample
from the random variables X ∽ 1 ...X ∽ k. Then the sample mean
x := 1
n
is an estimator for the population mean µ.
So generally, the result of estimating a parameter from one realization of the random
variables X ∽ 1,...,X ∽ k, k ≥ 1, is referred to as point estimation. If, in contrast,
the intervals are to be determined in which a parameter (or a function of which) lies
with a certain given probability, an interval estimation must be used. A point estimator
can be considered a zero-length interval estimator because this corresponds
to finding the (infinitesimal) interval of maximal probability. In the following, however,
it is only dealt with point estimation.
k�
i=1
Calculating the expectation E(X ∽ ) of the respective X ∽
E(X ∽ ) = E( 1
k
k�
i=1
X ∽ i) = 1
k
k�
i=1
xi
E(X ∽ i) = 1
k
k�
µ = µ = E(X
∽i), where i ∈ [1,k] � , shows that the estimator is unbiased, which means that no systematic
error is introduced. This is not natural: in order to estimate σ2 , the corrected
sample variance s2 x must be used, that is it must be divided by k − 1 rather than
by k
s 2 x := 1
k − 1
k�
(xi − x) 2 ,
otherwise an (asymptotically vanishing) bias would occur.
i=1
i=1