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7- Estimation of Parameters
The problem of estimating parameters is a decision problem
where one wishes to know some aspect of the state of nature. The
"action" is usually: to assign a value to the parameter. In geodesy,
the parameters to be estimated are e.g. the means of the observation
variates and (or) the standard deviations of their distributions.
The loss associated with a certain action is some function of the
difference between the estimate and the parameter to be estimated.
The strategy is: the estimator, which may e.g. be a least-squares
estimator. The standard deviation of the estimator can often be
used as an indication of the quality of the estimateif the standard
deviation is not known, it may in its turn be estimated and used
to construct a confidence interval. The set of available actions
(estimates) is the same as the set of possible states of nature (para
meter values).
If we denote the parameter to be estimated by 9, the estimator
by T and the observation variate by x, then T is a function of x:
in which C(6) represents some function of 0.
The expected regret, or the risk function when the function t
is used to compute the estimator, is
where Eq denotes the expectation if 0 is the "true value" of the
parameter. Chernoff and Moses [3] give the following motivation
for a regret function of the form (7.1):
1. Many smooth functions which are zero for T o can be
approximated by a quadratic function, especially for values
of T in the neighbourhood of 0.
2. Mathematically, this function is easy to work with.
3. Many estimators have distributions which are approximately
normal, and whose mean is 0. Their distribution is then
entirely determined by the variance, and the smaller the
variance, the better will be the estimator, whatever may be
the actual regret function (provided the regret increases
with |T-0|
Since (7.2) is small when Eq{(T-())2} is small, in the selection of
an estimator one will be interested in
T t(x)
A possible form of the regret function is
r(Q,T) C(0) (T-0)2
(7-i)
R(V,t) C(0) Eq{(T- 0)2}
(7-2)
£6{(T-0)2} a2r (E{T} - 0)2
which might be called a measure for the accuracy of the estimator.
