There is a quantity X, a calibration parameter of some machinery, we are trying to evaluate.
A number of estimates for X are given:
{X} = {X1, X2, X3 ... Xn)
To find which one is the best estimate we use another quantity f(X) = A - X for which we know A (A is a contant).
We know that <f(X)> = 0 and we also know the standard deviation σ of f(X).
We could we set X = A but it's not such a good idea, it's not precise enough.
It is better to work with the set of values X1, X2, X3 ... and compute X like this:
X = Σ (Wi.Xi) / Σ(Wi)
where the Wi are appropriate weights.
To help us do this we know we should give low weights to those values of X from the set (X} for which f(X) departs extravagantly from zero.
We could therefore use the weighting function W(Xi) = exp(-|A-Xi]/σ) and all this will -hopefully- result in a better estimate for the value of X.
The problem is this: The quantity A sometimes comes from a source of information of high variance and sometimes from a source of information of lower variance.
Let's say σ1 and σ2 < σ1. Every time we do know whence the A values came from (the high variance source or the low variance source) but we don't know about next time.
Sometimes it's the one, sometimes the other. So how to express the weight function ? The above form I gave for W(Xi) seems inconsistent so what do I do ?
A number of estimates for X are given:
{X} = {X1, X2, X3 ... Xn)
To find which one is the best estimate we use another quantity f(X) = A - X for which we know A (A is a contant).
We know that <f(X)> = 0 and we also know the standard deviation σ of f(X).
We could we set X = A but it's not such a good idea, it's not precise enough.
It is better to work with the set of values X1, X2, X3 ... and compute X like this:
X = Σ (Wi.Xi) / Σ(Wi)
where the Wi are appropriate weights.
To help us do this we know we should give low weights to those values of X from the set (X} for which f(X) departs extravagantly from zero.
We could therefore use the weighting function W(Xi) = exp(-|A-Xi]/σ) and all this will -hopefully- result in a better estimate for the value of X.
The problem is this: The quantity A sometimes comes from a source of information of high variance and sometimes from a source of information of lower variance.
Let's say σ1 and σ2 < σ1. Every time we do know whence the A values came from (the high variance source or the low variance source) but we don't know about next time.
Sometimes it's the one, sometimes the other. So how to express the weight function ? The above form I gave for W(Xi) seems inconsistent so what do I do ?