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Science Forum Index » Statistics - Math Forum » Standard Error...
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 Posted: Tue Jul 08, 2008 7:04 am |
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Guest
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Standard Error problem. It is probably written somewhere is
statistical
theory, but I haven't found anything yet
X = X_1,X_2,...,X_n are a vector of random variables
x = x_1,x_2,...,x_n are a vector of estimates of X
e = e_1,e_2,...,e_n are a vector of the std errors of the estimates
If, Y = f(X), and its estimate is y = f(x)
Then, what is the standard error of y.
I don't think it is f(e), or am I wrong?
Thanks, Richard |
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| Jack Tomsky... |
| Posted: Tue Jul 08, 2008 8:13 am |
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Quote: Standard Error problem. It is probably written
somewhere is
statistical
theory, but I haven't found anything yet
X = X_1,X_2,...,X_n are a vector of random variables
x = x_1,x_2,...,x_n are a vector of estimates of X
e = e_1,e_2,...,e_n are a vector of the std errors of
the estimates
If, Y = f(X), and its estimate is y = f(x)
Then, what is the standard error of y.
I don't think it is f(e), or am I wrong?
Thanks, Richard
By expanding Y = f(X) up to the linear terms, we have as an approximation
Var(Y) ~~ Sum[(df/dx_i)^2*Var(x_i)] + Sum[Sum[(df/dx_i)*(df/dx_j)*Cov(x_i,x_j)]],
where the sums are from i,j = 1, ..., n and the partial derivatives are evaluated at the expectations of X_i.
If the covariances are all zero, then it reduces to
Var(Y) ~~ Sum[(df/dx_i)^2*Var(x_i)] = Sum[(df/dx_i)^2*e_i^2]
Jack |
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