Hello,
I am building a test which reject the null hypothesis (H0) when the sum of
the observations exceed a value c, where c is calculated, as usually, such
that, under H0,
P[X1+X2+...Xn>c] = alpha.
The Xis are positive and independent random variables and the probability
density function (pdf) of Xi is proportional to
f(di+sqrt(c)) / sqrt(c)
where di is a constant specific to Xi, f(u) is the pdf of the standard
normal distribution (mean=0 and SD=1) and sqrt(c) is the square root of c.
If (d1, d2, .., dn) was equal to (0, 0, .., 0), the Xis would be independent
chi-square random variables with 1 degree of freedom. Therefore X1+X2+...Xn
would be a chi-square random variable with n degrees of freedom, and the
determination of c such that P[X1+X2+...Xn>c] = alpha would be very easy.
Unfortunately, (d1, d2, .., dn) is not equal to (0, 0, .., 0), and I don't
know the pdf of X1+X2+...Xn.
My question is:
Does somebody know the pdf of X1+X2+...Xn when (d1, d2, .., dn) is
not equal to (0, 0, .., 0) ?
Otherwise I will have to use very time consuming numerical methods to
approximate the integrals, or a simulation (like Gibbs Sampling).
Thank you very much
Regards
Tom.
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