Hello all,
I have m p-values and I bin them into B bins with the b^th bin having m_b
p-values. So that the sum of all the m_b's = m.
I now proceed to build a probability mass function as follows:
Bin 1 2 ....
B
Prob m_1/m m_2/m m_B/m
I call these probabilities: g_1, g_2, ....., g_B.
Next say I choose m_0 of the m p-values from m. These are the true null
p-values. so that m_0 <= m.
Also I once again bin the p-values into b bins with the b^th bin having m_0b
p-values. So that
the sum of all the m_0b pvalues = m_0.
The bining creating the following: probabilities:
Bin 1 2 ....
B
Prob m_01/m m_02/m m_0B/m
I call these probabilities: g_01, g_02, ....., g_0B.
So that the p-values is a mixture with pi_0 of the p-values being true
nulls. That is, m_0 = pi_0 * m.
My question is I want to prove that 1 - 0.5* sum( |g_b - g_0b|) is greater
than or equal to pi_0.
for both the continuous and discrete case (both coninuous and discrete
p-values).
In the continuous case, if f(p) is the distribution of p-values, then f(p)
= pi_0 * f_0(p) + (1 - pi_0) * f_1(p) where f_0(p) and f_1(p) are the
densities of the p-values under the null and alternative distributions
respectively.
Any help is greatly appreciated.
--
Thanks,
Jim.
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