Dear all,
I am trying to prove the following, without success:
Consider a family of distribution functions that satisfy the MLRP property, i.e.
. . . . .\(\displaystyle \dfrac{\partial}{\partial x}\, \dfrac{f_t (x)}{f_{t'} (x)}\, \geq\, 0\)
or, equivalently,
. . . . .\(\displaystyle F_{t}(x)\, >\, F_{t'}(x)\)
. . . . .\(\displaystyle \forall\, x\, \mbox{ and }\, \forall \, t\, >\, t'\)
(where \(\displaystyle f\) is the pdf, and is assumed to exist, and \(\displaystyle F\) is the cdf; \(\displaystyle x\) belongs to the non-negative reals).
I want to show that
. . . . .\(\displaystyle \dfrac{\partial}{\partial\, t}\, \dfrac{\int_a^b \, \left(F_t (x)\, -\, F_{t'} (x)\right)\, dx}{F_t (b)\, -\, F_{t'} (b)}\, \geq\, 0\)
. . . . .\(\displaystyle \forall \, a\, \geq \, 0\, \mbox{ and }\, b\,>\,a\)
thanks for any advice you might give me!
G
I am trying to prove the following, without success:
Consider a family of distribution functions that satisfy the MLRP property, i.e.
. . . . .\(\displaystyle \dfrac{\partial}{\partial x}\, \dfrac{f_t (x)}{f_{t'} (x)}\, \geq\, 0\)
or, equivalently,
. . . . .\(\displaystyle F_{t}(x)\, >\, F_{t'}(x)\)
. . . . .\(\displaystyle \forall\, x\, \mbox{ and }\, \forall \, t\, >\, t'\)
(where \(\displaystyle f\) is the pdf, and is assumed to exist, and \(\displaystyle F\) is the cdf; \(\displaystyle x\) belongs to the non-negative reals).
I want to show that
. . . . .\(\displaystyle \dfrac{\partial}{\partial\, t}\, \dfrac{\int_a^b \, \left(F_t (x)\, -\, F_{t'} (x)\right)\, dx}{F_t (b)\, -\, F_{t'} (b)}\, \geq\, 0\)
. . . . .\(\displaystyle \forall \, a\, \geq \, 0\, \mbox{ and }\, b\,>\,a\)
thanks for any advice you might give me!
G
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