About the invcdf function for discrete distributions
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Are we sure that the invcdf function works correctly for the case of
discrete distributions?
The inverse CDF method of generating a random variable using a
Uniform[0,1] is valid also for discrete rv's (by using the generalized
inverse), so I tried
<hansl>
series U = randgen(U,0,1) # generate Uniform(0,1)
series Xd = invcdf(P, 2, U) # generate Poisson with mean 2
</hansl>
but the Xd series did not have integer values, so it cannot be a Poisson rv.
--
Alecos Papadopoulos PhD
Athens University of Economics and Business
web: alecospapadopoulos.wordpress.com/
On Fri, 27 Sep 2019, Alecos Papadopoulos wrote:
Are we sure that the invcdf function works correctly for the case of
discrete
distributions?
The inverse CDF method of generating a random variable using a Uniform[0,1]
is valid also for discrete rv's (by using the generalized inverse), so I
tried
<hansl>
series U = randgen(U,0,1) # generate Uniform(0,1)
series Xd = invcdf(P, 2, U) # generate Poisson with mean 2
</hansl>
but the Xd series did not have integer values, so it cannot be a
Poisson rv.
Hmm, it seems there's something odd about the cephes function we're
using to give the answer here, namely pdtri(). For the most part the
cephes suite, written by Stephen Moshier, is fast and accurate, but
I agree that something looks to be amiss here. I'll try to find out
what.
(In case anyone is able to help: cephes' pdtri() takes two
arguments, an integer k and a double 0 < y < 1 (probability). And
the doc says "Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the given probability
y." So what's "k"? If it's the mean (= variance) it shouldn't
have
to be an integer, should it?)
Allin Cottrell
On Fri, 27 Sep 2019, Alecos Papadopoulos wrote:
Are we sure that the invcdf function works correctly for the case
of discrete distributions?
Thanks to Alecos for raising this question (and also to Frank
Benford for contributing to offlist discussion on the topic).
It turns out we were misinterpreting what the cephes functions pdtri
(Poisson) and bdtri (Binomial) are doing. They do "invert" the
respective CDFs, but with respect to a parameter (the Poisson rate
lambda, the Binomial success probability), not the value of the r.v.
That is, they solve for the parameter value that makes the CDF take
on a specified value given a value for the r.v.
The back-end to our invcdf() function now uses alternative code that
inverts the CDF in the sense that Alecos was expecting -- and that's
stated in the help for the function.
Allin
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Allin Cottrell