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Re: Mode of InverseChiSquareDistribution

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  • Subject: [mg128652] Re: Mode of InverseChiSquareDistribution
  • From: Bill Rowe <readnews at>
  • Date: Wed, 14 Nov 2012 01:28:06 -0500 (EST)
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On 11/13/12 at 12:02 AM, paulvonhippel at (paul) wrote:

>As a first step toward solving a more complicated problem, I would
>like to calculate the mode of the InverseChiSquareDistribution with
>D-1 degrees of freedom. The answer is 1/(D+1) but I am having
>trouble getting that expression from Mathematica.

>First I type
>PDF[InverseChiSquareDistribution[D - 1]]
>And then I cut and paste the function into ArgMax, imposing
>appropriate constraints:
>modeInverseChiSquare = ArgMax[{(2^((1 - D)/2) (1/x)^(1 + 1/2 (-1 +
>D)) E^(-(1/(2 x))))/ Gamma[1/2 (-1 + D)], x
>>0, D > 0, Element[D, Integers]}, x, Reals]

>But all ArgMax does is echo the input. If I evaluate the mode at a
>particular value of D I get the right answer -- e.g.,
>modeInverseChiSquare /. D -> 10 returns 1/11. But what I'd like
>Mathematica to do is tell me that the answer in general is 1/(D+1).

I get the same result with ArgMax. Not sure why. But here is a
way to get Mathematica to generate the general result

In[1]:= f =
   Assuming[x > 0,
    Simplify@PDF[InverseChiSquareDistribution[d - 1], x]];
Quiet@Solve[D[f, x] == 0, x]

Out[2]= {{x->1/(d+1)}}

Here, I've used Quiet to suppress the warning Solve generates
regarding using inverse functions.

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