R help - Dec 2009 - Regularized gamma function/ incomplete gamma function

Dear all,

I would be very grateful if you could help me with:

Given the regularized gamma function Reg=int_0^r (x^(k-1)e^(-x))dx/int_0^Inf
(x^(k-1)e^(-x))dx ; 0<r<Inf (which is eventually the ratio of the
Incomplete gamma function by the gamma function), does anyone know of a package
in R that would evaluate the derivative of the inverse of Reg with respect to k?
I am aware that the function "Rgamma.inv" of the package
"Zipfr" evaluates the inverse
of Reg and I'm wondering wether there is a function that would evaluate the
derivative of the inverse..

Alternatively, a good numerical integration package/ or simply a function that
could evaluate the integral int_0^r (log(x) x^(k-1) e^(-x))dx;  0<r<Inf
would be useful. I tried the function "int" of the package
"rmutil" but I'm not sure wether it is accurate especially for
small values of k. Does R have a powerful numerical integration package that can
deal with such functions especially when the limit close to zero in + or - Inf?

Many thanks for this opportunity to post our queries,

Amy



  ________________________________

The Open University is incorporated by Royal Charter (RC 000391), an exempt
charity in England & Wales and a charity registered in Scotland (SC 038302).

	[[alternative HTML version deleted]]

> I would be very grateful if you could help me with:
>
> Given the regularized gamma function Reg=int_0^r (x^(k-1)e^(-x))dx/ 
> int_0^Inf (x^(k-1)e^(-x))dx ; 0<r<Inf (which is eventually the ratio
> of the
> Incomplete gamma function by the gamma function), does anyone know  
> of a package in R that would evaluate the derivative of the inverse  
> of Reg with respect to k?
This doesn't look pleasant at all:

	http://en.wikipedia.org/wiki/Incomplete_gamma_function#Derivatives

It appears to be quite difficult to compute the derivative of the  
incomplete gamma function itself -- I'm not aware of any R  
implementation -- and the derivative of the inverse may be even harder.
> I am aware that the function "Rgamma.inv" of the package
"Zipfr"
> evaluates the inverse
> of Reg and I'm wondering wether there is a function that would  
> evaluate the derivative of the inverse..
For all I know, the inverse is calculated by an (optimized) Newton- 
type search algorithm, so it's not very likely that its derivative can  
easily be worked out (based on the current implementation).

Best,
Stefan

You can do this using the package "numDeriv".

require(zipfR)

require(numDeriv)

fn <- function(x, y) Rgamma.inv(x, y)

gRgamma.inv <- function(y, k) sapply(y, function(y) grad(x=k, func=fn, y=y))

plot(gRgamma.inv(y=seq(0,1, length=200), k=1), type="l",
xlab="x",
ylab="Derivative of Rgamma.inv w.r.t. `k' ")
lines(gRgamma.inv(seq(0,1, length=200), k=2), col=2)
lines(gRgamma.inv(seq(0,1, length=200), k=5), col=3)
lines(gRgamma.inv(seq(0,1, length=200), k=10), col=4)

I hope this helps,
Ravi.

----------------------------------------------------------------------------
-------

Ravi Varadhan, Ph.D.

Assistant Professor, The Center on Aging and Health

Division of Geriatric Medicine and Gerontology 

Johns Hopkins University

Ph: (410) 502-2619

Fax: (410) 614-9625

Email: rvaradhan at jhmi.edu

Webpage:
http://www.jhsph.edu/agingandhealth/People/Faculty_personal_pages/Varadhan.h
tml

 

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--------


-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On
Behalf Of A.Noufaily
Sent: Friday, December 11, 2009 11:13 AM
To: r-help at R-project.org
Subject: [R] Regularized gamma function/ incomplete gamma function



Dear all,

I would be very grateful if you could help me with:

Given the regularized gamma function Reg=int_0^r (x^(k-1)e^(-x))dx/int_0^Inf
(x^(k-1)e^(-x))dx ; 0<r<Inf (which is eventually the ratio of the
Incomplete gamma function by the gamma function), does anyone know of a
package in R that would evaluate the derivative of the inverse of Reg with
respect to k? I am aware that the function "Rgamma.inv" of the package
"Zipfr" evaluates the inverse
of Reg and I'm wondering wether there is a function that would evaluate the
derivative of the inverse..

Alternatively, a good numerical integration package/ or simply a function
that could evaluate the integral int_0^r (log(x) x^(k-1) e^(-x))dx; 
0<r<Inf
would be useful. I tried the function "int" of the package
"rmutil" but I'm
not sure wether it is accurate especially for small values of k. Does R have
a powerful numerical integration package that can deal with such functions
especially when the limit close to zero in + or - Inf?

Many thanks for this opportunity to post our queries,

Amy



  ________________________________

The Open University is incorporated by Royal Charter (RC 000391), an exempt
charity in England & Wales and a charity registered in Scotland (SC 038302).

	[[alternative HTML version deleted]]

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