topic how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),etc. in SAS Programming

how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),etc.

Gick — Sat, 19 Feb 2022 05:57:19 GMT

I need to create a cumulative distribution function (F) of the Gamma(a,b) distribution and then use this function to calculate Ws which I have noted profil_W in my program.
Here is the calculation formula for profil_W.

Profile_W=(k*F(k)+(k-2)*F(k-2))+(k-1)*F(k-1)+ab(2*F(k-1)-F( k-2)-F(k)).
The first part F follows the Gamma(a,b) law and the second part F follows the Gamma(a+1,b) law.
Note that the second part begins with ab(...).

Here is the code to calculate the 2 laws of Gamma followed by F that I noted "Fct_ab" the first and "Fct_A1b" the second.
k represents "Incid" in my formula. My problem is that I can't vary or calculate the functions F(k), F(k-1), F(k-2), etc.

Here is the code:

/*----------------------------------------------*/
proc contents date= tab_It;
run;
/*Fixed Gamma Law Parameter*/
%let a=1;
%let b=5;

/*Distribution of the Gamma(a,b) distribution*/
Data tab_W;
set tab_It;
if (%sysevalf(&b.)>0) and (%sysevalf(&b.)>0) ;
if (Incid >0);
Fct_ab=RAND('GAMMA', %sysevalf(&a.), %sysevalf(&b.));
Fct_A1b=RAND('GAMMA', %sysevalf(&a.)+1, %sysevalf(&b.));
profil_W=round((Incid*(Fct_ab)+(Incid-2)*Fct_ab-2*(Incid-1)*Fct_ab) +%sysevalf(&a.)*%sysevalf(&b.)*(2*Fct_A1b-Fct_A1b),0.01);
run;

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

ballardw — Sat, 19 Feb 2022 18:28:18 GMT

If a macro variable contains a value that is easily considered as a number you do not need all that %syseval junk.

Example:

%let a= 0.25;
%let b= 4;


data example;
   x = &a. * &b.;
run;

which results in the expected 1 for x. All macro variables are is text. If the value you have would be okay for a statement as is such as : x = 0.25 * 4 ; then that's all you need.

The function CDF in SAS returns a value from the cumulative distribution, the Quantile function returns a specific point from the distribution. Rand generates a random number from from the distribution so I doubt that is what you want.

"Generate a distribution" I would typically take to create fair portion of values from a distribution, not a single value as your code does. So I suspect that you need to provide more details. Possibly provide a link to what you mean by "Gamma law" as most of the internet search results I get are company names, gaming or image color correction related.

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

Gick — Sat, 19 Feb 2022 19:40:11 GMT

Here is the link that explains the gamma distribution. Source wikipedia. https://en.wikipedia.org/wiki/Gamma_distribution

For the macro variables created, I tried to fix values of a and b which are the parameters of the "gamma law". Once the calculation of the repartition function is good. I could change its parameter values.

In my code, I used %sysevalf() but the problem is to know if the calculation of my distribution functions are well calculated and also the profil_W variable.
Thank you for your comments and I am still on my hunger

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

Gick — Sat, 19 Feb 2022 19:50:08 GMT

Here is the link of a course that better explains the Gamma Law : https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

ballardw — Sun, 20 Feb 2022 03:28:55 GMT

Time to start over. I know what a Gamma distribution is. Neither of the links you provide have anything related to "Gamma Law"

Your request starts with

I need to create a cumulative distribution function (F) of the Gamma(a,b) distribution and then use this function to calculate

You do not need to create a CDF for the gamma function. SAS provides two functions, one of which is very likely what you want either CDF or QUANTILE.

CDF would involve writing code to implement and calculate integrals. Which you have not shown even a slight attempt to do so. I have done such code but refuse to do for functions that SAS Provides and except for nearly-trivial examples wouldn't on this forum as the amount of work is excessive for a volunteer.

Look at the documentation for the CDF function,

Returns a value from a cumulative probability distribution.

CDF('GAMMA', x, a <, λ>)

The CDF function for the gamma distribution returns the probability that an observation from a gamma distribution, with the shape parameter a and the scale parameter λ, is less than or equal to x.

X would be the point of the cumulative you are specifically interested in which could be a variable or fixed value, your choice.

If that is not what you want then likely the Quantile function is what you want.

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

Gick — Sun, 20 Feb 2022 06:12:21 GMT

Thank you for that answer. I was not aware of this function (CDF).

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

Gick — Sun, 20 Feb 2022 10:49:04 GMT

here is the calculation of the cumulative density function F(k) according to the Gamma distribution with parameter (a,b)
F_k=cdf('GAMMA', k, a, b);

of F(k-1), we have
F_k1=cdf('GAMMA', k-1, a, b);

Awaiting your comments

Re: how to calculate the distribution function of the Gamma(a,b) law then to calculate F(k), F(k-1),

ballardw — Mon, 21 Feb 2022 02:09:18 GMT

As long as your K has the appropriate value then that is likely the correct call.