fdistr.pas

{$MODESWITCH RESULT+}
{$GOTO ON}
(*************************************************************************
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier

Contributors:
    * Sergey Bochkanov (ALGLIB project). Translation from C to
      pseudocode.

See subroutines comments for additional copyrights.

>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the 
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses

>>> END OF LICENSE >>>
*************************************************************************)
unit fdistr;
interface
uses Math, Sysutils, Ap, gammafunc, normaldistr, ibetaf;

function FDistribution(a : AlglibInteger;
     b : AlglibInteger;
     x : Double):Double;
function FCDistribution(a : AlglibInteger;
     b : AlglibInteger;
     x : Double):Double;
function InvFDistribution(a : AlglibInteger;
     b : AlglibInteger;
     y : Double):Double;

implementation

(*************************************************************************
F distribution

Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density).  This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula

P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).


The arguments a and b are greater than zero, and x is
nonnegative.

ACCURACY:

Tested at random points (a,b,x).

               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
   IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
   IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
   IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************)
function FDistribution(a : AlglibInteger;
     b : AlglibInteger;
     x : Double):Double;
var
    w : Double;
begin
    Assert((a>=1) and (b>=1) and AP_FP_Greater_Eq(x,0), 'Domain error in FDistribution');
    w := a*x;
    w := w/(b+w);
    Result := IncompleteBeta(Double(0.5)*a, Double(0.5)*b, w);
end;


(*************************************************************************
Complemented F distribution

Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).


                     inf.
                      -
             1       | |  a-1      b-1
1-P(x)  =  ------    |   t    (1-t)    dt
           B(a,b)  | |
                    -
                     x


The incomplete beta integral is used, according to the
formula

P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).


ACCURACY:

Tested at random points (a,b,x) in the indicated intervals.
               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
   IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
   IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
   IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************)
function FCDistribution(a : AlglibInteger;
     b : AlglibInteger;
     x : Double):Double;
var
    w : Double;
begin
    Assert((a>=1) and (b>=1) and AP_FP_Greater_Eq(x,0), 'Domain error in FCDistribution');
    w := b/(b+a*x);
    Result := IncompleteBeta(Double(0.5)*b, Double(0.5)*a, w);
end;


(*************************************************************************
Inverse of complemented F distribution

Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.

This is accomplished using the inverse beta integral
function and the relations

     z = incbi( df2/2, df1/2, p )
     x = df2 (1-z) / (df1 z).

Note: the following relations hold for the inverse of
the uncomplemented F distribution:

     z = incbi( df1/2, df2/2, p )
     x = df2 z / (df1 (1-z)).

ACCURACY:

Tested at random points (a,b,p).

             a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between .001 and 1:
   IEEE     1,100       100000      8.3e-15     4.7e-16
   IEEE     1,10000     100000      2.1e-11     1.4e-13
 For p between 10^-6 and 10^-3:
   IEEE     1,100        50000      1.3e-12     8.4e-15
   IEEE     1,10000      50000      3.0e-12     4.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************)
function InvFDistribution(a : AlglibInteger;
     b : AlglibInteger;
     y : Double):Double;
var
    w : Double;
begin
    Assert((a>=1) and (b>=1) and AP_FP_Greater(y,0) and AP_FP_Less_Eq(y,1), 'Domain error in InvFDistribution');
    
    //
    // Compute probability for x = 0.5
    //
    w := IncompleteBeta(Double(0.5)*b, Double(0.5)*a, Double(0.5));
    
    //
    // If that is greater than y, then the solution w < .5
    // Otherwise, solve at 1-y to remove cancellation in (b - b*w)
    //
    if AP_FP_Greater(w,y) or AP_FP_Less(y,Double(0.001)) then
    begin
        w := InvIncompleteBeta(Double(0.5)*b, Double(0.5)*a, y);
        Result := (b-b*w)/(a*w);
    end
    else
    begin
        w := InvIncompleteBeta(Double(0.5)*a, Double(0.5)*b, Double(1.0)-y);
        Result := b*w/(a*(Double(1.0)-w));
    end;
end;


end.