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NAG Toolbox: nag_stat_prob_f_vector (g01sd)

Purpose

nag_stat_prob_f_vector (g01sd) returns a number of lower or upper tail probabilities for the FF or variance-ratio distribution with real degrees of freedom.

Syntax

[p, ivalid, ifail] = g01sd(tail, f, df1, df2, 'ltail', ltail, 'lf', lf, 'ldf1', ldf1, 'ldf2', ldf2)
[p, ivalid, ifail] = nag_stat_prob_f_vector(tail, f, df1, df2, 'ltail', ltail, 'lf', lf, 'ldf1', ldf1, 'ldf2', ldf2)

Description

The lower tail probability for the FF, or variance-ratio, distribution with uiui and vivi degrees of freedom, P( Fi fi : ui,vi) P( Fi fi :ui,vi) , is defined by:
fi
P( Fi fi : ui,vi) = ( uiui / 2 vivi / 2 Γ ((ui + vi) / 2) )/( Γ (ui / 2) Γ (vi / 2) )Fi (ui2) / 2 (uiFi + vi) (ui + vi) / 2 dFi,
0
P( Fi fi :ui,vi) = ui ui/2 vi vi/2 Γ ( ( ui + vi ) / 2 ) Γ (ui/2) Γ (vi/2) 0 fi Fi (ui-2) / 2 ( ui Fi + vi ) - ( ui + vi ) / 2 d Fi ,
for uiui, vi > 0vi>0, fi0fi0.
The probability is computed by means of a transformation to a beta distribution, Pβi( Bi βi : ai,bi) Pβi( Bi βi :ai,bi) :
P( Fi fi : ui,vi) = Pβi( Bi ( ui fi )/( ui fi + vi ) : ui / 2 , vi / 2 )
P( Fi fi :ui,vi) = Pβi( Bi ui fi ui fi + vi : ui / 2 , vi / 2 )
and using a call to nag_stat_prob_beta (g01ee).
For very large values of both uiui and vivi, greater than 105105, a normal approximation is used. If only one of uiui or vivi is greater than 105105 then a χ2χ2 approximation is used, see Abramowitz and Stegun (1972).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section [Vectorized s] in the G01 Chapter Introduction for further information.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Parameters

Compulsory Input Parameters

1:     tail(ltail) – cell array of strings
ltail, the dimension of the array, must satisfy the constraint ltail > 0ltail>0.
Indicates whether the lower or upper tail probabilities are required. For j = ((i1)  mod  ltail) + 1 j= ( (i-1) mod ltail ) +1 , for i = 1,2,,max (ltail,lf,ldf1,ldf2)i=1,2,,max(ltail,lf,ldf1,ldf2):
tail(j) = 'L'tailj='L'
The lower tail probability is returned, i.e., pi = P( Fi fi : ui,vi) pi = P( Fi fi :ui,vi) .
tail(j) = 'U'tailj='U'
The upper tail probability is returned, i.e., pi = P( Fi fi : ui,vi) pi = P( Fi fi :ui,vi) .
Constraint: tail(j) = 'L'tailj='L' or 'U''U', for j = 1,2,,ltailj=1,2,,ltail.
2:     f(lf) – double array
lf, the dimension of the array, must satisfy the constraint lf > 0lf>0.
fifi, the value of the FF variate with fi = f(j)fi=fj, j = ((i1)  mod  lf) + 1j=((i-1) mod lf)+1.
Constraint: f(j)0.0fj0.0, for j = 1,2,,lfj=1,2,,lf.
3:     df1(ldf1) – double array
ldf1, the dimension of the array, must satisfy the constraint ldf1 > 0ldf1>0.
uiui, the degrees of freedom of the numerator variance with ui = df1(j)ui=df1j, j = ((i1)  mod  ldf1) + 1j=((i-1) mod ldf1)+1.
Constraint: df1(j) > 0.0df1j>0.0, for j = 1,2,,ldf1j=1,2,,ldf1.
4:     df2(ldf2) – double array
ldf2, the dimension of the array, must satisfy the constraint ldf2 > 0ldf2>0.
vivi, the degrees of freedom of the denominator variance with vi = df2(j)vi=df2j, j = ((i1)  mod  ldf2) + 1j=((i-1) mod ldf2)+1.
Constraint: df2(j) > 0.0df2j>0.0, for j = 1,2,,ldf2j=1,2,,ldf2.

Optional Input Parameters

1:     ltail – int64int32nag_int scalar
Default: The dimension of the array tail.
The length of the array tail.
Constraint: ltail > 0ltail>0.
2:     lf – int64int32nag_int scalar
Default: The dimension of the array f.
The length of the array f.
Constraint: lf > 0lf>0.
3:     ldf1 – int64int32nag_int scalar
Default: The dimension of the array df1.
The length of the array df1.
Constraint: ldf1 > 0ldf1>0.
4:     ldf2 – int64int32nag_int scalar
Default: The dimension of the array df2.
The length of the array df2.
Constraint: ldf2 > 0ldf2>0.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     p( : :) – double array
Note: the dimension of the array p must be at least max (ltail,lf,ldf1,ldf2)max(ltail,lf,ldf1,ldf2).
pipi, the probabilities for the FF-distribution.
2:     ivalid( : :) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lf,ldf1,ldf2)max(ltail,lf,ldf1,ldf2).
ivalid(i)ivalidi indicates any errors with the input arguments, with
ivalid(i) = 0ivalidi=0
No error.
ivalid(i) = 1ivalidi=1
On entry,invalid value supplied in tail when calculating pipi.
ivalid(i) = 2ivalidi=2
On entry,fi < 0.0fi<0.0.
ivalid(i) = 3ivalidi=3
On entry,ui0.0ui0.0,
orvi0.0vi0.0.
ivalid(i) = 4ivalidi=4
The solution has failed to converge. The result returned should represent an approximation to the solution.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_stat_prob_f_vector (g01sd) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1ifail=1
On entry, at least one value of f, df1, df2 or tail was invalid, or the solution failed to converge.
Check ivalid for more information.
  ifail = 2ifail=2
Constraint: ltail > 0ltail>0.
  ifail = 3ifail=3
Constraint: lf > 0lf>0.
  ifail = 4ifail=4
Constraint: ldf1 > 0ldf1>0.
  ifail = 5ifail=5
Constraint: ldf2 > 0ldf2>0.
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

The result should be accurate to five significant digits.

Further Comments

For higher accuracy nag_stat_prob_beta_vector (g01se) can be used along with the transformations given in Section [Description].

Example

function nag_stat_prob_f_vector_example
f = [5.5; 39.9; 2.5];
df1 = [1.5; 1; 20.25];
df2 = [25.5; 1; 1];
tail = {'L'};
% calculate probability
[prob, ivalid, ifail] = nag_stat_prob_f_vector(tail, f, df1, df2);

fprintf('\n    F      DF1    DF2     PROB\n');
lf = numel(f);
ldf1 = numel(df1);
ldf2 = numel(df2);
ltail = numel(tail);
len = max ([lf, ldf1, ldf2, ltail]);
for i=0:len-1
 fprintf('%7.3f%8.3f%8.3f%8.3f  %5d\n', f(mod(i,lf)+1), df1(mod(i,ldf1)+1), ...
         df2(mod(i,ldf2)+1), prob(i+1), ivalid(i+1));
end
 

    F      DF1    DF2     PROB
  5.500   1.500  25.500   0.984      0
 39.900   1.000   1.000   0.900      0
  2.500  20.250   1.000   0.534      0

function g01sd_example
f = [5.5; 39.9; 2.5];
df1 = [1.5; 1; 20.25];
df2 = [25.5; 1; 1];
tail = {'L'};
% calculate probability
[prob, ivalid, ifail] = g01sd(tail, f, df1, df2);

fprintf('\n    F      DF1    DF2     PROB\n');
lf = numel(f);
ldf1 = numel(df1);
ldf2 = numel(df2);
ltail = numel(tail);
len = max ([lf, ldf1, ldf2, ltail]);
for i=0:len-1
 fprintf('%7.3f%8.3f%8.3f%8.3f  %5d\n', f(mod(i,lf)+1), df1(mod(i,ldf1)+1), ...
         df2(mod(i,ldf2)+1), prob(i+1), ivalid(i+1));
end
 

    F      DF1    DF2     PROB
  5.500   1.500  25.500   0.984      0
 39.900   1.000   1.000   0.900      0
  2.500  20.250   1.000   0.534      0


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Chapter Introduction
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