hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_chisq_vector (g01tc)

Purpose

nag_stat_inv_cdf_chisq_vector (g01tc) returns a number of deviates associated with the given probabilities of the χ2χ2-distribution with real degrees of freedom.

Syntax

[x, ivalid, ifail] = g01tc(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)
[x, ivalid, ifail] = nag_stat_inv_cdf_chisq_vector(tail, p, df, 'ltail', ltail, 'lp', lp, 'ldf', ldf)

Description

The deviate, xpixpi, associated with the lower tail probability pipi of the χ2χ2-distribution with νiνi degrees of freedom is defined as the solution to
xpi
P( Xi xpi : νi) = pi = 1/( 2νi / 2 Γ (νi / 2) )eXi / 2Xi vi / 2 1 dXi,  0xpi < ; ​νi > 0.
0
P( Xi xpi :νi) = pi = 1 2 νi/2 Γ (νi/2) 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi ,   0 xpi < ; ​ νi > 0 .
The required xpixpi is found by using the relationship between a χ2χ2-distribution and a gamma distribution, i.e., a χ2χ2-distribution with νiνi degrees of freedom is equal to a gamma distribution with scale parameter 22 and shape parameter νi / 2νi/2.
For very large values of νiνi, greater than 105105, Wilson and Hilferty's Normal approximation to the χ2χ2 is used; see Kendall and Stuart (1969).
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

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2χ2 distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

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 which tail the supplied probabilities represent. For j = ((i1)  mod  ltail) + 1 j= ( (i-1) mod ltail ) +1 , for i = 1,2,,max (ltail,lp,ldf)i=1,2,,max(ltail,lp,ldf):
tail(j) = 'L'tailj='L'
The lower tail probability, i.e., pi = P( Xi xpi : νi) pi = P( Xi xpi :νi) .
tail(j) = 'U'tailj='U'
The upper tail probability, i.e., pi = P( Xi xpi : νi) pi = P( Xi xpi :νi) .
Constraint: tail(j) = 'L'tailj='L' or 'U''U', for j = 1,2,,ltailj=1,2,,ltail.
2:     p(lp) – double array
lp, the dimension of the array, must satisfy the constraint lp > 0lp>0.
pipi, the probability of the required χ2χ2-distribution as defined by tail with pi = p(j)pi=pj, j = ((i1)  mod  lp) + 1j=((i-1) mod lp)+1.
Constraints:
  • if tail(k) = 'L'tailk='L', 0.0p(j) < 1.00.0pj<1.0;
  • otherwise 0.0 < p(j)1.00.0<pj1.0.
Where k = (i1)  mod  ltail + 1k=(i-1) mod ltail+1 and j = (i1)  mod  lp + 1j=(i-1) mod lp+1.
3:     df(ldf) – double array
ldf, the dimension of the array, must satisfy the constraint ldf > 0ldf>0.
νiνi, the degrees of freedom of the χ2χ2-distribution with νi = df(j)νi=dfj, j = ((i1)  mod  ldf) + 1j=((i-1) mod ldf)+1.
Constraint: df(j) > 0.0dfj>0.0, for j = 1,2,,ldfj=1,2,,ldf.

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:     lp – int64int32nag_int scalar
Default: The dimension of the array p.
The length of the array p.
Constraint: lp > 0lp>0.
3:     ldf – int64int32nag_int scalar
Default: The dimension of the array df.
The length of the array df.
Constraint: ldf > 0ldf>0.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     x( : :) – double array
Note: the dimension of the array x must be at least max (ltail,lp,ldf)max(ltail,lp,ldf).
xpixpi, the deviates for the χ2χ2-distribution.
2:     ivalid( : :) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lp,ldf)max(ltail,lp,ldf).
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 xpixpi.
ivalid(i) = 2ivalidi=2
On entry,invalid value for pipi.
ivalid(i) = 3ivalidi=3
On entry,νi0.0νi0.0.
ivalid(i) = 4ivalidi=4
pipi is too close to 0.00.0 or 1.01.0 for the result to be calculated.
ivalid(i) = 5ivalidi=5
The solution has failed to converge. The result should be a reasonable approximation.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and 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 tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.
  ifail = 2ifail=2
Constraint: ltail > 0ltail>0.
  ifail = 3ifail=3
Constraint: lp > 0lp>0.
  ifail = 4ifail=4
Constraint: ldf > 0ldf>0.
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

The results should be accurate to five significant digits for most parameter values. Some accuracy is lost for pipi close to 0.00.0 or 1.01.0.

Further Comments

For higher accuracy the relationship described in Section [Description] may be used and a direct call to nag_stat_inv_cdf_gamma_vector (g01tf) made.

Example

function nag_stat_inv_cdf_chisq_vector_example
tail = {'L'};
p = [0.01; 0.428; 0.869];
df = [20; 7.5; 45];
[x, ivalid, ifail] = nag_stat_inv_cdf_chisq_vector(tail, p, df);

fprintf('\n  TAIL     P       DF      X     IVALID\n');
ltail = numel(tail);
lp = numel(p);
ldf = numel(df);
len = max ([ltail, lp, ldf]);
for i=0:len-1
 fprintf('%5c%8.3f%8.3f%8.3f%8d\n',  cell2mat(tail(mod(i, ltail)+1)), ...
          p(mod(i,lp)+1), df(mod(i,ldf)+1), x(i+1), ivalid(i+1));
end
 

  TAIL     P       DF      X     IVALID
    L   0.010  20.000   8.260       0
    L   0.428   7.500   6.201       0
    L   0.869  45.000  55.738       0

function g01tc_example
tail = {'L'};
p = [0.01; 0.428; 0.869];
df = [20; 7.5; 45];
[x, ivalid, ifail] = g01tc(tail, p, df);

fprintf('\n  TAIL     P       DF      X     IVALID\n');
ltail = numel(tail);
lp = numel(p);
ldf = numel(df);
len = max ([ltail, lp, ldf]);
for i=0:len-1
 fprintf('%5c%8.3f%8.3f%8.3f%8d\n',  cell2mat(tail(mod(i, ltail)+1)), ...
          p(mod(i,lp)+1), df(mod(i,ldf)+1), x(i+1), ivalid(i+1));
end
 

  TAIL     P       DF      X     IVALID
    L   0.010  20.000   8.260       0
    L   0.428   7.500   6.201       0
    L   0.869  45.000  55.738       0


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013