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NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_students_t (g01fb)

Purpose

nag_stat_inv_cdf_students_t (g01fb) returns the deviate associated with the given tail probability of Student's tt-distribution with real degrees of freedom.

Syntax

[result, ifail] = g01fb(p, df, 'tail', tail)
[result, ifail] = nag_stat_inv_cdf_students_t(p, df, 'tail', tail)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 23: tail now optional (default 'l')
.

Description

The deviate, tptp associated with the lower tail probability, pp, of the Student's tt-distribution with νν degrees of freedom is defined as the solution to
tp
P(T < tp : ν) = p = (Γ((ν + 1) / 2))/(sqrt(νπ)Γ(ν / 2))(1 + (T2)/ν)(ν + 1) / 2dT,  ν1; ​ < tp < .
P(T<tp:ν)=p=Γ((ν+1)/2) νπΓ(ν/2) -tp (1+T2ν) -(ν+1)/2dT,  ν1; ​-<tp<.
For ν = 1​ or ​2ν=1​ or ​2 the integral equation is easily solved for tptp.
For other values of ν < 3ν<3 a transformation to the beta distribution is used and the result obtained from nag_stat_inv_cdf_beta (g01fe).
For ν3ν3 an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).

References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Hill G W (1970) Student's tt-distribution Comm. ACM 13(10) 617–619

Parameters

Compulsory Input Parameters

1:     p – double scalar
pp, the probability from the required Student's tt-distribution as defined by tail.
Constraint: 0.0 < p < 1.00.0<p<1.0.
2:     df – double scalar
νν, the degrees of freedom of the Student's tt-distribution.
Constraint: df1.0df1.0.

Optional Input Parameters

1:     tail – string (length ≥ 1)
Indicates which tail the supplied probability represents.
tail = 'U'tail='U'
The upper tail probability, i.e., P(Ttp : ν)P(Ttp:ν).
tail = 'L'tail='L'
The lower tail probability, i.e., P(Ttp : ν)P(Ttp:ν).
tail = 'S'tail='S'
The two tail (significance level) probability, i.e., P(T|tp| : ν) + P(T|tp| : ν)P(T|tp|:ν)+P(T-|tp|:ν).
tail = 'C'tail='C'
The two tail (confidence interval) probability, i.e., P(T|tp| : ν)P(T|tp| : ν)P(T|tp|:ν)-P(T-|tp|:ν).
Default: 'L''L'
Constraint: tail = 'U'tail='U', 'L''L', 'S''S' or 'C''C'.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     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_inv_cdf_students_t (g01fb) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
If ifail = 1ifail=1, 22 or 33 on exit, then nag_stat_inv_cdf_students_t (g01fb) returns zero.

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

  ifail = 1ifail=1
On entry,tail'U'tail'U', 'S''S', 'C''C' or 'L''L'.
  ifail = 2ifail=2
On entry,p0.0p0.0,
orp1.0p1.0.
  ifail = 3ifail=3
On entry,df < 1.0df<1.0.
W ifail = 5ifail=5
Convergence in the calculation of the inverse beta value was not achieved. However, the result should be a reasonable approximation to the correct value.

Accuracy

The results should be accurate to five significant digits, for most parameter values. The error behaviour for various parameter values is discussed in Hill (1970).

Further Comments

The value tptp may be calculated by using the transformation described in Section [Description] and using nag_stat_inv_cdf_beta (g01fe). This function allows you to set the required accuracy.

Example

function nag_stat_inv_cdf_students_t_example
p = 0.01;
df = 20;
[result, ifail] = nag_stat_inv_cdf_students_t(p, df, 'tail', 's')
 

result =

    2.8453


ifail =

                    0


function g01fb_example
p = 0.01;
df = 20;
[result, ifail] = g01fb(p, df, 'tail', 's')
 

result =

    2.8453


ifail =

                    0



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Chapter Introduction
NAG Toolbox

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