The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Vectorized Routines 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

Hill G W (1970) Student's

t

-distribution Comm. ACM 13(10) 617–619

Parameters

Compulsory Input Parameters

1: $tail (ltail)$ – cell array of strings

Indicates which tail the returned probabilities should represent. For

j = ((i - 1) mod ltail) + 1

, for

i = 1, 2, \dots, \max (ltail, lt, ldf)

$tail (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (T_{i} \leq t_{i} : ν_{i})$ .
$tail (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (T_{i} \geq t_{i} : ν_{i})$ .
$tail (j) ='C'$: The two tail (confidence interval) probability is returned,
i.e., $p_{i} = P (T_{i} \leq |t_{i}| : ν_{i}) - P (T_{i} \leq - |t_{i}| : ν_{i})$ .
$tail (j) ='S'$: The two tail (significance level) probability is returned,
i.e., $p_{i} = P (T_{i} \geq |t_{i}| : ν_{i}) + P (T_{i} \leq - |t_{i}| : ν_{i})$ .

Constraint:

tail (j) ='L'

'U'

'C'

'S'

, for

j = 1, 2, \dots, ltail

2: $t (lt)$ – double array

t_{i}

, the values of the Student's

t

variates with

t_{i} = t (j)

j = ((i - 1) mod lt) + 1

3: $df (ldf)$ – double array

ν_{i}

, the degrees of freedom of the Student's

t

-distribution with

ν_{i} = df (j)

j = ((i - 1) mod ldf) + 1

Constraint:

df (j) \geq 1.0

, for

j = 1, 2, \dots, ldf

Optional Input Parameters

1: $ltail$ – int64int32nag_int scalar: Default: the dimension of the array tail.
The length of the array tail.

Constraint: $ltail > 0$ .
2: $lt$ – int64int32nag_int scalar: Default: the dimension of the array t.
The length of the array t.

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

Constraint: $ldf > 0$ .

Output Parameters

1: $p (:)$ – double array

The dimension of the array p will be

\max (ltail, lt, ldf)

p_{i}

, the probabilities for the Student's

t

distribution.

2: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

\max (ltail, lt, ldf)

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

invalid value supplied in tail when calculating

p_{i}

$ivalid (i) = 2$

On entry,

ν_{i} < 1.0

3: $ifail$ – int64int32nag_int scalar

ifail = 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 $ifail = 1$: On entry, at least one value of tail or df was invalid.
Check ivalid for more information.

$ifail = 2$: Constraint: $ltail > 0$ .

$ifail = 3$: Constraint: $lt > 0$ .

$ifail = 4$: Constraint: $ldf > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than

10^{- 10}

), see Hastings and Peacock (1975).

Further Comments

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using nag_stat_prob_beta_vector (g01se). This function allows you to set the required accuracy.

Example

This example reads values from, and degrees of freedom for Student's

t

-distributions along with the required tail. The probabilities are calculated and printed.

Open in the MATLAB editor: g01sb_example

function g01sb_example


fprintf('g01sb example results\n\n');

t    = [0.85];
df   = [20];
tail = {'L'; 'S'; 'C'; 'U'};
% calculate probability
[prob, ivalid, ifail] = g01sb(...
                              tail, t, df);

fprintf('    t      df     prob  tail\n');
lt    = numel(t);
ldf   = numel(df);
ltail = numel(tail);
len   = max ([lt, ldf, ltail]);
for i=0:len-1
  fprintf('%7.3f%8.3f%8.4f  %c\n', t(mod(i,lt)+1), df(mod(i,ldf)+1), ...
          prob(i+1), tail{mod(i,ltail)+1});
end

g01sb example results

    t      df     prob  tail
  0.850  20.000  0.7973  L
  0.850  20.000  0.4054  S
  0.850  20.000  0.5946  C
  0.850  20.000  0.2027  U

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox