NAG Library Function Document
nag_prob_students_t (g01ebc) returns the lower tail, upper tail or two tail probability for the Student's -distribution with real degrees of freedom.
||nag_prob_students_t (Nag_TailProbability tail,
The lower tail probability for the Student's
degrees of freedom,
is defined by:
Computationally, there are two situations:
||when , a transformation of the beta distribution, is used
||when , an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).
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 -distribution Comm. ACM 13(10) 617–619
tail – Nag_TailProbabilityInput
: indicates which tail the returned probability should represent.
- The upper tail probability is returned, i.e., .
- The two tail (significance level) probability is returned, i.e., .
- The two tail (confidence interval) probability is returned, i.e., .
- The lower tail probability is returned, i.e., .
, , or .
t – doubleInput
On entry: , the value of the Student's variate.
df – doubleInput
On entry: , the degrees of freedom of the Student's -distribution.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
On entry, .
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
), see Hastings and Peacock (1975)
8 Parallelism and Performance
The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)
) and using nag_prob_beta_dist (g01eec)
. This function allows you to set the required accuracy.
This example reads values from, and degrees of freedom for Student's -distributions along with the required tail. The probabilities are calculated and printed until the end of data is reached.
10.1 Program Text
Program Text (g01ebce.c)
10.2 Program Data
Program Data (g01ebce.d)
10.3 Program Results
Program Results (g01ebce.r)