The lower tail probability for the Student's -distribution with degrees of freedom, is defined by:
Computationally, there are two situations:
(i)
when , a transformation of the beta distribution, is used
or
(ii)
when , an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).
4
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 -distribution Comm. ACM13(10) 617–619
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Arguments
1:
– Nag_TailProbabilityInput
On entry: 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., .
Constraint:
, , or .
2:
– doubleInput
On entry: , the value of the Student's variate.
3:
– doubleInput
On entry: , the degrees of freedom of the Student's -distribution.
Constraint:
.
4:
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
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Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INTERNAL_ERROR
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 for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_LT
On entry, .
Constraint: .
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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 ), see Hastings and Peacock (1975).
8
Parallelism and Performance
nag_prob_students_t (g01ebc) is not threaded in any implementation.
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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_prob_beta_dist (g01eec). This function allows you to set the required accuracy.
10
Example
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.