shannon (version 0.2.0)

Student's t distribution: Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Student's t distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Student's t distribution.

Usage

se_st(v)
re_st(v, delta)
hce_st(v, delta)
ae_st(v, delta)

Value

The functions se_st, re_st, hce_st, and ae_st provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Student's t distribution and \(\delta\).

Arguments

v

The strictly positive parameter of the Student's t distribution (\(v > 0\)), also called a degree of freedom.

delta

The strictly positive parameter (\(\delta > 0\)) and (\(\delta \ne 1\)).

Author

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[e-mail]>, Christophe Chesneau <[e-mail]> and Farrukh Jamal .

Details

The following is the probability density function of the Student t distribution: $$ f(x)=\frac{\Gamma(\frac{v+1}{2})}{\sqrt{v\pi}\Gamma(\frac{v}{2})}\left(1+\frac{x^{2}}{v}\right)^{-(v+1)/2}, $$ where \(x\in\left(-\infty,+\infty\right)\) and \(v > 0\), and \(\Gamma(a)\) is the standard gamma function.

References

Yang, Z., Fang, K. T., & Kotz, S. (2007). On the Student's t-distribution and the t-statistic. Journal of Multivariate Analysis, 98(6), 1293-1304.

Ahsanullah, M., Kibria, B. G., & Shakil, M. (2014). Normal and Student's t distributions and their applications (Vol. 4). Paris, France: Atlantis Press.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Inf. Control, 19, 181–194.

See Also

re_exp, re_gamma

Examples

se_st(4)
delta <- c(1.5, 2, 3)
re_st(4, delta)
hce_st(4, delta)
ae_st(4, delta)