Noncentral T Distribution Functions
Source:R/non_central_t_distribution.R
non_central_t_distribution.RdFunctions to compute the probability density function, cumulative distribution function, and quantile function for the Noncentral T distribution.
The noncentral T distribution is a generalization of the Student's t Distribution.
The PDF is:
$$f(x;\nu;\delta) = \frac{\nu^{\nu/2}\,\nu!} {2^{\nu} e^{\delta^{2}/2} (\nu+x^{2})^{\nu/2}\Gamma\!\left(\frac{\nu}{2}\right)} \left( \frac{\sqrt{2}\,\delta x\, {}_1F_1\!\left(\frac{\nu}{2}+1;\frac{3}{2}; \frac{\delta^{2}x^{2}}{2(\nu+x^{2})}\right)} {(\nu+x^{2})\Gamma\!\left(\frac{\nu+1}{2}\right)} + \frac{ {}_1F_1\!\left(\frac{\nu+1}{2};\frac{1}{2}; \frac{\delta^{2}x^{2}}{2(\nu+x^{2})}\right)} {\sqrt{\nu+x^{2}}\Gamma\!\left(\frac{\nu}{2}+1\right)} \right)$$
The CDF is:
$$F(t;\nu;\delta) = \Phi(-\delta) + \frac{1}{2} \sum_{i=0}^{\infty} \left( P_i\, I_x\!\left(i+\frac{1}{2},\frac{\nu}{2}\right) + \frac{\delta}{\sqrt{2}}\, Q_i\, I_x\!\left(i+1,\frac{\nu}{2}\right) \right)$$
Where:
$$P_i = e^{-\delta^{2}/2} \frac{\left(\delta^{2}/2\right)^i}{i!}, \qquad Q_i = e^{-\delta^{2}/2} \frac{\left(\delta^{2}/2\right)^i}{\Gamma\!\left(i+\frac{3}{2}\right)}, \qquad x = \frac{t^{2}}{\nu+t^{2}}$$
Usage
non_central_t_distribution(df, delta)
non_central_t_pdf(x, df, delta)
non_central_t_lpdf(x, df, delta)
non_central_t_cdf(x, df, delta)
non_central_t_lcdf(x, df, delta)
non_central_t_quantile(p, df, delta)Value
A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.
See also
Boost Documentation for more details on the mathematical background.
Examples
# Noncentral T distribution with 5 degrees of freedom and noncentrality parameter 1
dist <- non_central_t_distribution(5, 1)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.3021259
logcdf(dist, 0.5)
#> [1] -1.196912
pdf(dist, 0.5)
#> [1] 0.3360046
logpdf(dist, 0.5)
#> [1] -1.090631
hazard(dist, 0.5)
#> [1] 0.4814687
chf(dist, 0.5)
#> [1] 0.3597165
mean(dist)
#> [1] 1.189416
median(dist)
#> [1] 1.052851
mode(dist)
#> [1] 0.8781834
range(dist)
#> [1] -1.797693e+308 1.797693e+308
quantile(dist, 0.2)
#> [1] 0.165306
standard_deviation(dist)
#> [1] 1.385144
support(dist)
#> [1] -1.797693e+308 1.797693e+308
variance(dist)
#> [1] 1.918623
skewness(dist)
#> [1] 1.26633
kurtosis(dist)
#> [1] 13.32067
kurtosis_excess(dist)
#> [1] 10.32067
# Convenience functions
non_central_t_pdf(0, 5, 1)
#> [1] 0.2302431
non_central_t_lpdf(0, 5, 1)
#> [1] -1.46862
non_central_t_cdf(0, 5, 1)
#> [1] 0.1586553
non_central_t_lcdf(0, 5, 1)
#> [1] -1.841022
non_central_t_quantile(0.5, 5, 1)
#> [1] 1.052851