# Standardized Student t Distribution These functions calculate and differentiate a cumulative distribution function and density function of the standardized (to zero mean and unit variance) Student distribution. Quantile function and random numbers generator are also provided. ```r dt0(x, df = 10, log = FALSE, grad_x = FALSE, grad_df = FALSE) pt0(x, df = 10, log = FALSE, grad_x = FALSE, grad_df = FALSE, n = 10L) rt0(n = 1L, df = 10) qt0(x = 1L, df = 10) ``` ## Arguments - `x`: numeric vector of quantiles. - `df`: positive real value representing the number of degrees of freedom. Since this function deals with standardized Student distribution, argument `df` should be greater than `2` because otherwise variance is undefined. - `log`: logical; if `TRUE` then probabilities (or densities) p are given as log(p) and derivatives will be given respect to log(p). - `grad_x`: logical; if `TRUE` then function returns a derivative respect to `x`. - `grad_df`: logical; if `TRUE` then function returns a derivative respect to `df`. - `n`: positive integer. If `rt0` function is used then this argument represents the number of random draws. Otherwise `n` states for the number of iterations used to calculate the derivatives associated with `pt0` function via `pbetaDiff` function. ## Returns Function `rt0` returns a numeric vector of random numbers. Function `qt0` returns a numeric vector of quantiles. Functions `pt0` and `dt0` return a list which may contain the following elements: * `prob` - numeric vector of probabilities calculated for each element of `x`. Exclusively for `pt0` function. * `den` - numeric vector of densities calculated for each each element of `x`. Exclusively for `dt0` function. * `grad_x` - numeric vector of derivatives respect to `p` for each element of `x`. This element appears only if input argument `grad_x` is `TRUE`. * `grad_df` - numeric vector of derivatives respect to `q` for each element of `x`. This element appears only if input argument `grad_df` is `TRUE`. ## Details Standardized (to zero mean and unit variance) Student distribution has the following density and cumulative distribution functions: $$ f(x) = \frac{\Gamma\left(\frac{v + 1}{2}\right)}{\sqrt{(v - 2)\pi}\Gamma\left(\frac{v}{2}\right)}\left(1 + \frac{x^2}{v - 2}\right)^{-\frac{v+1}{2}}, $$ $$ F(x) =\begin{cases}1 - \frac{1}{2}I(\frac{v - 2}{x^2 + v - 2},\frac{v}{2}, \frac{1}{2})\text{, if }x \geq 0\\\frac{1}{2}I(\frac{v - 2}{x^2 + v - 2},\frac{v}{2}, \frac{1}{2})\text{, if }x < 0\end{cases}, $$ where $v > 2$ is the number of degrees of freedom `df` and $I(.)$ is a cumulative distribution function of beta distribution which is calculated by `pbeta` function. ## Examples ```r # Simple examples pt0(x = 0.3, df = 10, log = FALSE, grad_x = TRUE, grad_df = TRUE) dt0(x = 0.3, df = 10, log = FALSE, grad_x = TRUE, grad_df = TRUE) qt0(x = 0.3, df = 10) # Compare analytical and numeric derivatives delta <- 1e-6 x <- c(-2, -1, 0, 1, 2) df <- 5 # For probabilities out <- pt0(x, df = df, grad_x = TRUE, grad_df = TRUE) p0 <- out$prob # grad_x p1 <- pt0(x + delta, df = df)$prob data.frame(numeric = (p1 - p0) / delta, analytical = out$grad_x) # grad_df p1 <- pt0(x, df = df + delta)$prob data.frame(numeric = (p1 - p0) / delta, analytical = out$grad_df) # For densities out <- dt0(x, df = df, grad_x = TRUE, grad_df = TRUE) p0 <- out$den # grad_x p1 <- dt0(x + delta, df = df)$den data.frame(numeric = (p1 - p0) / delta, analytical = out$grad_x) # grad_df p1 <- dt0(x, df = df + delta)$den data.frame(numeric = (p1 - p0) / delta, analytical = out$grad_df) ```

stdt function