stdt function

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.

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:

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

# 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)