empirical function

The empirical distribution

Density, distribution function, quantile function, and raw moments for the empirical distribution.

dempirical(x, data, log = FALSE)

pempirical(q, data, log.p = FALSE, lower.tail = TRUE)

qempirical(p, data, lower.tail = TRUE, log.p = FALSE)

mempirical(r = 0, data, truncation = NULL, lower.tail = TRUE)

Arguments

• x, q: vector of quantiles
• data: data vector
• log, log.p: logical; if TRUE, probabilities p are given as log(p).
• lower.tail: logical; if TRUE (default), moments are $E[x^r|X \le y]$, otherwise, $E[x^r|X > y]$
• p: vector of probabilities
• r: rth raw moment of the Pareto distribution
• truncation: lower truncation parameter, defaults to NULL.

Returns

dempirical returns the density, pempirical the distribution function, qempirical the quantile function, mempirical gives the rth moment of the distribution or a function that allows to evaluate the rth moment of the distribution if truncation is NULL..

Details

The density function is a standard Kernel density estimation for 1e6 equally spaced points. The cumulative Distribution Function:

The y-bounded r-th raw moment of the empirical distribution equals:

Examples

#'
## Generate random sample to work with
x <- rlnorm(1e5, meanlog = -0.5, sdlog = 0.5)

## Empirical density
plot(x = seq(0, 5, length.out = 100), y = dempirical(x = seq(0, 5, length.out = 100), data = x))

# Compare empirical and parametric quantities
dlnorm(0.5, meanlog = -0.5, sdlog = 0.5)
dempirical(0.5, data = x)

plnorm(0.5, meanlog = -0.5, sdlog = 0.5)
pempirical(0.5, data = x)

qlnorm(0.5, meanlog = -0.5, sdlog = 0.5)
qempirical(0.5, data = x)

mlnorm(r = 0, truncation = 0.5, meanlog = -0.5, sdlog = 0.5)
mempirical(r = 0, truncation = 0.5, data = x)

mlnorm(r = 1, truncation = 0.5, meanlog = -0.5, sdlog = 0.5)
mempirical(r = 1, truncation = 0.5, data = x)

## Demonstration of log functionailty for probability and quantile function
quantile(x, 0.5, type = 1)
qempirical(p = pempirical(q = quantile(x, 0.5, type = 1), data = x, log.p = TRUE),
data = x, log.p = TRUE)

## The zeroth truncated moment is equivalent to the probability function
pempirical(q = quantile(x, 0.5, type = 1), data = x)
mempirical(truncation = quantile(x, 0.5, type = 1), data = x)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample,
#for large enough samples.
mean(x)
mempirical(r = 1, data = x, truncation = 0, lower.tail = FALSE)

sum(x[x > quantile(x, 0.1)]) / length(x)
mempirical(r = 1, data = x, truncation = quantile(x, 0.1), lower.tail = FALSE)
#'