distributionsrd (version 0.0.6)

empirical: The empirical distribution

Description

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

Usage

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.

Value

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:

$$F_n(x) = \frac{1}{n}\sum_{i=1}^{n}I_{x_i \leq x}$$

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

$$ \mu^{r}_{y} = \frac{1}{n}\sum_{i=1}^{n}I_{x_i \leq x}x^r$$

Examples

# NOT RUN {
#'
## 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)
#'
# }