Description Usage Arguments Details Value Examples
Density, distribution function, quantile function, and raw moments for the empirical distribution.
1 2 3 4 5 6 7 | 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)
|
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 ≤ 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. |
The density function is a standard Kernel density estimation for 1e6 equally spaced points. The cumulative Distribution Function:
F_n(x) = \frac{1}{n}∑_{i=1}^{n}I_{x_i ≤q x}
The y-bounded r-th raw moment of the empirical distribution equals:
μ^{r}_{y} = \frac{1}{n}∑_{i=1}^{n}I_{x_i ≤q x}x^r
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..
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | #'
## 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)
#'
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