sadists
Some Additional Distributions apparently not available in R.
-- Steven E. Pav, shabbychef@gmail.com
Installation
This package can be installed from CRAN, via drat, or from github:
# via CRAN:
install.packages("sadists")
# via drat:
if (require(drat)) {
drat:::add("shabbychef")
install.packages("sadists")
}
# via devtools (typically 'master' is stable):
if (require(devtools)) {
install_github("shabbychef/sadists")
}Testing distributions
First some functions to test the 'dpqr' functions:
testf <- function(dpqr, nobs, ...) {
require(ggplot2)
require(grid)
set.seed(3940071)
rv <- sort(dpqr$r(nobs, ...))
data <- data.frame(draws = rv, pvals = dpqr$p(rv,
...))
text.size <- 8 # sigh
# http://stackoverflow.com/a/5688125/164611
p1 <- qplot(rv, geom = "blank") + geom_line(aes(y = ..density..,
colour = "Empirical"), stat = "density") +
stat_function(fun = function(x) {
dpqr$d(x, ...)
}, aes(colour = "Theoretical")) + geom_histogram(aes(y = ..density..),
alpha = 0.3) + scale_colour_manual(name = "Density",
values = c("red", "blue")) + theme(text = element_text(size = text.size)) +
labs(title = "Density (tests dfunc)")
# Q-Q plot
p2 <- ggplot(data, aes(sample = draws)) + stat_qq(distribution = function(p) {
dpqr$q(p, ...)
}) + geom_abline(slope = 1, intercept = 0, colour = "red") +
theme(text = element_text(size = text.size)) +
labs(title = "Q-Q plot (tests qfunc)")
# empirical CDF of the p-values; should be uniform
p3 <- ggplot(data, aes(sample = pvals)) + stat_qq(distribution = qunif) +
geom_abline(slope = 1, intercept = 0, colour = "red") +
theme(text = element_text(size = text.size)) +
labs(title = "P-P plot (tests pfunc)")
# Define grid layout to locate plots and print each
# graph
pushViewport(viewport(layout = grid.layout(2, 2)))
print(p1, vp = viewport(layout.pos.row = 1, layout.pos.col = 1:2))
print(p2, vp = viewport(layout.pos.row = 2, layout.pos.col = 1))
print(p3, vp = viewport(layout.pos.row = 2, layout.pos.col = 2))
}Weighted sum of (non-central) chi-squares to power
This distribution is the weighted sum of independent non-central chi-square variates taken to some powers. The special case where the powers are all one half is related to the upsilon distribution. The special case where the powers are all one could be used to compute the distribution of the (doubly non-central) F distribution.
require(sadists)
wts <- c(-1, 1, 3, -3)
df <- c(100, 200, 100, 50)
ncp <- c(0, 1, 0.5, 2)
pow <- c(1, 0.5, 2, 1.5)
testf(list(d = dsumchisqpow, p = psumchisqpow, q = qsumchisqpow,
r = rsumchisqpow), nobs = 2^14, wts, df, ncp, pow)K-prime distribution
The K-prime distribution is the weighted sum of a standard normal and an independent central chi, all divided by another independent central chi. Depending on the degrees of freedom and the weights, the K-prime can appears as a Lambda-prime, a normal, or a central t.
require(sadists)
v1 <- 50
v2 <- 80
a <- 0.5
b <- 1.5
testf(list(d = dkprime, p = pkprime, q = qkprime, r = rkprime),
nobs = 2^14, v1, v2, a, b)Lambda prime distribution
A Lambda prime random variable is the sum of a standard normal and an independent, scaled central chi random variable.
require(sadists)
df <- 70
ts <- 2
testf(list(d = dlambdap, p = plambdap, q = qlambdap,
r = rlambdap), nobs = 2^14, df, ts)Upsilon distribution
An upsilon random variable is the sum of a standard normal and the weighted sum of several indpendent central chis.
require(sadists)
df <- c(30, 50, 100, 20, 10)
ts <- c(-3, 2, 5, -4, 1)
testf(list(d = dupsilon, p = pupsilon, q = qupsilon,
r = rupsilon), nobs = 2^14, df, ts)Doubly non-central t distribution
The doubly non-central t distribution generalizes the t distribution to the case where the denominator chi-square is non-central.
require(sadists)
df <- 75
ncp1 <- 2
ncp2 <- 3
testf(list(d = ddnt, p = pdnt, q = qdnt, r = rdnt),
nobs = 2^14, df, ncp1, ncp2)Doubly non-central F distribution
The doubly non-central F distribution generalizes the F distribution to the case where the denominator chi-square is non-central.
require(sadists)
df1 <- 40
df2 <- 80
ncp1 <- 1.5
ncp2 <- 2.5
testf(list(d = ddnf, p = pdnf, q = qdnf, r = rdnf),
nobs = 2^14, df1, df2, ncp1, ncp2)Doubly non-central Beta distribution
The doubly non-central Beta distribution can be viewed as a transformation of the doubly non-central F distribution.
require(sadists)
df1 <- 40
df2 <- 80
ncp1 <- 1.5
ncp2 <- 2.5
testf(list(d = ddnbeta, p = pdnbeta, q = qdnbeta, r = rdnbeta),
nobs = 2^14, df1, df2, ncp1, ncp2)Doubly non-central Eta distribution
The doubly non-central Eta distribution can be viewed as the square root of the doubly non-central Beta distribution. It is a transform of the doubly non-central t distribution.
require(sadists)
df <- 100
ncp1 <- 0.5
ncp2 <- 2.5
testf(list(d = ddneta, p = pdneta, q = qdneta, r = rdneta),
nobs = 2^14, df, ncp1, ncp2)Sum of logs of (non-central) chi-squares
This distribution is the sum of logs of independent non-central chi-square variates.
require(sadists)
wts <- c(5, -4, 10, -15)
df <- c(100, 200, 100, 50)
ncp <- c(0, 1, 0.5, 2)
testf(list(d = dsumlogchisq, p = psumlogchisq, q = qsumlogchisq,
r = rsumlogchisq), nobs = 2^14, wts, df, ncp)Product of doubly non-central F variates
This distribution is the product of independent doubly non-central F variates. The PDQ functions are computed by transformation on the sum of log chi-squares distribution.
require(sadists)
df1 <- c(10, 20, 5)
df2 <- c(1000, 500, 150)
ncp1 <- c(1, 0, 2.5)
ncp2 <- c(0, 1.5, 5)
testf(list(d = dproddnf, p = pproddnf, q = qproddnf,
r = rproddnf), nobs = 2^14, df1, df2, ncp1, ncp2)Product of (non-central) chi-squares to power
This distribution is the product of independent non-central chi-square variates taken to some powers. The PDQ functions are computed by transformation on the sum of log chi-squares distribution.
require(sadists)
df <- c(100, 200, 100, 50)
ncp <- c(0, 1, 0.5, 2)
pow <- c(1, 0.5, 2, 1.5)
testf(list(d = dprodchisqpow, p = pprodchisqpow, q = qprodchisqpow,
r = rprodchisqpow), nobs = 2^14, df, ncp, pow)