sadists (version 0.2.5)

upsilon: The upsilon distribution.

Description

Density, distribution function, quantile function and random generation for the upsilon distribution.

Usage

dupsilon(x, df, t, log = FALSE, order.max=6)
pupsilon(q, df, t, lower.tail = TRUE, log.p = FALSE, order.max=6)
qupsilon(p, df, t, lower.tail = TRUE, log.p = FALSE, order.max=6)
rupsilon(n, df, t)

Value

dupsilon gives the density, pupsilon gives the distribution function, qupsilon gives the quantile function, and rupsilon generates random deviates.

Invalid arguments will result in return value NaN with a warning.

Arguments

x, q

vector of quantiles.

df

the degrees of freedom in the chi square. a vector. we do not vectorize over this variable.

t

the scaling parameter on the chi. a vector. should be the same length as df. we do not vectorize over this variable.

log

logical; if TRUE, densities \(f\) are given as \(\mbox{log}(f)\).

order.max

the order to use in the approximate density, distribution, and quantile computations, via the Gram-Charlier, Edeworth, or Cornish-Fisher expansion.

p

vector of probabilities.

n

number of observations.

log.p

logical; if TRUE, probabilities p are given as \(\mbox{log}(p)\).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

Author

Steven E. Pav

Details

Suppose \(x_i \sim \chi^2\left(\nu_i\right)\) independently and independently of \(Z\), a standard normal. Then $$\Upsilon = Z + \sum_i t_i \sqrt{x_i/\nu_i}$$ takes an upsilon distribution with parameter vectors \([\nu_1, \nu_2, \ldots, \nu_k]', [t_1, t_2, ..., t_k]'\).

The upsilon distribution is used in certain tests of the Sharpe ratio for independent observations, and generalizes the lambda prime distribution, which can be written as \(Z + t \sqrt{x/\nu}\).

References

Lecoutre, Bruno. "Another look at confidence intervals for the noncentral t distribution." Journal of Modern Applied Statistical Methods 6, no. 1 (2007): 107--116. https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1128&context=jmasm

Lecoutre, Bruno. "Two useful distributions for Bayesian predictive procedures under normal models." Journal of Statistical Planning and Inference 79 (1999): 93--105.

Pav, Steven. "Inference on the Sharpe ratio via the upsilon distribution.' Arxiv (2015). https://arxiv.org/abs/1505.00829

See Also

lambda-prime distribution functions, dlambdap, plambdap, qlambdap, rlambdap. Sum of chi-squares to power distribution functions, dsumchisqpow, psumchisqpow, qsumchisqpow, rsumchisqpow.

Examples

mydf <- c(100,30,50)
myt <- c(-1,3,5)
rv <- rupsilon(500, df=mydf, t=myt)
d1 <- dupsilon(rv, df=mydf, t=myt)
# \donttest{
plot(rv,d1)
# }
p1 <- pupsilon(rv, df=mydf, t=myt)
# should be nearly uniform:
# \donttest{
plot(ecdf(p1))
# }
q1 <- qupsilon(ppoints(length(rv)),df=mydf,t=myt)
# \donttest{
qqplot(x=rv,y=q1)
# }
# \donttest{
if (require(SharpeR)) {
  ope <- 252
  n.sim <- 500
  n.term <- 3
  set.seed(234234)
  pp <- replicate(n.sim,{
    # these are population parameters 
    a <- rnorm(n.term) 
    psi <- 6 * rnorm(length(a)) / sqrt(ope)
    b <- sum(a * psi)
    df <- 100 + ceiling(200 * runif(length(psi)))
    comm <- 1 / sqrt(sum(a^2 / df))
    cdf <- df - 1
    # now independent draws from the SR distribution:
    x <- rsr(length(df), df, zeta=psi, ope=1)
    # now compute a p-value under the true null
    pupsilon(comm * b,df=cdf,t=comm*a*x) 
  })
  # ought to be uniform:
  plot(ecdf(pp)) 
}
# }