Title: Externally Studentized Midrange Distribution
Version: 2.1.0
Description: Computes the studentized midrange distribution (pdf, cdf and quantile) and generates random numbers.
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL: https://bendeivide.github.io/SMR/, https://github.com/bendeivide/SMR
BugReports: https://github.com/bendeivide/SMR/issues
RoxygenNote: 7.2.0
LinkingTo: Rcpp
Imports: Rcpp
NeedsCompilation: yes
Packaged: 2023-10-07 17:19:49 UTC; MABENLA
Author: Ben Deivide de Oliveira Batista ORCID iD [aut], Daniel Furtado Ferreira ORCID iD [aut, cre], Diego Arthur Bispo Justino de Oliveira [ctb], Matheus Fernando Rodrigues Santos [ctb]
Maintainer: Daniel Furtado Ferreira <danielff@ufla.br>
Repository: CRAN
Date/Publication: 2023-10-07 22:40:10 UTC

The externally studentized normal midrange distribution

Description

Computes the probability density, the cumulative distribution function and the quantile function and generates random samples for the externally studentized normal midrange distribution with the numbers means equal to size, the degrees of freedom equal to df and the number of points of the Gauss-Legendre quadrature equal to np.

Usage

dSMR(x, size, df, np=32, log = FALSE)
pSMR(q, size, df, np=32, lower.tail = TRUE, log.p = FALSE)
qSMR(p, size, df, np=32, eps = 1e-13, maxit = 5000, lower.tail = TRUE, log.p = FALSE)
rSMR(n, size,  df = Inf)

Arguments

x, q

vector of quantiles x \in R and q \in R.

p

vector of probabilities (0, 1).

size

sample size. Only for size > 1.

n

vector size to be simulated n > 1.

df

degrees of freedom df > 0.

np

number of points of the gaussian quadrature np > 2.

log, log.p

logical argument; if TRUE, the probabilities p are given as log(p).

lower.tail

logical argument; if TRUE, the probabilities are P[X \leq x] otherside, P[X \geq x].

eps

stopping criterion for Newton-Raphson's iteraction method.

maxit

maximum number of interaction in the Newton-Raphson method.

Details

Assumes np = 32 as default value for dSMR, pSMR and qSMR. If df is not specified, it assumes the default value Inf in rSMR. When df=1, the convergence of the routines requires np>250 to obtain the desired result accurately. The Midrange distribution has density

f(\overline{q};n,\nu) =\int^{\infty}_{0} \int^{x\overline{q}}_{-\infty}2n(n-1)x\phi(y) \phi(2x\overline{q}-y)[\Phi(2x\overline{q}-y)-\Phi(y)]^{n-2}f(x;\nu)dydx,

where, q is the quantile of externally studentized midrange distribution, n (size) is the sample size and \nu is the degrees of freedon.

The externally studentized midrange distribution function is given by

F(\overline{q};n,\nu)=\int^{\overline{q}}_{-\infty} \int^{\infty}_{0}\int^{x\overline{q}}_{-\infty}2n(n-1)x\phi(y) \phi(2xz-y)[\Phi(2xz-y)-\Phi(y)]^{n-2}f(x;\nu)dydxdz.

where, q is the quantile of externally studentized midrange distribution, n (size) is the sample size and \nu is the degrees of freedon.

Value

dSMR gives the density, pSMR gives the cumulative distribution function, qSMR gives the quantile function, and rSMR generates random deviates.

References

BATISTA, B. D. de O.; FERREIRA, D. F. SMR: An R package for computing the externally studentized normal midrange distribution. The R Journal, v. 6, n. 2, p. 123-136, dez. 2014.

Examples

library(SMR)

#example 1:
x  <- 2
q  <- 1
p  <- 0.9
n  <- 30
size  <- 5
df <- 3
np <- 32
dSMR(x, size, df, np)
pSMR(q, size, df, np)
qSMR(p, size, df, np)
rSMR(n, size, df)

#example 2:
x  <- c(-1, 2, 1.1)
q  <- c(1, 0, -1.5)
p  <- c(0.9, 1, 0.8)
n  <- 10
size  <- 5
df <- 3
np <- 32
dSMR(x, size, df, np)
pSMR(q, size, df, np)
qSMR(p, size, df, np)
rSMR(n, size, df)