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Maintainer:	Steven E. Pav <shabbychef@gmail.com>
Version:	0.2.5
Date:	2023-08-20
License:	LGPL-3
Title:	Some Additional Distributions
BugReports:	https://github.com/shabbychef/sadists/issues
Description:	Provides the density, distribution, quantile and generation functions of some obscure probability distributions, including the doubly non-central t, F, Beta, and Eta distributions; the lambda-prime and K-prime; the upsilon distribution; the (weighted) sum of non-central chi-squares to a power; the (weighted) sum of log non-central chi-squares; the product of non-central chi-squares to powers; the product of doubly non-central F variables; the product of independent normals.
Depends:	R (≥ 3.0.2)
Imports:	PDQutils (≥ 0.1.1), hypergeo, orthopolynom
Suggests:	SharpeR, shiny, testthat, ggplot2, xtable, formatR, knitr
URL:	https://github.com/shabbychef/sadists
VignetteBuilder:	knitr
Collate:	'cumulants.r' 'dnbeta.r' 'dneta.r' 'dnf.r' 'dnt.r' 'kprime.r''lambdap.r' 'moments.r' 'prodchisqpow.r' 'proddnf.r''prodnormal.r' 'runExample.r' 'sadists.r' 'sumchisqpow.r''sumlogchisq.r' 'upsilon.r' 'utils.r'
RoxygenNote:	7.1.1
NeedsCompilation:	no
Packaged:	2023-08-21 16:02:29 UTC; spav
Author:	Steven E. Pav [aut, cre]
Repository:	CRAN
Date/Publication:	2023-08-21 18:30:02 UTC

Some Additional Distributions

Description

Some Additional Distributions.

Details

A collection of distributions which can be approximated viaEdgeworth and Cornish-Fisher expansions

Sum of (non-central) chi-square to powers

LetX_i \sim \chi^2\left(\delta_i, \nu_i\right)be independently distributed non-central chi-squares, where\nu_iare the degrees of freedom, and\delta_i are thenon-centrality parameters. Letw_i andp_i be given constants. Suppose

Y = \sum_i w_i X_i^{p_i}.

ThenY follows a weighted sum of chi-squares to power distribution. The special case where all thep_i are one is a 'sum ofchi-squares' distribution; The special case where all thep_i are one half is a 'sum ofchis' distribution;

Lambda Prime

Introduced by Lecoutre, the lambda prime distributionfinds use in inference on the Sharpe ratio under normalreturns.Supposey \sim \chi^2\left(\nu\right), andZ is a standard normal.

T = Z + t \sqrt{y/\nu}

takes a lambda prime distribution with parameters\nu, t.A lambda prime random variable can be viewed as a confidencevariable on a non-central t because

t = \frac{Z' + T}{\sqrt{y/\nu}}

Upsilon

The upsilon distribution generalizes the lambda prime to thecase of the sum of multiple chi variables. That is,supposey_i \sim \chi^2\left(\nu_i\right)independently and independently ofZ, a standard normal. Then

T = Z + \sum_i t_i \sqrt{y_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 ofthe Sharpe ratio for independent observations.

K Prime

Introduced by Lecoutre, the K prime family of distributions generalizethe (singly) non-central t, and lambda prime distributions. Supposey \sim \chi^2\left(\nu_1\right), andx \sim t \left(\nu_2, a\sqrt{y/\nu_1}/b\right).Then the random variable

T = b x

takes a K prime distribution with parameters\nu_1, \nu_2, a, b. In Lecoutre's terminology,T \sim K'_{\nu_1, \nu_2}\left(a, b\right)

Equivalently, we can think of

T = \frac{b Z + a \sqrt{\chi^2_{\nu_1} / \nu_1}}{\sqrt{\chi^2_{\nu_2} / \nu_2}}

whereZ is a standard normal, and the normal and the (central) chi-squares areindependent of each other. Whena=0 we recovera central t distribution; when\nu_1=\infty we recover a rescaled non-central t distribution;whenb=0, we get a rescaled square root of a central Fdistribution; when\nu_2=\infty, we recover a Lambda prime distribution.

Doubly Noncentral t

The doubly noncentral t distribution generalizes the (singly)noncentral t distribution to the case where the numerator isthe square root of a scaled noncentral chi-square distribution.That is, ifX \sim \mathcal{N}\left(\mu,1\right) independentlyofY \sim \chi^2\left(k,\theta\right), thenthe random variable

T = \frac{X}{\sqrt{Y/k}}

takes a doubly non-central t distribution with parametersk, \mu, \theta.

Doubly Noncentral F

The doubly noncentral F distribution generalizes the (singly)noncentral F distribution to the case where the numerator isa scaled noncentral chi-square distribution.That is, ifX \sim \chi^2\left(n_1,\theta_1\right) independently ofY \sim \chi^2\left(n_2,\theta_2\right), thenthe random variable

T = \frac{X / n_1}{Y / n_2}

takes a doubly non-central F distribution with parametersn_1, n_2, \theta_1, \theta_2.

Parameter recycling

It should be noted that the functions provided by sadists donotrecycle their distribution parameters against thex, p, q orn parameters. This is in contrast to the common R idiom, and may cause some confusion. This is mostly for reasonsof performance, but also because some of the distributions have vector-valuedparameters; recycling over these would require the user to providelistsof parameters, which would be unpleasant.

Legal Mumbo Jumbo

sadists is distributed in the hope that it will be useful,but WITHOUT ANY WARRANTY; without even the implied warranty ofMERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See theGNU Lesser General Public License for more details.

Note

This package is maintained as a hobby.

Author(s)

Steven E. Pavshabbychef@gmail.com

Maintainer: Steven E. Pavshabbychef@gmail.com (ORCID)

References

Lecoutre, Bruno. "Two Useful distributions for Bayesian predictiveprocedures under normal models." Journal of Statistical Planning andInference 79, no. 1 (1999): 93-105.

Poitevineau, Jacques, and Lecoutre, Bruno. "Implementing Bayesian predictiveprocedures: The K-prime and K-square distributions." Computational Statistics and Data Analysis 54, no. 3 (2010): 724-731.https://arxiv.org/abs/1003.4890v1

Walck, C. "Handbook on Statistical Distributions for experimentalists."1996.https://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

See Also

Useful links:

• https://github.com/shabbychef/sadists

• Report bugs athttps://github.com/shabbychef/sadists/issues

The doubly non-central Beta distribution.

Description

Density, distribution function, quantile function and randomgeneration for the doubly non-central Beta distribution.

Usage

ddnbeta(x, df1, df2, ncp1, ncp2, log = FALSE, order.max=6)pdnbeta(q, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)qdnbeta(p, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)rdnbeta(n, df1, df2, ncp1, ncp2)

Arguments

Details

Supposex_i \sim \chi^2\left(\delta_i,\nu_i\right)be independent non-central chi-squares fori=1,2.Then

Y = \frac{x_1}{x_1 + x_2}

takes a doubly non-central Beta distribution with degrees of freedom\nu_1, \nu_2 and non-centrality parameters\delta_1,\delta_2.

Value

ddnbeta gives the density,pdnbeta gives the distribution function,qdnbeta gives the quantile function, andrdnbeta generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

See Also

(doubly non-central) F distribution functions,ddnf,pdnf,qdnf,rdnf.

Examples

rv <- rdnbeta(500, df1=100,df2=500,ncp1=1.5,ncp2=12)d1 <- ddnbeta(rv, df1=100,df2=500,ncp1=1.5,ncp2=12)plot(rv,d1)p1 <- ddnbeta(rv, df1=100,df2=500,ncp1=1.5,ncp2=12)# should be nearly uniform:plot(ecdf(p1))q1 <- qdnbeta(ppoints(length(rv)), df1=100,df2=500,ncp1=1.5,ncp2=12)qqplot(x=rv,y=q1)

The doubly non-central Eta distribution.

Description

Density, distribution function, quantile function and randomgeneration for the doubly non-central Eta distribution.

Usage

ddneta(x, df, ncp1, ncp2, log = FALSE, order.max=6)pdneta(q, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)qdneta(p, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)rdneta(n, df, ncp1, ncp2)

Arguments

Details

SupposeZ is a normal with mean\delta_1,and standard deviation 1, independent ofX \sim \chi^2\left(\delta_2,\nu_2\right),a non-central chi-square with\nu_2 degrees of freedomand non-centrality parameter\delta_2. Then

Y = \frac{Z}{\sqrt{Z^2 + X}}

takes a doubly non-central Eta distribution with\nu_2 degrees of freedom and non-centrality parameters\delta_1,\delta_2. Thesquare ofa doubly non-central Eta is a doubly non-central Beta variate.

Value

ddneta gives the density,pdneta gives the distribution function,qdneta gives the quantile function, andrdneta generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

See Also

(doubly non-central) t distribution functions,ddnt,pdnt,qdnt,rdnt.

(doubly non-central) Beta distribution functions,ddnbeta,pdnbeta,qdnbeta,rdnbeta.

Examples

rv <- rdneta(500, df=100,ncp1=1.5,ncp2=12)d1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12)plot(rv,d1)p1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12)# should be nearly uniform:plot(ecdf(p1))q1 <- qdneta(ppoints(length(rv)), df=100,ncp1=1.5,ncp2=12)qqplot(x=rv,y=q1)

The doubly non-central F distribution.

Description

Density, distribution function, quantile function and randomgeneration for the doubly non-central F distribution.

Usage

ddnf(x, df1, df2, ncp1, ncp2, log = FALSE, order.max=6)pdnf(q, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)qdnf(p, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)rdnf(n, df1, df2, ncp1, ncp2)

Arguments

Details

Supposex_i \sim \chi^2\left(\delta_i,\nu_i\right)be independent non-central chi-squares fori=1,2.Then

Y = \frac{x_1/\nu_1}{x_2/\nu_2}

takes a doubly non-central F distribution with degrees of freedom\nu_1, \nu_2 and non-centrality parameters\delta_1,\delta_2.

Value

ddnf gives the density,pdnf gives the distribution function,qdnf gives the quantile function, andrdnf generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

See Also

(singly non-central) F distribution functions,df,pf,qf,rf.

Examples

rv <- rdnf(500, df1=100,df2=500,ncp1=1.5,ncp2=12)d1 <- ddnf(rv, df1=100,df2=500,ncp1=1.5,ncp2=12)plot(rv,d1)p1 <- ddnf(rv, df1=100,df2=500,ncp1=1.5,ncp2=12)# should be nearly uniform:plot(ecdf(p1))q1 <- qdnf(ppoints(length(rv)), df1=100,df2=500,ncp1=1.5,ncp2=12)qqplot(x=rv,y=q1)

The doubly non-central t distribution.

Description

Density, distribution function, quantile function and randomgeneration for the doubly non-central t distribution.

Usage

ddnt(x, df, ncp1, ncp2, log = FALSE, order.max=6)pdnt(q, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)qdnt(p, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)rdnt(n, df, ncp1, ncp2)

Arguments

Details

LetZ \sim \mathcal{N}\left(\mu,1\right) independentlyofX \sim \chi^2\left(\theta,\nu\right). The random variable

T = \frac{Z}{\sqrt{X/\nu}}

takes adoubly non-central t distribution with parameters\nu, \mu, \theta.

Value

ddnt gives the density,pdnt gives the distribution function,qdnt gives the quantile function, andrdnt generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

References

Krishnan, Marakatha. "Series Representations of the Doubly Noncentralt-Distribution." Journal of the American Statistical Association 63, no. 323 (1968): 1004-1012.

See Also

t distribution functions,dt,pt,qt,rt

Examples

rvs <- rdnt(128, 20, 1, 1)dvs <- ddnt(rvs, 20, 1, 1)pvs.H0 <- pdnt(rvs, 20, 0, 1)pvs.HA <- pdnt(rvs, 20, 1, 1)plot(ecdf(pvs.H0))plot(ecdf(pvs.HA))# compare to singly non-centraldv1 <- ddnt(1, df=10, ncp1=5, ncp2=0, log=FALSE)dv2 <- dt(1, df=10, ncp=5, log=FALSE)pv1 <- pdnt(1, df=10, ncp1=5, ncp2=0, log.p=FALSE)pv11 <- pdnt(1, df=10, ncp1=5, ncp2=0.001, log.p=FALSE)v2 <- pt(1, df=10, ncp=5, log.p=FALSE)q1 <- qdnt(pv1, df=10, ncp1=5, ncp2=0, log.p=FALSE)

The K prime distribution.

Description

Density, distribution function, quantile function and randomgeneration for the K prime distribution.

Usage

dkprime(x, v1, v2, a, b = 1, order.max=6, log = FALSE)pkprime(q, v1, v2, a, b = 1, order.max=6, lower.tail = TRUE, log.p = FALSE)qkprime(p, v1, v2, a, b = 1, order.max=6, lower.tail = TRUE, log.p = FALSE)rkprime(n, v1, v2, a, b = 1)

Arguments

Details

Supposey \sim \chi^2\left(\nu_1\right), andx \sim t \left(\nu_2, a\sqrt{y/\nu_1}/b\right).Then the random variable

T = b x

takes a K prime distribution with parameters\nu_1, \nu_2, a, b. In Lecoutre's terminology,T \sim K'_{\nu_1, \nu_2}\left(a, b\right)

Equivalently, we can think of

T = \frac{b Z + a \sqrt{\chi^2_{\nu_1} / \nu_1}}{\sqrt{\chi^2_{\nu_2} / \nu_2}}

whereZ is a standard normal, and the normal and the (central) chi-squares areindependent of each other. Whena=0 we recovera central t distribution; when\nu_1=\infty we recover a rescaled non-central t distribution;whenb=0, we get a rescaled square root of a central Fdistribution; when\nu_2=\infty, we recover a Lambda prime distribution.

Value

dkprime gives the density,pkprime gives the distribution function,qkprime gives the quantile function, andrkprime generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

References

Lecoutre, Bruno. "Two Useful distributions for Bayesian predictiveprocedures under normal models." Journal of Statistical Planning andInference 79, no. 1 (1999): 93-105.

Poitevineau, Jacques, and Lecoutre, Bruno. "Implementing Bayesian predictiveprocedures: The K-prime and K-square distributions." Computational Statistics and Data Analysis 54, no. 3 (2010): 724-731.https://arxiv.org/abs/1003.4890v1

See Also

t distribution functions,dt,pt,qt,rt,lambda prime distribution functions,dlambdap,plambdap,qlambdap,rlambdap.

Examples

d1 <- dkprime(1, 50, 20, a=0.01)d2 <- dkprime(1, 50, 20, a=0.0001)d3 <- dkprime(1, 50, 20, a=0)d4 <- dkprime(1, 10000, 20, a=1)d5 <- dkprime(1, Inf, 20, a=1)

The lambda prime distribution.

Description

Density, distribution function, quantile function and randomgeneration for the lambda prime distribution.

Usage

dlambdap(x, df, t, log = FALSE, order.max=6)plambdap(q, df, t, lower.tail = TRUE, log.p = FALSE, order.max=6)qlambdap(p, df, t, lower.tail = TRUE, log.p = FALSE, order.max=6)rlambdap(n, df, t)

Arguments

Details

Supposey \sim \chi^2\left(\nu\right), andZ is a standard normal.

T = Z + t \sqrt{y/\nu}

takes a lambda prime distribution with parameters\nu, t.A lambda prime random variable can be viewed as a confidencelevel on a non-central t because

t = \frac{Z' + T}{\sqrt{y/\nu}}

Value

dlambdap gives the density,plambdap gives the distribution function,qlambdap gives the quantile function, andrlambdap generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

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.

See Also

t distribution functions,dt,pt,qt,rt,K prime distribution functions,dkprime,pkprime,qkprime,rkprime,upsilon distribution functions,dupsilon,pupsilon,qupsilon,rupsilon,

Examples

rv <- rlambdap(100, 50, t=0.01)d1 <- dlambdap(1, 50, t=0.01)pv <- plambdap(rv, 50, t=0.01)qv <- qlambdap(ppoints(length(rv)), 50, t=1)

The product of (non-central) chi-squares raised to powers distribution.

Description

Density, distribution function, quantile function and randomgeneration for the distribution of the product of non-centralchi-squares taken to powers.

Usage

dprodchisqpow(x, df, ncp=0, pow=1, log = FALSE, order.max=5)pprodchisqpow(q, df, ncp=0, pow=1, lower.tail = TRUE, log.p = FALSE, order.max=5)qprodchisqpow(p, df, ncp=0, pow=1, lower.tail = TRUE, log.p = FALSE, order.max=5)rprodchisqpow(n, df, ncp=0, pow=1)

Arguments

Details

LetX_i \sim \chi^2\left(\delta_i, \nu_i\right)be independently distributed non-central chi-squares, where\nu_iare the degrees of freedom, and\delta_i are thenon-centrality parameters. Letp_i be given constants. Suppose

Y = \prod_i X_i^{p_i}.

ThenY follows a product of chi-squares to power distribution.

Value

dprodchisqpow gives the density,pprodchisqpow gives the distribution function,qprodchisqpow gives the quantile function, andrprodchisqpow generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

The PDQ functions are computed by translation of the sum of log chi-squares distribution functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

References

Pav, Steven. Moments of the log non-central chi-square distribution.https://arxiv.org/abs/1503.06266

See Also

The sum of log of chi-squares distribution,dsumlogchisq,psumlogchisq,qsumlogchisq,rsumlogchisq,The upsilon distribution,dupsilon,pupsilon,qupsilon,rupsilon.The sum of chi-square powers distribution,dsumchisqpow,psumchisqpow,qsumchisqpow,rsumchisqpow.

Examples

df <- c(100,20,10)ncp <- c(5,3,1)pow <- c(1,0.5,1)rvs <- rprodchisqpow(128, df, ncp, pow)dvs <- dprodchisqpow(rvs, df, ncp, pow)qvs <- pprodchisqpow(rvs, df, ncp, pow)pvs <- qprodchisqpow(ppoints(length(rvs)), df, ncp, pow)

The product of multiple doubly non-central F's distribution.

Description

Density, distribution function, quantile function and randomgeneration for the product of multiple independent doubly non-central F variates.

Usage

dproddnf(x, df1, df2, ncp1, ncp2, log = FALSE, order.max=4)pproddnf(q, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=4)qproddnf(p, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=4)rproddnf(n, df1, df2, ncp1, ncp2)

Arguments

Details

Let

x_j \sim F\left(\delta_{1,j},\delta_{2,j},\nu_{1,j},\nu_{2,j}\right)

be independent doubly non-central F variates with non-centrality parameters\delta_{i,j} and degrees of freedom\nu_{i,j} fori=1,2,\ldots,I andj=1,2.Then

Y = \prod_j x_j

takes a product of doubly non-central F's distribution. We take theparameters of this distribution as the fourI length vectorsof the two degrees of freedom and the two non-centrality parameters.

Value

dproddnf gives the density,pproddnf gives the distribution function,qproddnf gives the quantile function, andrproddnf generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

The PDQ functions are computed by translation of the sum of log chi-squares distribution functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

References

Pav, Steven. Moments of the log non-central chi-square distribution.https://arxiv.org/abs/1503.06266

See Also

The sum of log of chi-squares distribution,dsumlogchisq,psumlogchisq,qsumlogchisq,rsumlogchisq.(doubly non-central) F distribution functions,ddnf,pdnf,qdnf,rdnf.

Examples

df1 <- c(10,20,5)df2 <- c(1000,500,150)ncp1 <- c(1,0,2.5)ncp2 <- c(0,1.5,5)rv <- rproddnf(500, df1=df1,df2=df2,ncp1=ncp1,ncp2=ncp2)d1 <- dproddnf(rv, df1=df1,df2=df2,ncp1=ncp1,ncp2=ncp2)plot(rv,d1)p1 <- pproddnf(rv, df1=df1,df2=df2,ncp1=ncp1,ncp2=ncp2)# should be nearly uniform:plot(ecdf(p1))q1 <- qproddnf(ppoints(length(rv)), df1=df1,df2=df2,ncp1=ncp1,ncp2=ncp2)qqplot(x=rv,y=q1)

The product of normal random variates.

Description

Density, distribution function, quantile function and randomgeneration for the distribution of the product of indepdendentnormal random variables.

Usage

dprodnormal(x, mu, sigma, log = FALSE, order.max=5)pprodnormal(q, mu, sigma, lower.tail = TRUE, log.p = FALSE, order.max=5)qprodnormal(p, mu, sigma, lower.tail = TRUE, log.p = FALSE, order.max=5)rprodnormal(n, mu, sigma)

Arguments

Details

LetZ_i \sim \mathcal{N}\left(\mu_i, \sigma_i^2\right)be independently distributed normal variates, with means\mu_iand variances\sigma_i^2.Suppose

Y = \prod_i Z_i.

ThenY follows a product of normals distribution.

Value

dprodnormal gives the density,pprodnormal gives the distribution function,qprodnormal gives the quantile function, andrprodnormal generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

Examples

mu <- c(100,20,10)sigma <- c(10,50,10)rvs <- rprodnormal(128, mu, sigma)dvs <- dprodnormal(rvs, mu, sigma)qvs <- pprodnormal(rvs, mu, sigma)pvs <- qprodnormal(ppoints(length(rvs)), mu, sigma)

Run Shiny Application

Description

Runs a shiny application which draws from the given distributions, thenillustrates the fidelity of the density, CDF, and quantile functions.

Usage

runExample(port=NULL,launch.browser=TRUE, host=getOption('shiny.host','127.0.0.1'),display.mode='auto')

Arguments

Details

Launches shiny applications, and optionally, your system's web browser.Draws are taken from the random variable, and d-d, q-q, and p-p plotsare available.

Author(s)

Steven E. Pavshabbychef@gmail.com

References

Attali, D. "Supplementing your R package with a shiny app."https://deanattali.com/2015/04/21/r-package-shiny-app/

Examples

## Not run: runExample(launch.browser=TRUE)## End(Not run)

News for package 'sadists'

Description

History of the 'sadists' package.

Version 0.2.5 (2023-08-20)

• CRAN emergency release.

Version 0.2.4 (2020-06-23)

• CRAN emergency release.

Version 0.2.3 (2017-03-19)

• add product of normals distribution.

• move github figures to location CRAN understands.

Version 0.2.2 (2016-03-03)

• work around badrchisq whendf=0=ncp (?)

• incompatibilities in vignette with ggplot2 release.

Version 0.2.1 (2015-06-12)

• shiny app (h/t Dean Attali).

Version 0.2.0 (2015-04-01)

• add doubly non-central Beta and Eta distributions.

• add (sum of) log chi-square distribution.

• have products of chi-square depend on transform of this distribution.

Initial Version 0.1.0 (2015-03-07)

• first CRAN release.

The sum of (non-central) chi-squares raised to powers distribution.

Description

Density, distribution function, quantile function and randomgeneration for the distribution of the weighted sum of non-centralchi-squares taken to powers.

Usage

dsumchisqpow(x, wts, df, ncp=0, pow=1, log = FALSE, order.max=6)psumchisqpow(q, wts, df, ncp=0, pow=1, lower.tail = TRUE, log.p = FALSE, order.max=6)qsumchisqpow(p, wts, df, ncp=0, pow=1, lower.tail = TRUE, log.p = FALSE, order.max=6)rsumchisqpow(n, wts, df, ncp=0, pow=1)

Arguments

Details

LetX_i \sim \chi^2\left(\delta_i, \nu_i\right)be independently distributed non-central chi-squares, where\nu_iare the degrees of freedom, and\delta_i are thenon-centrality parameters. Letw_i andp_i be given constants. Suppose

Y = \sum_i w_i X_i^{p_i}.

ThenY follows a weighted sum of chi-squares to power distribution.

Value

dsumchisqpow gives the density,psumchisqpow gives the distribution function,qsumchisqpow gives the quantile function, andrsumchisqpow generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

The 'sum of chisquare power' distribution doesnot generalizethe 'chi-bar-square' distribution, whosedensity is the sum ofchi-square densities.

Author(s)

Steven E. Pavshabbychef@gmail.com

See Also

The upsilon distribution,dupsilon,pupsilon,qupsilon,rupsilon.

Examples

wts <- c(1,-3,4)df <- c(100,20,10)ncp <- c(5,3,1)pow <- c(1,0.5,1)rvs <- rsumchisqpow(128, wts, df, ncp, pow)dvs <- dsumchisqpow(rvs, wts, df, ncp, pow)qvs <- psumchisqpow(rvs, wts, df, ncp, pow)pvs <- qsumchisqpow(ppoints(length(rvs)), wts, df, ncp, pow)

The sum of the logs of (non-central) chi-squares distribution.

Description

Density, distribution function, quantile function and randomgeneration for the distribution of the weighted sum of logs ofnon-central chi-squares.

Usage

dsumlogchisq(x, wts, df, ncp=0, log = FALSE, order.max=6)psumlogchisq(q, wts, df, ncp=0, lower.tail = TRUE, log.p = FALSE, order.max=6)qsumlogchisq(p, wts, df, ncp=0, lower.tail = TRUE, log.p = FALSE, order.max=6)rsumlogchisq(n, wts, df, ncp=0)

Arguments

Details

LetX_i \sim \chi^2\left(\delta_i, \nu_i\right)be independently distributed non-central chi-squares, where\nu_iare the degrees of freedom, and\delta_i are thenon-centrality parameters. Letw_i be given constants. Suppose

Y = \sum_i w_i \log X_i.

ThenY follows a weighted sum of log of chi-squares distribution.

Value

dsumlogchisq gives the density,psumlogchisq gives the distribution function,qsumlogchisq gives the quantile function, andrsumlogchisq generates random deviates.

Invalid arguments will result in return valueNaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

References

Pav, Steven. Moments of the log non-central chi-square distribution.https://arxiv.org/abs/1503.06266

See Also

The product of chi-squares to a power,dprodchisqpow,pprodchisqpow,qprodchisqpow,rprodchisqpow.

Examples

wts <- c(1,-3,4)df <- c(100,20,10)ncp <- c(5,3,1)rvs <- rsumlogchisq(128, wts, df, ncp)dvs <- dsumlogchisq(rvs, wts, df, ncp)qvs <- psumlogchisq(rvs, wts, df, ncp)pvs <- qsumlogchisq(ppoints(length(rvs)), wts, df, ncp)

The upsilon distribution.

Description

Density, distribution function, quantile function and randomgeneration 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)

Arguments

Details

Supposex_i \sim \chi^2\left(\nu_i\right)independently and independently ofZ, 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 ofthe Sharpe ratio for independent observations, and generalizesthe lambda prime distribution, which can be written asZ + t \sqrt{x/\nu}.

Value

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

Invalid arguments will result in return valueNaN with a warning.

Note

the PDF and CDF are approximated by an Edgeworth expansion; thequantile function is approximated by a Cornish-Fisher expansion.

The PDF, CDF, and quantile function are approximated, viathe Edgeworth or Cornish Fisher approximations, which maynot be terribly accurate in the tails of the distribution.You are warned.

The distribution parameters arenot recycledwith respect to thex, p, q orn parameters,for, respectively, the density, distribution, quantileand generation functions. This is for simplicity ofimplementation and performance. It is, however, in contrastto the usual R idiom for dpqr functions.

Author(s)

Steven E. Pavshabbychef@gmail.com

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)plot(rv,d1)p1 <- pupsilon(rv, df=mydf, t=myt)# should be nearly uniform:plot(ecdf(p1))q1 <- qupsilon(ppoints(length(rv)),df=mydf,t=myt)qqplot(x=rv,y=q1)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)) }