Package 'powdist'

Power and reversal power distributions

Description

The powdist package enables to compute the probability density function, cumulative distribution function, quantile function and generate random numbers for the following distributions: power Logistic (plogis), reversal power Logistic (rplogis), power Normal (pnorm), reversal power Normal (rpnorm), power Cauchy (pcauchy), reversal power Cauchy (rpcauchy), power reversal-Gumbel (prgumbel), power Student T (pt), reversal power Student T (rpt), power Laplace (plaplace), reversal power Laplace (rplaplace), power exponential power (pexpow) and reversal power exponential power (rpexpow).

---

The Exponential Power Distribution

Description

Density, distribution function, quantile function and random generation for the exponential power distribution with parameters mu, sigma and k.

Usage

dexpow(x, mu = 0, sigma = 1, k = 0, log = FALSE)

pexpow(q, mu = 0, sigma = 1, k = 0, lower.tail = TRUE, log.p = FALSE)

qexpow(p, mu = 0, sigma = 1, k = 0, lower.tail = TRUE, log.p = FALSE)

rexpow(n, mu = 0, sigma = 1, k = 0)

Arguments

x, q	vector of quantiles.
mu, sigma	location and scale parameters.
k	shape parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The Exponential distribution has density

$f\left(x\right)=\left[\frac{e^{-\left(\frac{x-\mu}{\sigma}\right)}}{\left(1+e^{-\left(\frac{x-\mu}{\sigma}\right)}\right)^{2}}\right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and k the shape parameter.

References

Lemonte A. and Bazán J.L.

Examples

dexpow(1, 3, 4, 1)
pexpow(1, 3, 4, 1)
qexpow(0.2, 3, 4, 1)
rexpow(5, 3, 4, 1)

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The Gumbel Distribution

Description

Density, distribution function, quantile function and random generation for the Gumbel distribution with parameters mu and sigma.

Usage

dgumbel(x, mu = 0, sigma = 1, log = FALSE)

pgumbel(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qgumbel(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rgumbel(n, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The Gumbel distribution has density

$f(x)=\left[\frac{1}{\sigma}e^{\left(-\frac{x-\mu}{\sigma}\right)-e^{\left(-\frac{x-\mu}{\sigma}\right)}}\right]$,

where $-\infty<\mu<\infty$ is the location paramether and $\sigma^2>0$ is the scale parameter.

Examples

dgumbel(1, 3, 4)
pgumbel(1, 3, 4)
qgumbel(0.2, 3, 4)
rgumbel(5, 3, 4)

---

The Power Cauchy Distribution

Description

Density, distribution function, quantile function and random generation for the power Cauchy distribution with parameters mu, sigma and lambda.

Usage

dpcauchy(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

ppcauchy(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qpcauchy(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rpcauchy(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power Cauchy distribution has density

$f(x)=\lambda\left [\frac{1}{\pi}\arctan\left ( \frac{x-\mu}{\sigma} \right )+\frac{1}{2} \right ]^{\lambda -1} \left[ \frac{1}{\pi\sigma\left( 1+\left (\frac{x-\mu}{\sigma} \right )^{2} \right)} \right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Examples

dpcauchy(1, 1, 3, 4)
ppcauchy(1, 1, 3, 4)
qpcauchy(0.2, 1, 3, 4)
rpcauchy(5, 2, 3, 4)

---

The Power Exponential Power Distribution

Description

Density, distribution function, quantile function and random generation for the power exponential power distribution with parameters mu, sigma, lambda and k.

Usage

dpexpow(x, lambda = 1, mu = 0, sigma = 1, k = 0, log = FALSE)

ppexpow(q, lambda = 1, mu = 0, sigma = 1, k = 0, lower.tail = TRUE,
  log.p = FALSE)

qpexpow(p, lambda = 1, mu = 0, sigma = 1, k = 0, lower.tail = TRUE,
  log.p = FALSE)

rpexpow(n, lambda = 1, mu = 0, sigma = 1, k = 0)

Arguments

x, q	vector of quantiles.
mu, sigma	location and scale parameters.
k, lambda	shape parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power exponential power distribution has density

$f\left(x\right)=\frac{\lambda}{\sigma}\left[\frac{e^{-\left(\frac{x-\mu}{\sigma}\right)}}{\left(1+e^{-\left(\frac{x-\mu}{\sigma}\right)}\right)^{2}}\right]\left[\frac{e^{\left(\frac{x-\mu}{\sigma}\right)}}{1+e^{\left(\frac{x-\mu}{\sigma}\right)}}\right]^{\lambda-1}$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ and k the shape parameters.

References

Lemonte A. and Bazán J.L.

Examples

dpexpow(1, 1, 3, 4, 1)
ppexpow(1, 1, 3, 4, 1)
qpexpow(0.2, 1, 3, 4, 1)
rpexpow(5, 2, 3, 4, 1)

---

The Power Laplace Distribution

Description

Density, distribution function, quantile function and random generation for the power Laplace distribution with parameters mu, sigma and lambda.

Usage

dplaplace(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

pplaplace(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qplaplace(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rplaplace(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power Laplace distribution has density

$f(x)=\lambda\left[\frac{1}{2}+\frac{\left(1-e^{-\frac{\left|x-\mu\right|}{\sigma}}\right)}{2}\textrm{sign}\left(\frac{x-\mu}{\sigma}\right)\right]^{\lambda-1}\left[\frac{e^{-\frac{\left|x-\mu\right|}{\sigma}}}{2\sigma}\right]$, where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

Examples

dplaplace(1, 1, 3, 4)
pplaplace(1, 1, 3, 4)
qplaplace(0.2, 1, 3, 4)
rplaplace(5, 2, 3, 4)

---

The Power Logistic Distribution

Description

Density, distribution function, quantile function and random generation for the power logistic distribution with parameters mu, sigma and lambda.

Usage

dplogis(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

pplogis(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qplogis(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rplogis(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power Logistic distribution has density

$f(x)=\lambda \left [\frac{1}{1+e^{-\left ( \frac{x-\mu}{\sigma} \right )} }\right ]^{\lambda-1}\left[\frac{ e^{-\left ( \frac{x-\mu}{\sigma} \right )} }{\sigma\left ( 1+e^{-\left ( \frac{x-\mu}{\sigma} \right )} \right )^2}\right]$, where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 16. Wiley, New York.

Lemonte, A. J. and Bazán, J. L. (2017) New links for binary regression: an application to coca cultivation in Peru. TEST.

Nadarajah, S. (2009) The skew logistic distribution. AStA Advances in Statistical Analysis, 93, 187-203.

Prentice, R. L. (1976) A Generalization of the probit and logit methods for dose-response curves. Biometrika, 32, 761-768.

Examples

dplogis(1, 1, 3, 4)
pplogis(1, 1, 3, 4)
qplogis(0.2, 1, 3, 4)
rplogis(5, 2, 3, 4)

---

The Power Normal Distribution

Description

Density, distribution function, quantile function and random generation for the power normal distribution with parameters mu, sigma and lambda.

Usage

dpnorm(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

ppnorm(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qpnorm(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rpnorm(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power Normal distribution has density

$f(x)=\lambda \left [ \Phi \left ( \frac{x-\mu}{\sigma} \right ) \right]^{\lambda - 1} \left[\frac{e^{ -\frac{1}{2}\left ( \frac{x-\mu}{\sigma} \right )^2}}{\sigma\sqrt{2\pi}} \right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Bazán, J. L., Romeo, J. S. and Rodrigues, J. (2014) Bayesian skew-probit regression for binary response data. Brazilian Journal of Probability and Statistics. 28(4), 467–482.

Gupta, R. D. and Gupta, R. C. (2008) Analyzing skewed data by power normal model. Test 17, 197–210.

Kundu, D. and Gupta, R. D. (2013) Power-normal distribution. Statistics 47, 110–125.

Examples

dpnorm(1, 1, 3, 4)
ppnorm(1, 1, 3, 4)
qpnorm(0.2, 1, 3, 4)
rpnorm(5, 2, 3, 4)

---

The Power Reversal-Gumbel Distribution

Description

Density, distribution function, quantile function and random generation for the power Reversal-Gumbel distribution with parameters mu, sigma and lambda.

Usage

dprgumbel(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

pprgumbel(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qprgumbel(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rprgumbel(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power reverlsa-Gumbel distribution has density

$f(x)=\lambda \left[1-e^{-e^{\left(\frac{x-\mu}{\sigma}\right)}}\right]^{\lambda-1}\left[\frac{1}{\sigma}e^{\left(\frac{x-\mu}{\sigma}\right)-e^{\left(\frac{x-\mu}{\sigma}\right)}} \right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Abanto -Valle, C. A., Bazán, J. L. and Smith, A. C. (2014) State space mixed models for binary responses with skewed inverse links using JAGS. Rio de Janeiro, Brazil.

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Examples

dprgumbel(1, 1, 3, 4)
pprgumbel(1, 1, 3, 4)
qprgumbel(0.2, 1, 3, 4)
rprgumbel(5, 2, 3, 4)

---

The Power Student t Distribution

Description

Density, distribution function, quantile function and random generation for the power Student t distribution with parameters mu, sigma, lambda and df.

Usage

dpt(x, lambda = 1, mu = 0, sigma = 1, df, log = FALSE)

ppt(q, lambda = 1, mu = 0, sigma = 1, df, lower.tail = TRUE,
  log.p = FALSE)

qpt(p, lambda = 1, mu = 0, sigma = 1, df, lower.tail = TRUE,
  log.p = FALSE)

rpt(n, lambda = 1, mu = 0, sigma = 1, df)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
df	degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The power Student t distribution has density

$f(x)=[\lambda/\sigma][f((x-\mu)/\sigma)][F((x-\mu)/\sigma)] ^(\lambda-1)$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Lemonte A. and Bazán J.L.

Examples

dpt(1, 1, 3, 4, 1)
ppt(1, 1, 3, 4, 1)
qpt(0.2, 1, 3, 4, 1)
rpt(5, 2, 3, 4, 1)

---

The Reversal-Gumbel Distribution

Description

Density, distribution function, quantile function and random generation for the Reversal-Gumbel distribution with parameters mu and sigma.

Usage

drgumbel(x, mu = 0, sigma = 1, log = FALSE)

prgumbel(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qrgumbel(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rrgumbel(n, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal-Gumbel distribution has density

$f(x)=\left[\frac{1}{\sigma}e^{\left(\frac{x-\mu}{\sigma}\right)-e^{\left(\frac{x-\mu}{\sigma}\right)}}\right]$,

where $-\infty<\mu<\infty$ is the location paramether and $\sigma^2>0$ is the scale parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Examples

drgumbel(1, 3, 4)
prgumbel(1, 3, 4)
qrgumbel(0.2, 3, 4)
rprgumbel(5, 3, 4)

---

The Reversal Power Cauchy Distribution

Description

Density, distribution function, quantile function and random generation for the reversal power Cauchy distribution with parameters mu, sigma and lambda.

Usage

drpcauchy(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

prpcauchy(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qrpcauchy(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rrpcauchy(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power Cauchy distribution has density

$f(x)=\lambda\left [\frac{1}{\pi}\arctan\left (-\frac{x-\mu}{\sigma} \right )+\frac{1}{2} \right ]^{\lambda -1}\left[ \frac{1}{\pi\sigma\left( 1+\left (\frac{x-\mu}{\sigma} \right )^{2} \right)} \right]$

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Examples

drpcauchy(1, 1, 3, 4)
prpcauchy(1, 1, 3, 4)
qrpcauchy(0.2, 1, 3, 4)
rrpcauchy(5, 2, 3, 4)

---

The Reversal Power Exponential Power Distribution

Description

Density, distribution function, quantile function and random generation for the reversal power exponential power distribution with parameters mu, sigma, lambda and k.

Usage

drpexpow(x, lambda = 1, mu = 0, sigma = 1, k = 0, log = FALSE)

prpexpow(q, lambda = 1, mu = 0, sigma = 1, k = 0, lower.tail = TRUE,
  log.p = FALSE)

qrpexpow(p, lambda = 1, mu = 0, sigma = 1, k = 0, lower.tail = TRUE,
  log.p = FALSE)

rrpexpow(n, lambda = 1, mu = 0, sigma = 1, k = 0)

Arguments

x, q	vector of quantiles.
mu, sigma	location and scale parameters.
k, lambda	shape parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power exponential power distribution has density

$f(x)=[\lambda/\sigma][f((x-\mu)/\sigma)][F((x-\mu)/\sigma)] ^(\lambda-1)$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ and k the shape parameters.

Examples

drpexpow(1, 1, 3, 4, 1)
prpexpow(1, 1, 3, 4, 1)
qrpexpow(0.2, 1, 3, 4, 1)
rrpexpow(5, 2, 3, 4, 1)

---

The Power Reversal Laplace Distribution

Description

Density, distribution function, quantile function and random generation for the power reversal Laplace distribution with parameters mu, sigma and lambda.

Usage

drplaplace(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

prplaplace(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qrplaplace(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rrplaplace(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power Laplace distribution has density

$f(x)=\lambda\left[\frac{1}{2}+\frac{\left(1-e^{\frac{\left|x-\mu\right|}{\sigma}}\right)}{2}\textrm{sign}\left(-\frac{x-\mu}{\sigma}\right)\right]^{\lambda-1}\left[\frac{e^{-\frac{\left|x-\mu\right|}{\sigma}}}{2\sigma}\right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

Examples

drplaplace(1, 1, 3, 4)
prplaplace(1, 1, 3, 4)
qrplaplace(0.2, 1, 3, 4)
rrplaplace(5, 2, 3, 4)

---

The Reversal Power Logistic Distribution

Description

Density, distribution function, quantile function and random generation for the reversal power logistic distribution with parameters mu, sigma and lambda.

Usage

drplogis(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

prplogis(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qrplogis(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rrplogis(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power Logistic distribution has density

$f(x)=\lambda \left [\frac{1}{1+e^{\left ( \frac{x-\mu}{\sigma} \right )} }\right ]^{\lambda-1}\left[\frac{ e^{-\left ( \frac{x-\mu}{\sigma} \right )} }{\sigma\left ( 1+e^{-\left ( \frac{x-\mu}{\sigma} \right )} \right )^2}\right]$, where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 16. Wiley, New York.

Nagler J. (1994) Scobit: an alternative estimator to logit and probit. American Journal Political Science, 38(1), 230-255.

Prentice, R. L. (1976) A Generalization of the probit and logit methods for dose-response curves. Biometrika, 32, 761-768.

Examples

drplogis(1, 1, 3, 4)
prplogis(1, 1, 3, 4)
qrplogis(0.2, 1, 3, 4)
rrplogis(5, 2, 3, 4)

---

The Reversal Power Normal Distribution

Description

Density, distribution function, quantile function and random generation for the reversal power normal distribution with parameters mu, sigma and lambda.

Usage

drpnorm(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

prpnorm(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qrpnorm(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rrpnorm(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power Normal distribution has density

$f(x)=\lambda \left [ \Phi \left ( -\frac{x-\mu}{\sigma} \right ) \right ]^{\lambda - 1} \left[\frac{e^{ -\frac{1}{2}\left ( \frac{x-\mu}{\sigma} \right )^2}}{\sigma\sqrt{2\pi}} \right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Bazán, J. L., Romeo, J. S. and Rodrigues, J. (2014) Bayesian skew-probit regression for binary response data. Brazilian Journal of Probability and Statistics. 28(4), 467–482.

Examples

drpnorm(1, 1, 3, 4)
prpnorm(1, 1, 3, 4)
qrpnorm(0.2, 1, 3, 4)
rrpnorm(5, 2, 3, 4)

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The Reversal Power Reversal-Gumbel Distribution

Description

Density, distribution function, quantile function and random generation for the reversal power reversal-Gumbel distribution with parameters mu, sigma and lambda.

Usage

drprgumbel(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

prprgumbel(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qrprgumbel(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rrprgumbel(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power reversal-Gumbel distribution has density

$f(x)=\lambda \left[1-e^{-e^{\left(-\frac{x-\mu}{\sigma}\right)}}\right]^{\lambda-1}\left[\frac{1}{\sigma}e^{\left(\frac{x-\mu}{\sigma}\right)-e^{\left(\frac{x-\mu}{\sigma}\right)}} \right]$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Examples

drprgumbel(1, 1, 3, 4)
prprgumbel(1, 1, 3, 4)
qrprgumbel(0.2, 1, 3, 4)
rrprgumbel(5, 2, 3, 4)

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The Power Reversal Student t Distribution

Description

Density, distribution function, quantile function and random generation for the power reversal Student t distribution with parameters mu, sigma, lambda and df.

Usage

drpt(x, lambda = 1, mu = 0, sigma = 1, df, log = FALSE)

prpt(q, lambda = 1, mu = 0, sigma = 1, df, lower.tail = TRUE,
  log.p = FALSE)

qrpt(p, lambda = 1, mu = 0, sigma = 1, df, lower.tail = TRUE,
  log.p = FALSE)

rrpt(n, lambda = 1, mu = 0, sigma = 1, df)

Arguments

x, q	vector of quantiles.
lambda	shape parameter.
mu, sigma	location and scale parameters.
df	degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x ]$, otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

Details

The reversal power Student t distribution has density

$f(x)=[\lambda/\sigma][f((x-\mu)/\sigma)][F((x-\mu)/\sigma)] ^(\lambda-1)$,

where $-\infty<\mu<\infty$ is the location paramether, $\sigma^2>0$ the scale parameter and $\lambda>0$ the shape parameter.

Examples

drpt(1, 1, 3, 4, 1)
prpt(1, 1, 3, 4, 1)
qrpt(0.2, 1, 3, 4, 1)
rrpt(5, 2, 3, 4, 1)

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