Generated Probability Distribution Models

Description

Computes the probability density function (pdf), cumulative distribution function (cdf), quantile function (qf) and generates random values (rg) for the following general models : mixture models, composite models, folded models, skewed symmetric models and arc tan models.

Details

All the models use parent distribution(s) and thus flexible to incorporate many exisiting probability distributions.

Author(s)

Shaiful Anuar Abu Bakar

Maintainer: Shaiful Anuar Abu Bakar <saab@um.edu.my>

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Gomez-Deniz, E., & Calderin-Ojeda, E. Modelling insurance data with the pareto arctan distribution. ASTIN Bulletin, 1-22.
Cooray, K., & Ananda, M. M. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321-334.
Scollnik, D. P. (2007). On composite lognormal-Pareto models. Scandinavian Actuarial Journal, 2007(1), 20-33.
Nadarajah, S., & Bakar, S. A. A. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014(2), 180-187.
Bakar, S. A., Hamzah, N. A., Maghsoudi, M., & Nadarajah, S. (2015). Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146-154.
Brazauskas, V., & Kleefeld, A. (2011). Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011(1), 59-74.
Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 71-110.
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

Probabilty density function of arc tan model.

Description

Computes pdf of the arc tan model.

Usage

darctan(x, alpha, spec, arg, log = FALSE)

Arguments

Details

The pdf of arc tan model with parameter \alpha has a general form of:

f(x) = \frac{1}{\arctan(\alpha)} \frac{\alpha g(x)}{1 + (\alpha (1-G(x)))^{2}}

for a\leq x\leq b where a and b follow the support of g(x). \arctan denote the inverse function of tangent. g(x) and G(x) are the pdf and cdf of parent distribution, respectively. Note also that \alpha>0.

Value

An object of the same length as x, giving the pdf values computed at x.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Gomez-Deniz, E., & Calderin-Ojeda, E. Modelling insurance data with the pareto arctan distribution. ASTIN Bulletin, 1-22.

Examples

x=runif(10, min=0, max=1)
y=darctan(x, alpha=0.5, spec="lnorm", arg=list(meanlog=1,sdlog=2) )

Probabilty density function of composite model.

Description

Computes pdf of the composite model.

Usage

dcomposite(x, spec1, arg1, spec2, arg2, initial = 1, log = FALSE)

Arguments

Details

The pdf of composite model has a general form of:

f(x) = \frac{1}{1+\phi} f_{1}^{*}(x), \mbox{ if} \quad x \leq \theta,

= \frac{\phi}{1+\phi} f_{2}^{*}(x), \mbox{ if} \quad x > \theta,

whereby \phi is the weight component, \theta is the threshold and f_{i}^{*}(x) for i=1,2 are the truncated pdfs correspond to head and tail parent distributions defined by

f_{1}^{*}(x) = \frac{f_{1}(x)}{F_{1}(\theta)}

and

f_{2}^{*}(x) = \frac{f_{2}(x)}{1-F_{2}(\theta)}

respectively.

Value

An object of the same length as x, giving the pdf values computed at x.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Cooray, K., & Ananda, M. M. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321-334.
Scollnik, D. P. (2007). On composite lognormal-Pareto models. Scandinavian Actuarial Journal, 2007(1), 20-33.
Nadarajah, S., & Bakar, S. A. A. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014(2), 180-187.
Bakar, S. A., Hamzah, N. A., Maghsoudi, M., & Nadarajah, S. (2015). Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146-154.

Examples

x=runif(10, min=0, max=1)
y=dcomposite(x, spec1="lnorm", arg1=list(meanlog=0.1,sdlog=0.2), spec2="exp", 
             arg2=list(rate=0.5) )

Probabilty density function of folded model.

Description

Computes pdf of the folded model.

Usage

dfolded(x, spec, arg, log = FALSE)

Arguments

Details

The pdf of folded model has a general form of:

f(x) = g(x) + g(-x) \quad x>0

where G(x) is the cdf of parent distribution.

Value

An object of the same length as x, giving the pdf values computed at x.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Brazauskas, V., & Kleefeld, A. (2011). Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011(1), 59-74.

Examples

x=runif(10, min=0, max=1)
y=dfolded(x, spec="norm", arg=list(mean=1,sd=2) )

Probabilty density function of mixture model.

Description

Computes pdf of the mixture model.

Usage

dmixt(x, phi, spec1, arg1, spec2, arg2, log = FALSE)

Arguments

Details

The pdf of mixture model with parameter phi has a general form of:

f(x) = \frac{1}{1+\phi} \left( g_{1}(x) + \phi g_{2}(x)\right)

where x follows the support of parent distributions, \phi is the weight component and g_{i}(x) for i=1,2 are the pdfs of first and second parent distributions, respectively.

Value

An object of the same length as x, giving the pdf values computed at x.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 71-110.

Examples

x=runif(10, min=0, max=1)
y=dmixt(x, phi=0.5, spec1="lnorm", arg1=list(meanlog=1,sdlog=2), spec2="exp", 
        arg2=list(rate=2) )

Probabilty density function of skewed symmetric model.

Description

Computes pdf of the skewed symmetric model.

Usage

dskew(x, spec1, arg1, spec2, arg2, log = FALSE)

Arguments

Details

The pdf of skewed symmetric model has a general form of:

f(x) = 2h(x)G(x), \quad -\infty < x < \infty

where h(x) and G(x) are the pdf and cdf of parent distributions, respectively.

Value

An object of the same length as x, giving the pdf values computed at x.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

Examples

x=runif(10, min=0, max=1)
y=dskew(x, spec1="norm", arg1=list(mean=0,sd=1), spec2="logis", 
        arg2=list(location=0,scale=2) )

Cumulative distribution function of arc tan model.

Description

Computes cdf of the arc tan model.

Usage

parctan(q, alpha, spec, arg, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The cdf of arc tan model with parameter \alpha has a general form of:

F(q) = 1- \frac{\arctan(\alpha (1-G(q)) )}{\arctan(\alpha)}

for a\leq x\leq b where a and b follow the support of g(q). \arctan denote the inverse function of tangent. g(q) and G(q) are the pdf and cdf of parent distribution, respectively. Note also that \alpha>0.

Value

An object of the same length as q, giving the cdf values computed at q.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Gomez-Deniz, E., & Calderin-Ojeda, E. Modelling insurance data with the pareto arctan distribution. ASTIN Bulletin, 1-22.

Examples

x=runif(10, min=0, max=1)
y=parctan(x, alpha=0.5, spec="lnorm", arg=list(meanlog=1,sdlog=2) )

Cumulative distribution function of composite model.

Description

Computes cdf of the composite model.

Usage

pcomposite(q, spec1, arg1, spec2, arg2, initial = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The cdf of composite model has a general form of:

F(x) = \frac{1}{1+\phi} \frac{F_{1}(x)}{F_{1}(\theta)} \mbox{ if } \quad x \leq \theta,

= \frac{1}{1+\phi} \left( 1 + \phi \frac{F_{2}(x)-F_{2}(\theta)}{1-F_{2}(\theta)} \right) \mbox{ if } \quad x > \theta,

whereby \phi is the weight component, \theta is the threshold and F_{i}(x) for i=1,2 are the cdfs correspond to head and tail parent distributions, respectively.

Value

An object of the same length as q, giving the cdf values computed at q.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Cooray, K., & Ananda, M. M. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321-334.
Scollnik, D. P. (2007). On composite lognormal-Pareto models. Scandinavian Actuarial Journal, 2007(1), 20-33.
Nadarajah, S., & Bakar, S. A. A. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014(2), 180-187.
Bakar, S. A., Hamzah, N. A., Maghsoudi, M., & Nadarajah, S. (2015). Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146-154.

Examples

x=runif(10, min=0, max=1)
y=pcomposite(x, spec1="lnorm", arg1=list(meanlog=0.1,sdlog=0.2), spec2="exp", 
             arg2=list(rate=0.5) )

Cumulative distribution function of folded model.

Description

Computes cdf of the folded model.

Usage

pfolded(q, spec, arg, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The cdf of folded model has a general form of:

F(x) = G(x) - G(-x) \quad x>0

where G(x) is the cdf of parent distribution.

Value

An object of the same length as q, giving the cdf values computed at q.

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Brazauskas, V., & Kleefeld, A. (2011). Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011(1), 59-74.

Examples

x=runif(10, min=0, max=1)
y=pfolded(x, spec="norm", arg=list(mean=1,sd=2) )

Cumulative distribution function of mixture model.

Description

Computes cdf of the mixture model.

Usage

pmixt(q, phi, spec1, arg1, spec2, arg2, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The cdf of mixture model has a general form of:

F(x) = \\frac{1}{1+\phi} \left(G_{1}(x) + \phi G_{2}(x) \right)

where x follows the support of parent distributions, \phi is the weight component and G_{i}(x) for i=1,2 are the cdfs of first and second parent distributions, respectively.

Value

An object of the same length as q, giving the cdf values computed at q.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 71-110.

Examples

x=runif(10, min=0, max=1)
y=pmixt(x, phi=0.5, spec1="lnorm", arg1=list(meanlog=1,sdlog=2), spec2="exp", 
        arg2=list(rate=2) )

Cumulative distribution function of skewed symmetric model.

Description

Computes cdf of the skewed symmetric model.

Usage

pskew(q, spec1, arg1, spec2, arg2, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The cdf of skewed symmetric model has a general form of:

F(x) = \int_{-\infty}^{x} 2 h(y) G(y) dy, \quad -\infty < x < \infty

where h(x) and G(x) are the pdf and cdf of parent distributions, respectively.

Value

An object of the same length as q, giving the cdf values computed at q.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

Examples

x=runif(10, min=0, max=1)
y=pskew(x, spec1="norm", arg1=list(mean=0,sd=1), spec2="logis", 
        arg2=list(location=0,scale=2) )

Quantile function of arc tan model.

Description

Computes qf of the arc tan model.

Usage

qarctan(p, alpha, spec, arg, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The qf of arc tan model with parameter \alpha has a general form of:

Q(p) = G^{-1}\left(1-\frac{1}{\alpha} \tan( (1-p)\arctan(\alpha) )\right)

for a\leq x\leq b where a and b follow the support of G(x). \arctan denote the inverse function of tangent and G^{-1} is the inverse cdf of parent distribution, respectively. Note also that \alpha>0.

Value

An object of the same length as p, giving the qf values computed at p.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Gomez-Deniz, E., & Calderin-Ojeda, E. Modelling insurance data with the pareto arctan distribution. ASTIN Bulletin, 1-22.

Examples

x=runif(10, min=0, max=1)
y=qarctan(x, alpha=0.5, spec="lnorm", arg=list(meanlog=1,sdlog=2) )

Quantile function of composite model.

Description

Computes qf of the composite model.

Usage

qcomposite(p, spec1, arg1, spec2, arg2, initial = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The qf of composite model has a general form of:

Q(p) = Q_{1}(p(1+\phi)F_{1}(\theta)) \mbox{ if } \quad p \leq \frac{1}{1+\phi},

= Q_{2} \left( F_{2}(\theta) + (1-F_{2}(\theta)) \left( \frac{p(1+\phi)-1}{\phi} \right)\right) \mbox{ if } \quad p > \frac{1}{1+\phi}

whereby \phi is the weight component, \theta is the threshold and F_{i}(x) for i=1,2 are the qfs correspond to head and tail parent distributions, respectively.

Value

An object of the same length as p, giving the qf values computed at p.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Cooray, K., & Ananda, M. M. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321-334.
Scollnik, D. P. (2007). On composite lognormal-Pareto models. Scandinavian Actuarial Journal, 2007(1), 20-33.
Nadarajah, S., & Bakar, S. A. A. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014(2), 180-187.
Bakar, S. A., Hamzah, N. A., Maghsoudi, M., & Nadarajah, S. (2015). Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146-154.

Examples

x=runif(10, min=0, max=1)
y=qcomposite(x, spec1="lnorm", arg1=list(meanlog=0.1,sdlog=0.2), spec2="exp", 
             arg2=list(rate=0.5) )

Quantile function of folded model.

Description

Computes cdf of the folded model.

Usage

qfolded(p, spec, arg, interval = c(0, 100), lower.tail = TRUE, log.p = FALSE)

Arguments

Value

An object of the same length as p, giving the qf values computed at p.

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Brazauskas, V., & Kleefeld, A. (2011). Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011(1), 59-74.

Examples

x=runif(10, min=0, max=1)
y=qfolded(x, spec="norm", arg=list(mean=1,sd=2), interval=c(0,100) )

Quantile function of mixture model.

Description

Computes qf of the mixture model.

Usage

qmixt(p, phi, spec1, arg1, spec2, arg2, interval = c(0, 100),
      lower.tail = TRUE, log.p = FALSE)

Arguments

Value

An object of the same length as p, giving the qf values computed at p.

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 71-110.

Examples

x=runif(10, min=0, max=1)
y=qmixt(x, phi=0.5, spec1="lnorm", arg1=list(meanlog=0.1,sdlog=0.2), spec2="exp", 
        arg2=list(rate=0.5))

Quantile function of skewed symmetric model.

Description

Computes qf of the skewed symmetric model.

Usage

qskew(p, spec1, arg1, spec2, arg2, interval = c(1, 10), lower.tail = TRUE, log.p = FALSE)

Arguments

Value

An object of the same length as p, giving the qf values computed at p.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

Examples

x=runif(10, min=0, max=1)
y=qskew(x, spec1="norm", arg1=list(mean=0,sd=0.1), spec2="logis", 
        arg2=list(location=0,scale=0.2))

Random generation of arc tan model.

Description

Computes rg of the arc tan model.

Usage

rarctan(n, alpha, spec, arg)

Arguments

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Gomez-Deniz, E., & Calderin-Ojeda, E. Modelling insurance data with the pareto arctan distribution. ASTIN Bulletin, 1-22.

Examples

y=rarctan(10, alpha=0.5, spec="lnorm", arg=c(meanlog=1,sdlog=2) )

Random generation of composite model.

Description

Computes rg of the composite model.

Usage

rcomposite(n, spec1, arg1, spec2, arg2, initial = 1)

Arguments

Value

An object of the length n, giving the random generated values for the composite model.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Cooray, K., & Ananda, M. M. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321-334.
Scollnik, D. P. (2007). On composite lognormal-Pareto models. Scandinavian Actuarial Journal, 2007(1), 20-33.
Nadarajah, S., & Bakar, S. A. A. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014(2), 180-187.
Bakar, S. A., Hamzah, N. A., Maghsoudi, M., & Nadarajah, S. (2015). Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146-154.

Examples

y=rcomposite(10, spec1="lnorm", arg1=list(meanlog=0.1,sdlog=0.2), spec2="exp", 
             arg2=list(rate=0.5))

Random generation of folded model.

Description

Computes rg of the folded model.

Usage

rfolded(n, spec, arg, interval = c(0, 100))

Arguments

Value

An object of the length n, giving the random generated values for the folded model.

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Brazauskas, V., & Kleefeld, A. (2011). Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011(1), 59-74.

Examples

y=rfolded(10, spec="norm", arg=list(mean=1,sd=2), interval=c(0,100) )

Random generation of mixture model.

Description

Computes rg of the mixture model.

Usage

rmixt(n, phi, spec1, arg1, spec2, arg2, interval = c(0, 100))

Arguments

Value

An object of the length n, giving the random generated values for the mixture model.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 71-110.

Examples

y=rmixt(10, phi=0.5, spec1="lnorm", arg1=list(meanlog=0.1,sdlog=0.2), spec2="exp", 
        arg2=list(rate=0.5) )

Random generation of skewed symmetric model.

Description

Computes rg of the skewed symmetric model.

Usage

rskew(n, spec1, arg1, spec2, arg2, interval = c(1, 10))

Arguments

Value

An object of the length n, giving the random generated values for the skewed symmetric model.

Author(s)

Shaiful Anuar Abu Bakar

References

Abu Bakar, S. A., Nadarajah, S., Adzhar, Z. A. A. K., & Mohamed, I. (2016). gendist: An R package for generated probability distribution models. PloS one, 11(6).
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

Examples

y=rskew(10, spec1="norm", arg1=list(mean=0,sd=0.1), spec2="logis", 
        arg2=list(location=0,scale=0.2))