Package 'unifed'

Car insurance claims

Description

This data set is based on one-year vehicle insurance policies taken out in 2004 or 2005. There are 67856 policies, of which 4624 (6.8%) had at least one claim.

Usage

car.insurance

Format

A data frame with 67,856 rows and 11 columns:

veh_value

vehicle value, in $10,000s

exposure

Values between 0 and 1

clm

occurrence of claim (0 = no, 1 = yes)

numclaims

number of claims

claimcst0

claim amount (0 if no claim)

veh_body

vehicle body, coded as

BUS

CONVT

convertible

COUPE

HBACK

hatchback

HDTOP

hardtop

MCARA

motorized caravan

MIBUS

minibus

PANVN

panel van

RDSTR

roadster

SEDAN

STNWG

station wagon

TRUCK

UTE

utility

veh_age

age of vehicle: 1 (youngest), 2, 3, 4

gender

gender of driver: M, F

area

driver's area of residence: A, B, C, D, E, F

agecat

driver's age category: 1 (youngest), 2, 3, 4, 5, 6

Source

http://www.businessandeconomics.mq.edu.au/our_departments/Applied_Finance_and_Actuarial_Studies/research/books/GLMsforInsuranceData

References

De Jong, P., and G.Z. Heller. 2008. Generalized Linear Models for Insurance Data. Cambridge University Press. http://dx.doi.org/10.1017/CBO9780511755408

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Irwin-Hall density

Description

Irwin-Hall density

Usage

dirwin.hall(x, n, log = FALSE)

Arguments

x	A number between 0 and n.
n	Number of uniform distributions in the unit interval to sum.
log	If it evaluates to TRUE it returns the log of the density instead of the density.

Details

Gives the density of the Irwin-Hall distribution. It is the density of the sum of n uniform distributions on the interval (0,1).

$h(y;n) = \frac{1}{(n-1)!}\sum_{k=0}^{ \left\lfloor y \right\rfloor } (-1)^k {n \choose k} (y-k)^{n-1}$

where $x \in [0,1]$ and $n$ is a positive integer.

This function is not numerically stable. The examples have some cases of this.

Examples

dirwin.hall(2,5)

# Numerically unstable example
# Run the following one after the other
# See how it goes from positive to negative (which means overflowing )
dirwin.hall(35,50)
dirwin.hall(36,50)
dirwin.hall(37,50)
dirwin.hall(38,50)

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The unifed distribution

Description

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

Usage

dunifed(x, theta)

unifed.lcdf(x, theta)

punifed(q, theta)

qunifed(p, theta)

runifed(n, theta)

Arguments

x	A vector of quantiles. They must be numbers between 0 and 1.
theta	The value of the canonical parameter. It must be of length one.
q	A vector of quantiles.
p	A vector of probabilities.
n	number of observations

Value

dunifed gives the density function.

unifed.lcdf returns the log of the cumulative distribution function of the unifed.

punifed gives the distribution function.

qunifed gives the quantile function.

runifed generates random observations.

References

Quijano Xacur, O.A. The unifed distribution. J Stat Distrib App 6, 13 (2019). doi:10.1186/s40488-019-0102-6.

Examples

dunifed( c(0.1,0.3,0.7), 10)


x <- c(0.3,0.6,0.9)
unifed.lcdf(x,5)


x <- c(0.1,0.4,0.7,1)
punifed(x,-5)


p <- 1:9/10
qunifed(p,5)


runifed(20,-3.3)

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Summarizing Generalized Linear Model Fits

Description

Wrapper function for summary.glm.

Usage

summary_unifed_glm(object, ...)

Arguments

object	an object of class "glm".
...	Other arguments for stats::summary.glm. This wrapper function was created in order to automatically set to 1 the dispersion parameter of a fitted unifed GLM. When the package is loaded the summary method of the glm class is rewritten using this function.

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Family object for the unifed distribution

Description

Family object for the unifed distribution

Usage

unifed(link = "logit", ...)

quasiunifed(link = "logit", ...)

unifed.canonical.link()

Arguments

link	a specification for the model link function. This can be a name/expression, a literal character string, a length-one character vector or an object of class ‘"link-glm"’ (such as generated by ‘make.link’) provided it is not specified via one of the accepted names. The unifed family accepts the links (as names) 'canonical', 'logit', 'probit', 'cloglog' and 'cauchit'.
...	Optional tol and maxit arguments for unifed.unit.deviance.

Details

The link 'canonical' is not part of the standard names accepted by make.link() from the stats package. It corresponds to the canonical link function for the unifed distribution, which is the inverse of the derivative of its cumulant generator. There is no explicit formula for it. The function unifed.kappa.prime.inverse() implements it using the Newthon-Raphson method.

This function is used inside of unifed() when the link parameter is set to "canonical". It returns the link function, inverse link function, the derivative dmu/deta and a function for domain checking for the unifed distribution canonical link.

Value

unifed returns a family object for using the unifed distribution with the glm function.

The quasiunifed family differs from the unifed only in that the dispersion parameter is not fixed to one.

An object of class "link-glm".

References

Jørgensen, Bent (1992). The Theory of Exponential Dispersion Models and Analysis of Deviance. Instituto de Matemática Pura e Aplicada, (IMPA), Brazil.

Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method. Biometrika. 61 (3): 439–447.

McCullagh, Peter; Nelder, John (1989). Generalized Linear Models (second ed.). London: Chapman and Hall.

See Also

Gamma unifed.kappa.prime.inverse

make.link

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Deviance of the unifed distribution

Description

Deviance of the unifed distribution

Usage

unifed.deviance(y.v, mu.v, wt = 1, ...)

unifed.unit.deviance(y, mu, tol = 1e-07, maxit = 50)

Arguments

y.v	A numeric vector with values between 0 and 1
mu.v	A numeric vector with values between 0 and 1
wt	(default value: 1) The weight vector. It contains the weight of each observation. It must contain positive integers only.
...	Additional parameters of unifed.kappa.prime.inverse.one
y	A vector with values between 0 and 1.
mu	A vector with values between 0 and 1.
tol	Tolerance level for the Newton-Raphson algorithm for computing the inverse of the derivative of the cumulant generator of the family.
maxit	Maximum number of iterations for the Newton-Raphson algorithm for computing the inverse of the derivative of the cumulant generator of the family.

Details

unifed.unit.deviance uses the following expression for the deviance of regular exponential dispersion families

$d(y,\mu)=2\left[y\{\dot{\kappa}^{-1}(y)-\dot{\kappa}^{-1}(\mu)\}-\kappa(\dot{\kappa}^{-1}(y))+\kappa(\dot{\kappa}^{-1}(\mu))\right]$

$\dot{\kappa}^{-1}$ is computed with the function unifed.kappa.prime.inverse from this package.

Value

unifed.deviance returns the deviance of a GLM with a unifed response distribution. This is

$D(\bm{y},\bm{\mu})=\sum_{i=1}^m w_i d(y_i,\mu_i)$

Where $d(y_i,\mu_i)$ is the unit deviance of the unifed distribution between the i-th entry of $\bm{y}$ and $\bm{\mu}$. $w_i$ is the i-th entry of the weight vector. unifed.unit.deviance is used to get the value of $d$.

unifed.unit.deviance

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Cumulant generator of the unifed distribution

Description

Cumulant generator of the unifed distribution

Usage

unifed.kappa(theta)

unifed.kappa.prime(theta)

unifed.kappa.double.prime(theta)

unifed.kappa.prime.inverse(mu, ...)

unifed.kappa.prime.inverse.one(mu, tol = 1e-07, maxit = 1e+07)

Arguments

theta	A numeric vector.
mu	A vector of numbers between 0 and 1
...	Other parameters of unifed.kappa.prime.inverse.one
tol	Tolerance level. The algorithm stops if the proportional difference between the new and old value of an iteration is less or equal than this number.
maxit	Maximum number of iterations of the algorithm to look for convergence.

Details

The cumulant generator of the unifed distribution is defined as

$\kappa(\theta)=\left\{ \begin{array}{ll} \log\left(\frac{e^\theta-1}{\theta}\right)& if \theta \neq 0\\ 0 & \mbox{if }\theta=0 \end{array} \right..$

unifed.kappa.prime.inverse.one uses the Newthon-Raphson method for finding the inverse of unifed.kappa.prime for a single value.

Value

unifed.kappa returns a vector that contains the cumulant generator of the unifed distribution applied to each element of theta.

unifed.kappa.prime returns a vector that contains the derivative of the cumulant generator of the unifed distribution for each element of theta.

unifed.kappa.double.prime returns a vector that contains the second derivative of the cumulant generator of the unifed distribution for each element of theta.

unifed.kappa.prime.inverse returns a vector with unifed.kappa.prime.inverse.one evaluated at every entry of mu.

unifed.kappa.prime.inverse.one if the tolerance level is reached within maxit iterations, the function returns the value of the last iteration. Otherwise it returns NA.

References

Quijano Xacur, O.A. The unifed distribution. J Stat Distrib App 6, 13 (2019). doi:10.1186/s40488-019-0102-6.

Jørgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall, London.

Examples

unifed.kappa(1)
unifed.kappa(-5:5)


unifed.kappa.prime(4.5)


unifed.kappa.double.prime(0)


unifed.kappa.prime.inverse(0.5)
unifed.kappa.prime.inverse(c(0.1,0.7,0.9))

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Maximum Likelihood Estimate for the unifed distribution

Description

Maximum Likelihood Estimate for the unifed distribution

Usage

unifed.mle(x)

Arguments

x	A numeric vector with values in the interval [0,1]. Computes the maximum likelihood estimator of the canonical parameter of the unifed distribution. It is assumed that the elements of x come from independent and identically distributed unifed random variables.

Examples

a.unifed.sample <- runifed(1000,10)
theta.mle <- unifed.mle(a.unifed.sample)

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Stan functions for working with the unifed distribution

Description

Stan functions for working with the unifed distribution

Details

A script with stan functions of the unifed is provided. The script can be included in stan code. The full path to the script can be obtained with the function unifed.stan.path. The following list are the names of functions that take one real value:

real unifed_kappa(real theta)

Computes the cumulant generator of the unifed distribution.

real unifed_kappa_prime(real theta)

Computes the first derivative of the cumulant generator.

real unifed_kappa_double_prime(real theta)

Computes the second derivative of the cumulant generator.

real unifed_lpdf(real x,real theta)

Computes the logarithm of the probability density function of a unifed distribution. theta is the value of the canonical parameter of the unifed and x if the value where the density is evaluated.

real unifed_quantile(real p,real theta)

Returns the p-th quantile of a unifed distribution with canonical parameter theta

.

real unifed_rng(real theta)

Returns a simulated value of a unifed distribution with canonical parameter theta.

real unifed_lcdf(real x,real theta)

Computes the logarithm of the cumulative density function of a unifed distribution. theta is the value of the canonical parameter of the unifed and x if the value where the density is evaluated.

real unifed_kappa_prime_inverse(real mu)

Returns the inverse of the derivative of the unifed cumulant generator

real unifed_unit_deviance(real y,real mu)

Unit deviance function of the unifed.

The following functions take vectors as arguments

vector unifed_kappa_v(vector theta)

Vectorized version of unifed_kappa.

vector unifed_kappa_prime_inverse_v(vector mu)

Vectorized version of unifed_kappa_prime_inverse.

void unifed_glm_lp(vector y, vector theta, vector weights)

Adds to the Log Probability Accumulator the logarithm of the likelihood function of a GLM with observed response y, estimated canonical parameter theta and weights weights.

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Unifed Stan function paths

Description

The unifed.stan provided by the file contains functions for using the unifed distribution in stan. The file can be included (with #include) insided the functions block of a stan program or its contents can be copied and pasted.

Usage

unifed.stan.path()

unifed.stan.folder()

Value

The full path to the unifed.stan file provided by the package.

unifed.stan.folder returns a string containing the path to the folder containing the unifed.stan file. This can be used as the isystem parameter in stan functions.

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Variance function of the unifed distribution

Description

Variance function of the unifed distribution

Usage

unifed.varf(mu)

Arguments

mu	A vector with numbers between 0 and 1.

Value

It returns unifed.kappa.double.prime(unifed.kappa.prime.inverse(mu)).

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