Discrete Inverse Weibull Distribution

Description

Probability mass function, distribution function, quantile function, random generation and parameter estimation for the discrete inverse Weibull distribution

Details

Author(s)

Alessandro Barbiero <alessandro.barbiero@unimi.it>, Riccardo Inchingolo <dott.inchingolo_r@libero.it>, Mario Samigli <giulio.samigli@gmail.com>

References

Jazi M.A., Lai C.-D., Alamatsaz M.H. (2010) A discrete inverse Weibull distribution and estimation of its parameters, Statistical Methodology 7: 121-132

Khan M.S., Pasha G.R., Pasha A.H. (2008) Theoretical Analysis of Inverse Weibull Distribution, WSEAS Trabsactions on Mathematics 2(7): 30-38

Drapella A. (1993) Complementary Weibull distribution: unknown or just forgotten, Quality Reliability Engineering International 9: 383-385

Dutang, C., Goulet, V., Pigeon, M. (2008) actuar: An R package for actuarial science, Journal of Statistical Software 25(7): 1-37

The discrete inverse Weibull distribution

Description

Probability mass function, distribution function, quantile function and random generation for the discrete inverse Weibull distribution with parameters q and \beta

Usage

ddiweibull(x, q, beta)
pdiweibull(x, q, beta)
qdiweibull(p, q, beta)
rdiweibull(n, q, beta)

Arguments

Details

The discrete inverse Weibull distribution has probability mass function given by P(X=x;q,\beta)=q^{(x)^{-\beta}}-q^{(x-1)^{\beta}}, x=1,2,3,\ldots, 0<q<1, \beta>0. Its cumulative distribution function is F(x; q, \beta)=q^{x^{-\beta}}

Value

ddiweibull gives the probability, pdiweibull gives the distribution function, qdiweibull gives the quantile function, and rdiweibull generates random values. See the reference below for the continuous inverse Weibull distribution.

References

Dutang, C., Goulet, V., Pigeon, M. (2008) actuar: An R package for actuarial science, Journal of Statistical Software 25(7): 1-37

Examples

# Ex.1
x<-1:10
q<-0.6
beta<-0.8
ddiweibull(x, q, beta)
t<-qdiweibull(0.99, q, beta)
t
pdiweibull(t, q, beta)
# Ex.2
q<-0.4
beta<-1.7
n<-100
x<-rdiweibull(n, q, beta)
tabulate(x)/sum(tabulate(x))
y<-1:round(max(x))
# compare with
ddiweibull(y, q, beta)

First and second order moments

Description

First and second order moments of the discrete inverse Weibull distribution

Usage

Ediweibull(q, beta, eps = 1e-04, nmax = 1000)

Arguments

Details

For a discrete inverse Weibull distribution we have E(X;q,\beta)=\sum_{x=0}^{+\infty} 1-F(x;q, \beta) and E(X^2;q,\beta)=2\sum_{x=1}^{+\infty} x(1-F(x;q, \beta))+E(X;q, \beta). The expected values are numerically computed considering a truncated support: integer values smaller than or equal to \min(nmax;F^{-1}(1-eps;q,\beta)), where F^{-1} is the inverse of the cumulative distribution function (implemented by the function qdiweibull). Increasing the value of nmax or decreasing the value of eps improves the approximation, but slows down the calculation speed

Value

a list comprising the (approximate) first and second order moments of the discrete inverse Weibull distribution. Note that the first moment is finite iff \beta is greater than 1; the second order moment is finite iff \beta is greater than 2

References

Khan M.S., Pasha G.R., Pasha A.H. (2008) Theoretical Analysis of Inverse Weibull Distribution, WSEAS Trabsactions on Mathematics 2(7): 30-38

Examples

# Ex.1
q<-0.75
beta<-1.25
Ediweibull(q, beta)
# Ex.2 
q<-0.5
beta<-2.5
Ediweibull(q, beta)
# Ex.3
q<-0.4
beta<-4
Ediweibull(q, beta)

Alternative hazard rate function

Description

Alternative hazard rate function for the discrete inverse Weibull distribution

Usage

ahrdiweibull(x, q, beta)

Arguments

Details

The alternative hazard rate function is defined as h(x)=\log(P(X>x-1)/P(X>x))=\log[(1-q^{(x-1)^{-\beta}})/(1-q^{x^{-\beta}})]

Value

the value of the alternative hazard rate function in the x values

See Also

hrdiweibull

Examples

q<-0.5
beta<-2
x<-1:10
y<-ahrdiweibull(x, q, beta)
y
plot(x,y,ylab="alt.hazard rate")

Estimation of parameters

Description

Sample estimation of the parameters of the discrete inverse Weibull distribution

Usage

estdiweibull(x, method="P", control=list())

Arguments

Details

For a description of the methods, have a look at the reference. Note that they may be not applicable to some specific samples. For examples, the method of proportion cannot be applied if there are no 1s in the samples; it cannot be applied for estimating \beta if all the sample values are \leq 2. The method of moments cannot be applied for estimating \beta if all the sample values are \leq 2; besides, it may return unreliable results since the first and second moments can be computed only if \beta>2. The heuristic method cannot be applied for estimating \beta if all the sample values are \leq 2.

Value

a vector containing the two estimates of q and \beta

See Also

heuristic, Ediweibull

Examples

n<-100 
q<-0.5 
beta<-2.5
# generation of a sample
x<-rdiweibull(n, q, beta)
# sample estimation through each of the implemented methods
estdiweibull(x, method="P") 
estdiweibull(x, method="M")
estdiweibull(x, method="H")
estdiweibull(x, method="PP")

Heuristic method of estimation

Description

Heuristic method for the estimation of parameters of the discrete inverse Weibull

Usage

heuristic(x, beta1=1, z = 0.1, r = 0.1, Leps = 0.01)

Arguments

Details

For a detailed description of the method, have a look at the reference

Value

a list containig the two estimates of q and \beta

References

Jazi M.A., Lai C.-D., Alamatsaz M.H. (2010) A discrete inverse Weibull distribution and estimation of its parameters, Statistical Methodology, 7: 121-132

Drapella A. (1993) Complementary Weibull distribution: unknown or just forgotten, Quality Reliability Engineering International 9: 383-385

See Also

estdiweibull

Examples

n<-50
q<-0.25
beta<-1.5
x<-rdiweibull(n, q, beta)
# estimates using the heuristic algorithm
par0<-heuristic(x)
par0
# change the default values of some working parameters...
par1<-heuristic(x, beta1=2)
par1
par2<-heuristic(x, z=0.5)
par2
par3<-heuristic(x, r=0.2)
par3
par4<-heuristic(x, Leps=0.1)
par4
# ...there should be just light differences among the estimates...
# ... and among the corresponding values of the loglikelihood functions
loglikediw(x, par0[1], par0[2])
loglikediw(x, par1[1], par1[2])
loglikediw(x, par2[1], par2[2])
loglikediw(x, par3[1], par3[2])
loglikediw(x, par4[1], par4[2])

Hazard rate function

Description

Hazard rate function for the discrete inverse Weibull distribution

Usage

hrdiweibull(x, q, beta)

Arguments

Details

The hazard rate function is defined as r(x)=P(X=x)/P(X\ge x)=(q^{x^{-\beta}}-q^{(x-1)^{-\beta}})/(1-q^{(x-1)^{-\beta}})

Value

the hazard rate function computed on the x values

See Also

ahrdiweibull

Examples

q<-0.5
beta<-2.5
x<-1:10
hrdiweibull(x, q, beta)

likelihood function

Description

Log-likelihood function of the discrete inverse Weibull

Usage

loglikediw(x, q, beta)

Arguments

Value

the value of the log-likelihood function (changed in sign) of the discrete inverse Weibull distribution with parameters q and \beta computed on a sample x

See Also

heuristic

Examples

n<-100
q<-0.4
beta<-2
x<-rdiweibull(n, q, beta)
# loglikelihood function (changed in sign) computed on the true values
loglikediw(x, q, beta)
par<-estdiweibull(x, method="H")
par
# loglikelihood function (changed in sign) computed on the ML estimates
loglikediw(x, par[1], par[2])
# it should be smaller than before...

Loss function

Description

Quadratic loss function for the method of moments

Usage

lossdiw(x, par, eps = 1e-04, nmax=1000)

Arguments

Value

the value of the quadratic loss function L(x; q, \beta)=(E(X; q, \beta)-m_1)^2+(E(X^2; q, \beta)-m_2)^2 where m_1 and m_2 are the first and second order sample moments.

See Also

Ediweibull

Examples

n<-100
q<-0.5
beta<-2.5
x<-rdiweibull(n, q, beta)
# loss function computed on the true values
lossdiw(x, c(q, beta))
par<-estdiweibull(x, method="M")
# estimates of the parameters through the method of moments
par
# loss function computed on the estimates derived through
# the method of moments
lossdiw(x, par)
# it should be zero (however, smaller than before...)