Discrete Weibull Distributions (Type 1 and 3)

Description

Probability mass function, distribution function, quantile function, random generation and parameter estimation for the type I and III discrete Weibull distributions. In the present version, some modifications have been made on the functions in Edweibull for the computation of the moments of the type I discrete Weibull distribution.

Details

Author(s)

Alessandro Barbiero <alessandro.barbiero@unimi.it>

References

T. Nakagawa and S. Osaki (1975) The discrete Weibull distribution, IEEE Transactions on Reliability, 24(5), pp. 300-301

D.N.P. Murthy, M. Xie, R Jiang (2004) Weibull models, John Wiley & Sons: Hoboken, New Jersey

W.J. Padgett and J.D. Spurrier (1985) Discrete failure models, IEEE Transactions on Reliability, 34(3), pp. 253-256

H. Rinne (2008) The Weibull Distribution: A Handbook. CRC Press: Boca Raton, Florida

M. S. A. Khan, A. Khalique, and A. M. Abouammoh (1989) On estimating parameters in a discrete Weibull distribution, IEEE Transactions on Reliability, 38(3), pp. 348-350

See Also

ddweibull, estdweibull, Edweibull, ddweibull3, estdweibull3, Edweibull3

The type 1 discrete Weibull distribution

Description

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

Usage

ddweibull(x, q, beta, zero = FALSE)
pdweibull(x, q, beta, zero = FALSE)
qdweibull(p, q, beta, zero = FALSE)
rdweibull(n, q, beta, zero = FALSE)

Arguments

Details

The discrete Weibull distribution has probability mass function given by P(X=x;q,\beta)=q^{(x-1)^{\beta}}-q^{x^{\beta}}, x=1,2,3,\ldots, if zero=FALSE; or P(X=x;q,\beta)=q^{x^{\beta}}-q^{(x+1)^{\beta}}, x=0,1,2,\ldots, if zero=TRUE. The cumulative distribution function is F(x;q,\beta)=1-q^{x^{\beta}} if zero=FALSE; F(x;q,\beta)=1-q^{(x+1)^{\beta}} otherwise

Value

ddweibull gives the probability function, pdweibull gives the distribution function, qdweibull gives the quantile function, and rdweibull generates random values.

Author(s)

Alessandro Barbiero

Examples

# Ex.1
x <- 1:10
q <- 0.6
beta <- 0.8
ddweibull(x, q, beta)
t <- qdweibull(0.99, q, beta)
t
pdweibull(t, q, beta)
# 
x <- 0:10
ddweibull(x, q, beta, zero=TRUE)
t <- qdweibull(0.99, q, beta, zero=TRUE)
t
pdweibull(t, q, beta, zero=TRUE)

# Ex.2
q <- 0.4
beta <- 0.7
n <- 100
x <- rdweibull(n, q, beta)
tabulate(x)/sum(tabulate(x))
y <- 1:round(max(x))
# compare with
ddweibull(y, q, beta)

The type 3 discrete Weibull distribution

Description

Probability mass function, distribution function, quantile function, random generation, and hazard function for the type 3 discrete Weibull distribution with parameters c and \beta

Usage

ddweibull3(x, c, beta)
pdweibull3(x, c, beta)
qdweibull3(p, c, beta)
rdweibull3(n, c, beta)
hdweibull3(x, c, beta)

Arguments

Details

The type 3 discrete Weibull distribution is characterized by the following cumulative distribution function: F(x;c,\beta)=1-\exp(-c\sum_{j=0}^{x+1}j^{\beta}), for x=0,1,2,\dots, with c>0 and \beta>=-1.

Value

ddweibull3 gives the probability function, pdweibull3 gives the distribution function, qdweibull3 gives the quantile function, hdweibull3 gives the hazard function, and rdweibull generates random values.

Author(s)

Alessandro Barbiero

Examples

# ddweibull3
x <- 0:10
c <- 0.3
beta <- 0.75
p <- ddweibull3(x, c, beta)
p
plot(x, p, type="b", ylab=expression(P(X)==x))
# pdweibull3
x <- 0:10
c <- 0.5
beta <- 0.5
p <- pdweibull3(x, c, beta)
p
cumsum(ddweibull3(x, c, beta))
plot(x, p, type="s", ylab=expression(P(X<=x)))
# qdweibull3
p <- c(1:9)/10
p
c <- 0.1
beta <- 0.5
qdweibull3(p, c, beta)
pdweibull3(10, c, beta)
pdweibull3(9, c, beta)
# rdweibull3
n <- 20
c <- 0.25
beta <- -0.25
x <- rdweibull3(n, c, beta)
x
beta <- 0
x <- rdweibull3(n, c, beta)
x
beta <- 0.25
x <- rdweibull3(n, c, beta)
x
n <- 1000
x <- rdweibull3(n, c, beta)
obs <- c(sum(x==0), tabulate(x))
obs <- obs/sum(obs)
theo <- ddweibull3(min(x):max(x), c, beta)
barplot(rbind(obs, theo), beside=TRUE, names.arg=min(x):max(x),
ylab="relative frequency/probability", col=1:2)
legend(24, 0.1, c("observed", "theoretical"), pch=15, col=1:2)
#hdweibull3
x <- 0:15
c <- 0.5
hn<-hdweibull3(x, c, beta = -0.5)
h0<-hdweibull3(x, c, beta = 0)
hp<-hdweibull3(x, c, beta = 0.5)
plot(x, hn, type="b", ylim = c(0, 1), ylab="hazard rate")
points(x, h0, type = "b", col=2)
points(x, hp, type = "b", col=3)
legend(11, 0.5, c("beta<0", "beta=0", "beta>0"), col=1:3, pch=21)

Expected values

Description

First and second order moments, variance and expected value of the reciprocal for the type 1 discrete Weibull distribution

Usage

Edweibull(q, beta, eps = 1e-04, nmax = 1000, zero = FALSE)
E2dweibull(q, beta, eps = 1e-04, nmax = 1000, zero = FALSE)
Vdweibull(q, beta, eps = 1e-04, nmax = 1000, zero = FALSE)
ERdweibull(q, beta, eps = 1e-04, nmax = 1000)

Arguments

Details

The expected value is numerically computed considering a truncated support: integer values smaller than or equal to 2F^{-1}(1-eps;q,\beta) are considered, where F^{-1} is the inverse of the cumulative distribution function (implemented by the function qdweibull). However, if such value is greater than nmax, the expected value is computed recalling the formula of the expected value of the corresponding continuous Weibull distribution (see the reference), adding 0.5. Similar arguments apply to the other moments.

Value

the (approximate) expected values of the discrete Weibull distribution: Edweibull gives the first order moment, E2dweibull the second order moment, Vdweibull the variance, ERdweibull the expected value of the reciprocal (only if zero is FALSE)

Author(s)

Alessandro Barbiero

References

M. S. A. Khan, A. Khalique, and A. M. Abouammoh (1989) On estimating parameters in a discrete Weibull distribution, IEEE Transactions on Reliability, 38(3), pp. 348-350

Examples

q <- 0.75
beta <- 1.25
Edweibull(q, beta)
E2dweibull(q, beta)
Vdweibull(q, beta)
ERdweibull(q, beta)
# if beta=0.75...
beta <- 0.75
Edweibull(q, beta)
Edweibull(q, beta, nmax=100)
# here above, the approximation through the continuous model intervenes
# if beta=1...
beta <- 1
Edweibull(q, beta)
# which equals...
1/(1-q)

Expected values

Description

First and second order moments for the type 3 discrete Weibull distribution

Usage

Edweibull3(c, beta, eps = 1e-04)
E2dweibull3(c, beta, eps = 1e-04)

Arguments

Details

The expected values are numerically computed considering a truncated support: integer values smaller than or equal to 2F^{-1}(1-eps;c,\beta)), where F^{-1} is the inverse of the cumulative distribution function (implemented by the function qdweibull3)

Value

the (approximate) expected values of the discrete Weibull distribution: Edweibull3 gives the first order moment, E2dweibull3 the second order moment

Author(s)

Alessandro Barbiero

Examples

c <- 0.4
beta <- 0.25
Edweibull3(c,beta)
c <- 0.4
beta <- -0.75
Edweibull3(c, beta) # may require too much time
Edweibull3(c, beta, eps=0.001) # try with a smaller eps->worse approximation
c <- rep(0.1, 11)
beta <- (0:10)/10
Edweibull3(c, beta)
c <- rep(0.5, 11)
beta <- (-5:5)/10
Edweibull3(c,beta)
# E2dweibull3
c <- 0.4
beta <- 0.25
E2dweibull3(c, beta)
c <- rep(0.1, 11)
beta <- (0:10)/10
Edweibull3(c, beta)
c <- rep(0.8, 11)
beta <- (-5:5)/11
E2dweibull3(c, beta)

Estimation of parameters

Description

Estimation of the parameters of the type 1 discrete Weibull distribution

Usage

estdweibull(x, method = "ML", zero = FALSE, eps = 1e-04, nmax=1000)

Arguments

Value

the vector of the estimates of q and \beta

Author(s)

Alessandro Barbiero

See Also

ddweibull

Examples

# Ex1
n <- 10
q <- 0.5
beta <- 0.8
x <- rdweibull(n, q, beta)
estdweibull(x, "ML") # maximum likelihood method
# it may return some harmless warnings
# that depend on the optimization function used in the maximization routine
estdweibull(x, "M") # method of moments
estdweibull(x, "P") # method of proportion
# the estimates provided by the three methods may be quite different
# from the true values... and to each other
# change the sample size
n <- 50
q <- 0.5
beta <- 0.8
x <- rdweibull(n, q, beta)
estdweibull(x, "ML") # maximum likelihood method
estdweibull(x, "M") # method of moments
estdweibull(x, "P") # method of proportion
# the estimates should be (on average) closer to the true values
# ...and to each other

# When the estimation methods fail...
# Ex2
# only 1s and 2s
x <- c(1,1,1,1,1,1,2,2,2,2)
estdweibull(x, "ML") # fails!
estdweibull(x, "M") # fails!
estdweibull(x, "P") # fails!

# Ex3
# no 1s
x <- c(2,2,3,4,5,5,5,6,6,8,10)
estdweibull(x, "ML") # works
estdweibull(x, "M") # works
estdweibull(x, "P") # fails!

Estimation of parameters

Description

Estimation of the parameters of the type 3 discrete Weibull distribution

Usage

estdweibull3(x, method = "P", eps = 1e-04)

Arguments

Value

the vector of the estimates of c and \beta

Author(s)

Alessandro Barbiero

See Also

ddweibull3

Examples

# Ex1
x <- c(0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6)
estdweibull3(x, "P")
estdweibull3(x, "ML")
estdweibull3(x, "M")
# Ex 2
n <- 20
c <- 1/3
beta <- 2/3
x <- rdweibull3(n, c, beta)
estdweibull3(x, "P")
par <- estdweibull3(x, "ML")
par
-loglikedw3(par, x)
par <- estdweibull3(x, "M")
par
lossdw3(par, x)
n <- 50
x <- rdweibull3(n, c, beta)
estdweibull3(x, "P")
estdweibull3(x, "ML")
estdweibull3(x, "M")
n <- 100
x <- rdweibull3(n, c, beta)
estdweibull3(x, "P")
estdweibull3(x, "ML")
estdweibull3(x, "M")
# Ex 3: a piece of simulation study
nSim <- 50
n <- 50
c <- 0.2
beta <- 0.7
par <- matrix(0, nSim, 2)
for(i in 1:nSim)
{
x <- rdweibull3(n, c, beta)
par[i,] <- estdweibull3(x, "ML")
}
op <- par(mfrow = c(1,2))
boxplot(par[,1], xlab=expression(hat(c)[ML]))
abline(h = c)
boxplot(par[,2], xlab=expression(hat(beta)[ML]))
abline(h = beta)
op <- par()

Loglikelihood function

Description

Loglikelihood function (changed in sign) for the type 1 discrete Weibull distribution

Usage

loglikedw(par, x, zero = FALSE)

Arguments

Value

the value of the loglikelihood function (changed in sign) for the observed sample x under the parameters par

Author(s)

Alessandro Barbiero

See Also

estdweibull

Examples

x <- c(1,1,1,2,2,2,2,2,2,3,4,4,5,6,8)
-loglikedw(c(0.8, 1), x) # loglikelihood function for q=0.8 and beta=1
-loglikedw(c(0.4, 2), x) # loglikelihood function for q=0.4 and beta=2
par <- estdweibull(x, "ML")# parameter estimates derived by the ML method
par
-loglikedw(par, x) # the maximum value of the loglikelihood function

Loglikelihood function

Description

Loglikelihood function (changed in sign) for the type 3 discrete Weibull distribution

Usage

loglikedw3(par, x)

Arguments

Value

the value of the loglikelihood function (changed in sign) for the observed sample x under the parameters par

Author(s)

Alessandro Barbiero

See Also

estdweibull3

Examples

n <- 20
c <- 1/3
beta <- 2/3
x <- rdweibull3(n, c, beta)
par <- estdweibull3(x, "ML")
par
-loglikedw3(par, x)

Loss function

Description

Loss function for the method of moments (type 1 discrete Weibull)

Usage

lossdw(par, x, zero = FALSE, eps = 1e-04, nmax=1000)

Arguments

Details

The loss function is given by L(x;q,\beta)=[m_1-\mathrm{E}(X;q,\beta)]^2+[m_2-\mathrm{E}(X^2;q,\beta)]^2, where \mathrm{E}(\cdot) denotes the expected value, m_1 and m_2 are the first and second order sample moments respectively.

Value

the value of the quadratic loss function

Author(s)

Alessandro Barbiero

See Also

Edweibull

Examples

x <- c(1,1,1,1,1,2,2,2,3,4)
lossdw(c(0.5, 1), x)
par <- estdweibull(x, "M") # parameter estimates derived by the method of moments
par
lossdw(par, x) # the loss is zero using these estimates

Loss function

Description

Loss function for the method of moments (type 3 discrete Weibull)

Usage

lossdw3(par, x, eps = 1e-04)

Arguments

Details

The loss function is given by L(x;c,\beta)=[m_1-\mathrm{E}(X;c,\beta)]^2+[m_2-\mathrm{E}(X^2;c,\beta)]^2, where \mathrm{E}(\cdot) denotes the expected value, m_1 and m_2 are the first and second order sample moments respectively.

Value

the value of the quadratic loss function

Author(s)

Alessandro Barbiero

See Also

Edweibull3

Examples

n <- 25
c <- 1/3
beta <- 2/3
x <- rdweibull3(n, c, beta)
par <- estdweibull3(x, "M")
par
lossdw3(par, x)

Observed Fisher information matrix

Description

Observed Fisher information matrix on a sample from the type 1 discrete Weibull distribution

Usage

varFisher(x, zero = FALSE)

Arguments

Value

a list of two matrices: the observed Fisher information matrix, and its inverse, which contains asymptotic variances and covariances of the maximum likelihood estimators of q and \beta

Author(s)

Alessandro Barbiero

See Also

estdweibull

Examples

x <- rdweibull(100, 2/3, 3/2)
estdweibull(x, "ML")
IF <- varFisher(x)[[2]]
diag(IF)
diag(IF)/length(x)
# asymptotic variances of the ML estimators directly estimated from the sample