Package 'DiscreteWeibull' reference manual

Title:	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.
Authors:	Alessandro Barbiero
Maintainer:	Alessandro Barbiero <[email protected]>
License:	GPL
Version:	1.1
Built:	2024-12-13 06:29:54 UTC
Source:	CRAN

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

Package:	DiscreteWeibull
Type:	Package
Version:	1.1
Date:	2015-10-17
License:	GPL
LazyLoad:	yes
Depends:	Rsolnp

Author(s)

Alessandro Barbiero <[email protected]>

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

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)
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

`x`	vector of quantiles
`p`	vector of probabilities
`q`	first parameter
`beta`	second parameter
`zero`	`TRUE`, if the support contains $0$ ; `FALSE` otherwise
`n`	sample size

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

Arguments

`x`	vector of values/quantiles
`p`	vector of probabilities
`c`	first parameter
`beta`	second parameter
`n`	sample size

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)
# 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)
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

`q`	first parameter
`beta`	second parameter
`eps`	error threshold for the numerical computation of the expected value
`nmax`	maximum value considered for the numerical approximate computation of the expected value;
`zero`	`TRUE`, if the support contains $0$ ; `FALSE` otherwise

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

Arguments

`c`	first parameter
`beta`	second parameter
`eps`	error threshold for the numerical computation of the expected value

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

Arguments

`x`	the vector of sample values
`method`	`"ML"` for the maximum likelihood method; `"M"` for the method of moments; `"P"` for the method of proportions
`zero`	`TRUE`, if the support contains $0$ ; `FALSE` otherwise
`eps`	error threshold for the computation of the moments of the distribution
`nmax`	maximum value considered for the numerical computation of the expected value

Value

the vector of the estimates of $q$ and $\beta$

Author(s)

Alessandro Barbiero

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

Arguments

`x`	the vector of sample values
`method`	`"ML"` for the maximum likelihood method; `"M"` for the method of moments; `"P"` for the method of proportions
`eps`	error threshold for the computation of the moments of the distribution

Value

the vector of the estimates of $c$ and $\beta$

Author(s)

Alessandro Barbiero

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()
# 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)
loglikedw(par, x, zero = FALSE)

Arguments

`par`	the vector of parameters, $q$ and $\beta$
`x`	the vector of sample values
`zero`	`TRUE`, if the support contains $0$ ; `FALSE` otherwise

Value

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

Author(s)

Alessandro Barbiero

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
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)
loglikedw3(par, x)

Arguments

`par`	the vector of parameters, $c$ and $\beta$
`x`	the vector of sample values

Value

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

Author(s)

Alessandro Barbiero

Examples

n <- 20
c <- 1/3
beta <- 2/3
x <- rdweibull3(n, c, beta)
par <- estdweibull3(x, "ML")
par
-loglikedw3(par, x)
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)
lossdw(par, x, zero = FALSE, eps = 1e-04, nmax=1000)

Arguments

`par`	vector of parameters $q$ and $\beta$
`x`	the vector of sample values
`zero`	`TRUE`, if the support contains $0$ ; `FALSE` otherwise
`eps`	error threshold for the numerical computation of the expected value
`nmax`	maximum value considered for the numerical computation of the expected value

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

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

Arguments

`par`	vector of parameters $q$ and $\beta$
`x`	the vector of sample values
`eps`	error threshold for the numerical computation of the expected value

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

Examples

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

Arguments

`x`	a vector of sample values
`zero`	`TRUE`, if the support contains $0$ ; `FALSE` otherwise

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

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
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

Package 'DiscreteWeibull'

Help Index

Discrete Weibull Distributions (Type 1 and 3)

Description

Details

Author(s)

References

See Also

The type 1 discrete Weibull distribution

Description

Usage

Arguments

Details

Value

Author(s)

Examples

The type 3 discrete Weibull distribution

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Expected values

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Expected values

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Estimation of parameters

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Estimation of parameters

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Loglikelihood function

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Loglikelihood function

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Loss function

Description

Usage

Arguments

Details

Value

Author(s)