The Weibull distribution is an absolute continuous probability distribution, parameterized by a shape parameter \(k > 0\) and a scale parameter \(\lambda > 0\).
Usage
Weib(shape = 1, scale = 1)
# S4 method for class 'Weib,numeric'
d(distr, x, log = FALSE)
# S4 method for class 'Weib,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)
# S4 method for class 'Weib,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)
# S4 method for class 'Weib,numeric'
r(distr, n)
# S4 method for class 'Weib'
mean(x)
# S4 method for class 'Weib'
median(x)
# S4 method for class 'Weib'
mode(x)
# S4 method for class 'Weib'
var(x)
# S4 method for class 'Weib'
sd(x)
# S4 method for class 'Weib'
skew(x)
# S4 method for class 'Weib'
kurt(x)
# S4 method for class 'Weib'
entro(x)
llweibull(x, shape, scale)
# S4 method for class 'Weib,numeric'
ll(distr, x)
eweibull(x, type = "mle", ...)
# S4 method for class 'Weib,numeric'
mle(
distr,
x,
par0 = "lme",
method = "L-BFGS-B",
lower = 1e-05,
upper = Inf,
na.rm = FALSE
)
# S4 method for class 'Weib,numeric'
me(distr, x, par0 = "lme", lower = 0.5, upper = Inf, na.rm = FALSE)Arguments
- shape, scale
numeric. The non-negative distribution parameters.
- distr
an object of class
Weib.- x
For the density function,
xis a numeric vector of quantiles. For the moments functions,xis an object of classWeib. For the log-likelihood and the estimation functions,xis the sample of observations.- log, log.p
logical. Should the logarithm of the probability be returned?
- q
numeric. Vector of quantiles.
- lower.tail
logical. If TRUE (default), probabilities are \(P(X \leq x)\), otherwise \(P(X > x)\).
- p
numeric. Vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.- type
character, case ignored. The estimator type (mle, me or lme).
- ...
extra arguments.
- par0, method, lower, upper
arguments passed to optim for the mle and me optimization. See Details.
- na.rm
logical. Should the
NAvalues be removed?
Value
Each type of function returns a different type of object:
Distribution Functions: When supplied with one argument (
distr), thed(),p(),q(),r(),ll()functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distrandx), they evaluate the aforementioned functions directly.Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The
moments()function returns a list with all the available methods.Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.
Details
The probability density function (PDF) of the Weibull distribution is: $$ f(x; k, \lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda} \right)^{k - 1} \exp\left[-\left(\frac{x}{\lambda}\right)^k\right], \quad x \geq 0 .$$
For the parameter estimation, both the MLE and the ME cannot be explicitly
derived. However, the L-moment estimator (type = "lme") is available, and
is used as initialization for the numerical approximation of the MLE and the
ME.
The MLE and ME of the Weibull distribution parameters is not available in closed form and has to be approximated numerically. The optimization can be performed on the shape parameter \(k\in(0,+\infty)\).
For the MLE, this is done with optim(). The default method used is the
L-BFGS-B method with lower bound 1e-5 and upper bound Inf. The par0
argument can either be a numeric (satisfying lower <= par0 <= upper) or a
character specifying the closed-form estimator to be used as initialization
for the algorithm ("lme" - the default value).
For the ME, this is done with uniroot(). Again, the par0 argument can
either be a numeric (satisfying lower <= par0 <= upper) or a character
specifying the closed-form estimator to be used as initialization for the
algorithm ("mle" or "lme" - the default value). The lower and upper
bounds are set by default to 0.5 and Inf, respectively. Note that the
ME equations involve the \(\Gamma(1 + 1 \ k)\), which can become unreliable
for small values of k, hence the 0.5 lower bound. Specifying a lower
bound below 0.5 will result in a warning and be ignored.
References
Kim, H. M., Jang, Y. H., Arnold, B. C., & Zhao, J. (2024). New efficient estimators for the Weibull distribution. Communications in Statistics-Theory and Methods, 53(13), 4576-4601.
See also
Functions from the stats package: dweibull(), pweibull(), qweibull(),
rweibull()
Examples
# -----------------------------------------------------
# Weibull Distribution Example
# -----------------------------------------------------
# Create the distribution
a <- 3 ; b <- 5
D <- Weib(a, b)
# ------------------
# dpqr Functions
# ------------------
d(D, c(0.3, 2, 10)) # density function
#> [1] 0.0021595335 0.0900484800 0.0008051103
p(D, c(0.3, 2, 10)) # distribution function
#> [1] 0.0002159767 0.0619950005 0.9996645374
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 3.996939 5.859512
x <- r(D, 100) # random generator function
# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#> [1] 0.213216484 0.189095707 0.223099907 0.171570313 0.183506911 0.235044754
#> [7] 0.230584643 0.176772122 0.230333687 0.207085245 0.167936595 0.198665143
#> [13] 0.161516267 0.227530990 0.214915556 0.207129065 0.124968588 0.203969106
#> [19] 0.200518032 0.056385202 0.209026726 0.218628586 0.228204805 0.233139174
#> [25] 0.144983652 0.213969519 0.133126941 0.145349466 0.233569584 0.176998852
#> [31] 0.233913242 0.233568640 0.011617491 0.183448743 0.141567297 0.191559224
#> [37] 0.222189145 0.233343469 0.186499957 0.117952698 0.141628034 0.133815768
#> [43] 0.156439565 0.219399362 0.234203286 0.209956745 0.190507331 0.165846503
#> [49] 0.205451427 0.118625567 0.183711919 0.160501655 0.233070706 0.165627489
#> [55] 0.048790685 0.226325065 0.076753493 0.190647485 0.045573016 0.231022590
#> [61] 0.201478393 0.035674815 0.232670668 0.229377728 0.210088231 0.192001303
#> [67] 0.234201175 0.217805481 0.235080104 0.002620994 0.234898813 0.185030856
#> [73] 0.146714918 0.154330941 0.159883015 0.218050643 0.072927834 0.234298139
#> [79] 0.234894547 0.089766721 0.146003765 0.219905987 0.231509449 0.160623031
#> [85] 0.149186864 0.235034641 0.218020589 0.147624730 0.119072433 0.234959975
#> [91] 0.170872230 0.170564649 0.122234950 0.217872452 0.101972743 0.094784734
#> [97] 0.057743329 0.211975329 0.207145840 0.232091664
# ------------------
# Moments
# ------------------
mean(D) # Expectation
#> [1] 4.464898
median(D) # Median
#> [1] 4.424985
mode(D) # Mode
#> [1] 4.367902
var(D) # Variance
#> [1] 2.633322
sd(D) # Standard Deviation
#> [1] 1.622751
skew(D) # Skewness
#> [1] 0.1681028
kurt(D) # Excess Kurtosis
#> [1] -0.2705364
entro(D) # Entropy
#> [1] 4.844159
# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 4.464898
# ------------------
# Point Estimation
# ------------------
ll(D, x)
#> [1] -183.4598
llweibull(x, a, b)
#> [1] -183.4598
eweibull(x, type = "mle")
#> $shape
#> [1] 3.231992
#>
#> $scale
#> [1] 5.006414
#>
eweibull(x, type = "me")
#> $shape
#> [1] 3.193606
#>
#> $scale
#> [1] 4.972298
#>
eweibull(x, type = "lme")
#> $shape
#> [1] 3.231992
#>
#> $scale
#> [1] 4.969405
#>
mle(D, x)
#> $shape
#> [1] 3.231992
#>
#> $scale
#> [1] 5.006414
#>
me(D, x)
#> $shape
#> [1] 3.193606
#>
#> $scale
#> [1] 4.972298
#>
e(D, x, type = "mle")
#> $shape
#> [1] 3.231992
#>
#> $scale
#> [1] 5.006414
#>
mle("weib", x) # the distr argument can be a character
#> $shape
#> [1] 3.231992
#>
#> $scale
#> [1] 5.006414
#>