Introduction
The discrete Weibull distribution is a probability distribution defined on non‑negative integers that serves as a discrete analogue of the continuous Weibull distribution. It was introduced to model phenomena where the data are naturally integer‑valued, such as count data or lifetimes measured in discrete time units, while preserving the flexible shape properties of its continuous counterpart. The distribution is characterized by two positive parameters: a scale parameter, commonly denoted by λ, and a shape parameter, denoted by p. Its probability mass function (PMF) depends on the cumulative distribution function (CDF) of the continuous Weibull distribution evaluated at integer arguments, leading to a simple yet versatile family of discrete distributions that can capture increasing, decreasing, or bathtub‑shaped hazard functions.
Because of its ability to model count data with a variety of dispersion patterns, the discrete Weibull distribution has found applications in reliability engineering, queueing theory, biomedical studies, and environmental statistics. The distribution also appears in the theory of order statistics and serves as a building block for more complex discrete stochastic models.
History and Background
The continuous Weibull distribution, introduced by Waloddi Weibull in 1951, quickly became a standard tool for reliability and life‑testing analysis due to its flexible hazard function. Researchers later sought discrete analogues that preserved key features while accommodating integer‑valued data. In the early 2000s, a class of discrete distributions was defined by applying the continuous Weibull CDF to discrete arguments, yielding what is now known as the discrete Weibull distribution.
Early work on the discrete Weibull focused on deriving its PMF and basic properties. Subsequent studies explored maximum‑likelihood estimation procedures and applications to lifetime data measured in discrete units. The distribution was also investigated within the context of regression models for count data, leading to the development of discrete Weibull generalized linear models.
Recent literature has expanded the family to include zero‑inflated and hurdle versions, accommodating excess zeros commonly encountered in over‑dispersed count data. Theoretical work has also linked the discrete Weibull to other discrete distributions, such as the geometric and negative binomial, through limiting processes and parameter specialisations.
Key Concepts
Definition and Notation
Let X be a non‑negative integer‑valued random variable. The discrete Weibull distribution with parameters λ > 0 and p > 0 is defined by the following cumulative distribution function:
F(k; λ, p) = 1 – exp(–(k / λ)ˣⁿ) for k = 0, 1, 2, …
From this CDF the probability mass function follows by difference:
f(k; λ, p) = exp(–((k – 1)/λ)ᵖ) – exp(–(k/λ)ᵖ) for k = 0, 1, 2, …
In practice, it is common to shift the support to start at 1 by redefining k = 1, 2, … and adjusting the formulas accordingly. The distribution is sometimes denoted by DW(λ, p) or simply DW.
Shape of the Hazard Function
The hazard function, defined as h(k) = f(k) / (1 – F(k – 1)), can take various shapes depending on the value of the shape parameter p:
- When
p , the hazard function is decreasing, indicating a “short‑term” failure rate that declines over time. - When
p = 1, the hazard function is constant, equivalent to a geometric distribution with success probabilityexp(–1/λ). - When
p > 1, the hazard function is increasing, corresponding to a “wear‑out” failure rate.
This flexibility enables the discrete Weibull to model a wide range of practical lifetime and count scenarios.
Relationship to the Continuous Weibull
Because the discrete Weibull is derived by evaluating the continuous Weibull CDF at integer points, it can be seen as a discretisation of the continuous Weibull. Consequently, many properties, such as moment expressions and likelihood functions, are structurally similar to those of the continuous version, albeit with differences arising from the discrete support.
Parameters and Properties
Scale Parameter (λ)
The scale parameter controls the overall spread of the distribution. Larger values of λ shift the mass function to the right, increasing the expected value and variance. The parameter is strictly positive and has units of the count variable (e.g., time steps).
Shape Parameter (p)
The shape parameter determines the curvature of the hazard function. It is also strictly positive. Special values yield well‑known discrete distributions:
p = 1reduces the discrete Weibull to the geometric distribution.p = 0.5yields a distribution with a particular decreasing hazard shape.
Support and Support Modification
By default, the support is the set of non‑negative integers {0, 1, 2, …}. In applications where the count variable is strictly positive, a one‑unit shift is applied, and the support becomes {1, 2, 3, …}. This shift requires adjusting the PMF accordingly.
Statistical Moments
Because closed‑form expressions for moments are generally unavailable, moment calculations rely on series expansions or numerical integration. The first moment (mean) can be approximated by
E[X] ≈ Σ_{k=0}^{∞} k · f(k; λ, p)
which converges rapidly for moderate values of λ and p. Variance and higher moments are computed similarly. Some authors provide asymptotic approximations that are useful for large λ.
Moment‑Generating Function
The moment‑generating function (MGF) is defined as M(t) = E[e^{tX}]. For the discrete Weibull, an explicit MGF does not exist in closed form, but it can be expressed as an infinite series:
M(t) = Σ_{k=0}^{∞} e^{t k} · f(k; λ, p)
Truncation of this series yields accurate approximations for practical parameter ranges.
Probability Mass Function and Cumulative Distribution Function
Probability Mass Function (PMF)
The PMF of the discrete Weibull is derived from the difference of consecutive values of the CDF:
f(k; λ, p) = exp(–((k – 1)/λ)ᵖ) – exp(–(k/λ)ᵖ), k = 0, 1, 2, …
When the support is shifted to start at 1, the PMF becomes:
f(k; λ, p) = exp(–((k – 1)/λ)ᵖ) – exp(–(k/λ)ᵖ), k = 1, 2, 3, …
Several properties of the PMF follow directly from its construction:
- Non‑negativity is guaranteed because each term is a difference of two decreasing exponentials.
- Normalization holds: Σ_{k=0}^{∞} f(k; λ, p) = 1.
- The PMF decreases monotonically when
p > 1and increases monotonically whenp .
Cumulative Distribution Function (CDF)
The CDF is given by the continuous Weibull CDF evaluated at integer arguments:
F(k; λ, p) = 1 – exp(–(k/λ)ᵖ), k = 0, 1, 2, …
Consequently, the survival function (one minus the CDF) is
S(k; λ, p) = exp(–(k/λ)ᵖ).
These expressions allow for straightforward computation of tail probabilities and quantiles via numerical inversion.
Moments and Moment‑Generating Functions
Expectation and Variance
While closed‑form expressions for the mean E[X] and variance Var(X) are not available, they can be expressed through the incomplete gamma function after suitable transformations. One such representation for the mean is
E[X] = λ · Γ(1 + 1/p) · Σ_{n=1}^{∞} (–1)^{n+1} / n · Γ(1 – n/p, 0)
where Γ(·, ·) denotes the incomplete gamma function. Truncating the series after a modest number of terms yields highly accurate approximations. The variance can be obtained analogously using the second moment.
Raw Moments via Series Expansion
The r-th raw moment is given by
E[X^r] = Σ_{k=0}^{∞} k^r · f(k; λ, p)
Numerical summation or specialized series expansions are employed to evaluate this expression. For small values of r, the series converges rapidly, enabling practical computation of skewness and kurtosis.
Moment‑Generating Function (MGF)
The MGF for the discrete Weibull has no simple closed form, but can be expressed as an infinite series:
M(t) = Σ_{k=0}^{∞} e^{t k} · (exp(–((k – 1)/λ)ᵖ) – exp(–(k/λ)ᵖ))
When t , the series converges quickly. For t > 0, truncation must be applied with care to ensure numerical stability. The MGF is useful for deriving moments by differentiation with respect to t at zero.
Characteristic Function
The characteristic function, ϕ(t) = E[e^{itX}], follows a similar series representation:
ϕ(t) = Σ_{k=0}^{∞} e^{i t k} · f(k; λ, p)
Its use is mainly theoretical, providing insights into the distribution’s asymptotic behaviour and facilitating proofs of convergence in the central limit theorem for sums of independent discrete Weibull variables.
Relationship to the Continuous Weibull
Discretisation Procedure
The discrete Weibull can be viewed as a natural discretisation of the continuous Weibull by evaluating the continuous CDF at integer points. This procedure preserves the hazard shape and allows for the transfer of intuition from the continuous to the discrete setting. The resulting discrete CDF is
F_d(k) = F_c(k) = 1 – exp(–(k/λ)ᵖ)
where F_c denotes the continuous Weibull CDF.
Limiting Behaviour
As the scale parameter λ grows large, the discrete Weibull approaches a normal distribution with mean λ · Γ(1 + 1/p) and variance λ² · (Γ(1 + 2/p) – Γ(1 + 1/p)²). This result follows from a second‑order Taylor expansion of the exponential terms and demonstrates the distribution’s compatibility with the central limit theorem.
Special Parameter Values
For p = 1, the discrete Weibull reduces to the geometric distribution with success probability q = exp(–1/λ). Consequently, many properties of the geometric distribution - such as memorylessness and simple moments - hold in this special case. When p = 2, the discrete Weibull resembles a discrete Rayleigh distribution in shape.
Related Distributions
Geometric Distribution
As noted, the discrete Weibull coincides with the geometric distribution when the shape parameter equals one. This connection provides a convenient baseline for comparing more general discrete Weibull models to the memoryless case.
Negative Binomial Distribution
In the limit as the scale parameter approaches zero while keeping the shape parameter fixed, the discrete Weibull can be approximated by a negative binomial distribution. This relationship arises from the expansion of the exponential terms in the PMF and can be used to derive approximate inference procedures in over‑dispersed data contexts.
Discrete Log‑Normal Distribution
Under certain parameter transformations, the discrete Weibull can approximate a discretised log‑normal distribution. Specifically, for large λ and p close to one, the log‑scale of the discrete Weibull resembles that of a log‑normal, enabling comparative studies between the two families.
Hurdle and Zero‑Inflated Models
To accommodate excess zeros, two extensions of the discrete Weibull have been proposed:
- Zero‑inflated discrete Weibull adds a point mass at zero with probability
πwhile preserving the discrete Weibull form for positive counts. - Hurdle discrete Weibull separates the generation of zeros from positive counts, allowing for different parameterisations.
These extensions are particularly useful in ecological and health‑economics data where zeros arise from distinct mechanisms.
Applications
Reliability Engineering
Discrete lifetime data, such as the number of operating cycles until failure, can be modelled with the discrete Weibull. The flexible hazard function allows practitioners to capture both early‑life failures and wear‑out failures within a single framework.
Queueing Theory
In discrete‑time queueing models, the number of customers served in a given time slot often follows a discrete Weibull distribution, especially when service rates vary over time. The hazard structure provides a convenient way to incorporate time‑dependent service behaviours.
Biomedical Studies
Clinical trials that record the number of treatment cycles until remission or relapse can be analysed with discrete Weibull regression models. The shape parameter can capture whether the risk of relapse increases or decreases over successive treatment cycles.
Environmental Statistics
Count data such as the number of rainfall events per month or the number of flood occurrences per year can be modelled with discrete Weibull. The distribution’s ability to model over‑dispersion and heavy tails makes it suitable for rare‑event analysis.
Economic and Insurance Modelling
In actuarial science, the number of claims filed within a policy period is sometimes modelled with the discrete Weibull to reflect varying claim probabilities across policyholders. The hazard function can reflect increasing or decreasing claim risk due to changing risk exposure.
Social Sciences
Survey data that capture the number of repeated behaviours (e.g., alcohol consumption episodes) over time can be modelled with discrete Weibull. The regression framework permits the examination of covariate effects on both the count and the risk trajectory.
Estimation and Inference
Maximum Likelihood Estimation (MLE)
The log‑likelihood function for a sample {x₁, x₂, …, x_n} is
ℓ(λ, p) = Σ_{i=1}^{n} [–( (x_i/λ)ᵖ ) + ( ( (x_i – 1)/λ )ᵖ ) + log( x_i! ) – log( ((x_i/λ)^p – ((x_i – 1)/λ)^p) )]
Numerical optimisation routines - such as Newton‑Raphson or quasi‑Newton methods - are employed to solve for the MLEs. The log‑likelihood’s gradient and Hessian can be derived analytically, improving convergence rates.
Bayesian Inference
Bayesian estimation for the discrete Weibull uses conjugate‑like priors for the scale and shape parameters. A common choice is a Gamma prior for λ and a truncated normal prior for p. Markov Chain Monte Carlo (MCMC) methods, specifically Gibbs sampling with Metropolis‑Hastings updates, are used to sample from the posterior distribution.
Regression Modelling
In a discrete Weibull regression, the parameters λ and p are modelled as functions of covariates. Common link functions include:
- Log‑link for λ ensures positivity of the scale parameter.
- Log‑log link for p ensures positivity of the shape parameter.
The likelihood contributions for each observation are then multiplied, and standard generalized linear model software can be extended to incorporate the discrete Weibull link functions.
Model Selection and Goodness‑of‑Fit
Goodness‑of‑fit tests for discrete Weibull models rely on likelihood‑ratio tests comparing nested models, such as the discrete Weibull against the geometric. Information criteria - Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) - are used to select between alternative parameterisations and model extensions.
Estimation Techniques
Maximum Likelihood Estimation (MLE)
The MLE of the discrete Weibull parameters is obtained by solving the score equations derived from the log‑likelihood. Numerical solutions are typically found using the Newton‑Raphson algorithm, which requires the evaluation of the first and second derivatives of the log‑likelihood with respect to λ and p. The derivatives are straightforwardly computed because the PMF involves exponential functions.
Method of Moments
When an analytical MLE is difficult to compute - such as in small‑sample situations - method‑of‑moments estimators can be employed. These estimators match the sample mean and variance to their theoretical counterparts approximated by series expansions, yielding initial parameter values for subsequent MLE refinement.
Bayesian Estimation
Bayesian inference uses a posterior density obtained by combining a prior with the likelihood. For the discrete Weibull, a common prior for the scale parameter is λ ~ Gamma(a, b), and for the shape parameter p ~ TruncatedNormal(μ, σ², 0, ∞). MCMC sampling is performed via a Metropolis‑Hastings algorithm that proposes new values for λ and p from suitable proposal distributions (often log‑normal for positivity). Acceptance probabilities are calculated using the ratio of posterior densities.
Robust Estimation
Because the discrete Weibull can generate heavy tails, robust estimation techniques such as M‑estimation or trimmed likelihood approaches are sometimes applied. These methods reduce the influence of extreme observations on parameter estimates, improving model stability in contaminated data sets.
Model Diagnostics
Diagnostic plots - such as the empirical probability plot and the quantile‑quantile (Q‑Q) plot against the theoretical discrete Weibull - are used to assess fit. Residual analysis based on the discrete Weibull regression model can identify systematic departures from the assumed hazard structure.
Software and Implementation
R Packages
Several R packages provide functions for the discrete Weibull and its extensions:
dweibullimplements the PMF, CDF, and random sampling for the discrete Weibull.dwglmoffers regression modelling capabilities, including zero‑inflated and hurdle extensions.- Additional functions for estimation - such as
mleDW- utilise numerical optimisation to compute MLEs.
Python Libraries
Python implementations are available through the scipy.stats module, which includes discrete_weibull with functions for PDF, CDF, and random generation. Custom modules for zero‑inflated discrete Weibull are available on GitHub, allowing for direct integration into data‑analysis pipelines.
MATLAB Toolboxes
The Statistics and Machine Learning Toolbox in MATLAB provides the dwib function, which supports random number generation, PDF, and CDF calculations. MATLAB also offers optimization routines such as fminsearch for MLE of discrete Weibull parameters.
Stata Commands
Stata users can employ the dwprob command to compute discrete Weibull probabilities and the dwreg command for regression analysis. The dwtest command provides goodness‑of‑fit tests for the discrete Weibull in specified models.
Java and C++ Libraries
Statistical libraries in Java (e.g., Apache Commons Math) and C++ (e.g., Boost.Math) include functions for the discrete Weibull distribution. These libraries are valuable for high‑performance simulation studies and for integration into industrial‑scale applications.
See Also
To broaden the understanding of the discrete Weibull distribution, consider exploring the following topics:
- Discrete probability distributions
- Generalised linear models
- Survival analysis
- Zero‑inflated and hurdle models
- Statistical inference for count data
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