Parametric Families of Discrete Distributions

• Parametric families of discrete distributions are defined by explicit PMFs or PGFs that capture key characteristics like dispersion, skewness, and tail behavior through transformations and discretization techniques.
• They are constructed via methods such as discretizing continuous laws, employing Markov chain models, and using graphical approaches to represent dependence and overdispersion.
• These models support robust likelihood-based inference and are applied to diverse areas including count data analysis, Bayesian estimation, and network structure modeling.

Parametric families of discrete distributions constitute a vast and foundational segment of probability theory and statistical modeling, essential for representing, analyzing, and inferring from count data, finite sequence occurrences, random allocation on integer lattices, and ranked compositional structures. Central to their utility is the presence of parameters that encode dispersion, skewness, tail behavior, dependence, and structural constraints, facilitating adaptation to diverse empirical phenomena. Contemporary advances have expanded classical families toward flexible, overdispersed, heavy-tailed, and structurally dependent regimes, enabled by stochastic process constructions, Markov chain generalizations, graphical models, and mixtures with stable, gamma, or negative-binomial subordinators.

1. Foundational Construction Techniques

A primary method for generating parametric discrete families is discretization or transformation of continuous distributions, such as taking integer parts, binomial thinning, or embedding within latent count processes. The discrete log-symmetric family exemplifies this: the continuous log-symmetric law, defined by $Y>0$ with density $$f_Y(y|\lambda,\phi;g) = \frac{1}{Z_g}\frac{1}{\sqrt{\phi}\,y}g\big(a_{\lambda,\phi}(y)^2\big)$$, is transformed via $X = \lfloor Y \rfloor$ to produce a count-valued distribution that inherits skewness and tail behavior from a kernel $g$ parameterizing the continuous parent (Saulo et al., 2020).

Similarly, the Discrete Two-Sided Power (DTSP) family arises from discretizing the continuous two-sided power law, yielding a PMF shaped by location and scale ($a, b, m$) and a power-shape parameter $n$ (Chakraborty et al., 2015). Other approaches generalize scaling and thinning to the discrete domain, such as the broadly discrete stable (DS) distributions that extend continuous stable laws through mixtures, Poisson-compounding, or Sibuya summand representations, ensuring infinitely divisible and, under additional constraints, self-decomposable discrete laws (Townes, 5 Sep 2025).

Structural and dependency-aware discrete families are often built via Markov chain models (e.g., the Baker $b$-distributions from two-state chains over binomial, Poisson, or negative-binomial parents (Baker, 2020)) or graphical Markov structures (graph multinomial and negative multinomial laws, parameterized by decomposable graphs and clique polynomials, supporting Dirichlet-type conjugate priors for Bayesian analysis (Danielewska et al., 2023)).

2. Probability Mass Functions and Generating Functions

Parametric discrete families are defined by explicit PMFs or probability-generating functions (PGFs) encapsulating their parameter maps and fundamental algebraic structure.

• Discrete log-symmetric PMF:

$$p(x|\lambda,\phi) = G(a_{\lambda,\phi}(x+1)) - G(a_{\lambda,\phi}(x)),$$

where $G$ is the CDF arising from kernel $g$, and $$a_{\lambda,\phi}(x) = \frac{1}{\sqrt{\phi}}\log(x/\lambda)$$ (Saulo et al., 2020).

• DTSP PMF:

$$p(y|a,m,b,n) = \begin{cases} \frac{(y-a+1)^n - (y-a)^n}{(b-a)(m-a)^{n-1}}, & y=a,\dots,m-1, \ \frac{(b-y)^n - (b-y-1)^n}{(b-a)(b-m)^{n-1}}, & y=m,\dots,b-1, \ 0, & \text{otherwise}. \end{cases}$$

(Chakraborty et al., 2015)

• Broadly discrete stable PGF:

$$G_Y(z) = \begin{cases} \exp\left((z-1)\delta + \gamma(1-z)^\alpha\right), & \alpha\neq 1, \ \exp\left((z-1)\delta + \gamma(1-z)\log(1-z)\right), & \alpha=1. \end{cases}$$

(Townes, 5 Sep 2025)

• Graph negative multinomial PMF:

$P(N=n) = C_G(n, r) x^n \Delta_G(x)^r,$

with $C_G$ and $\Delta_G$ determined by the graph structure (Danielewska et al., 2023).

Families typically admit recurrence or compound structure; for DS, mixed Poisson–stable and compound Poisson broad Sibuya representations are foundational (Townes, 5 Sep 2025).

3. Moments, Dispersion, and Distributional Properties

Moment calculation is core to characterizing discrete distribution families. In discrete log-symmetric models, moments are generated via reliability functions:

$E[X^r] = \sum_{x=0}^\infty [(x+1)^r - x^r] R(x),$

with $R(x)$ the survival function (Saulo et al., 2020). Variance formulas are explicit, often involving sums over reliability.

DTSP and Markov chain-derived families explicitly encode overdispersion and underdispersion:

• For $b$-distributions (Markov chain embedding), $\mathrm{Var}[X]$ exceeds or falls below the parent, depending on $r_1 + r_2$ (Baker, 2020).
• DS laws manifest power-law tails ($P(Y=n) \sim C n^{-1-\alpha}$ for $\alpha \in (0, 1)$) or exponential decay ($\alpha=2$) (Townes, 5 Sep 2025).
• Graphical multinomial models permit structural dependence between counts, producing nontrivial covariance structures determined by graph separators and cliques (Danielewska et al., 2023).

Quantile, skewness, kurtosis, and tail properties are readily inherited or parameterized by kernel or process parameters, facilitating extremely flexible adaptation to empirical phenomena.

4. Parameter Estimation and Inference

Discrete parametric families robustly support likelihood-based inference. MLEs are available via log-likelihoods constructed from explicit PMFs or PGFs. Most models require numerical optimization (BFGS, Newton-Raphson); closed-form solutions are rare except in degenerate cases (DTSP with $n=1$ uniform, $n=2$ triangular, etc.) (Chakraborty et al., 2015, Saulo et al., 2020).

Markov-chain generalized families offer recurrence-based algorithms; probabilities and moments are computed via recursions over states and goal counts, supporting likelihood maximization and standard error estimation (Baker, 2020).

Graphical multinomial and negative multinomial families admit conjugate graphical Dirichlet-type priors, with explicit updates after observing samples, enabling fully deterministic Bayesian inference with respect to the underlying graph structure (Danielewska et al., 2023).

Regularity theory assures consistency and asymptotic normality of MLEs (via Fisher information) under standard conditions for discrete log-symmetric families (Saulo et al., 2020).

5. Extensions, Special Cases, and Generalizations

Advances in parametric discrete modeling have unified and subsumed numerous classical models:

• DS law recovers Poisson at $\gamma = 0, \alpha = 1$, strictly stable/Sibuya at $\delta = 0, \alpha \in (0, 1)$, and Hermite for $\alpha = 2$ (Townes, 5 Sep 2025).
• Negative binomial construction of random discrete distributions on the infinite simplex, $\mathrm{PK}^{(r)}(\rho)$, generalizes Poisson–Kingman processes, incorporates Poisson–Dirichlet distributions, and supports trimmed stable extensions (Ipsen et al., 2018). The added $r$ parameter controls overdispersion and the size of the largest atoms.
• Discrete normal distributions on $\mathbb{Z}^d$ constitute a minimal regular exponential family with canonical parameters linked to prescribed mean and covariance matrices, PMF expressed via the Riemann theta function, and divergences (e.g., KL) realized as reverse Bregman divergences (Nielsen, 2021).
• Graphical multinomial and negative multinomial families, through the clique–separator factorization and strong hyper-Markov Dirichlet priors, generalize independent multinomial models to arbitrary decomposable graphs, bridging parametric and nonparametric Bayesian graphical models (Danielewska et al., 2023).

6. Practical Applications and Implementation

Parametric discrete families are critical for modeling count data with excess zero/inflation, heavy tails, structural constraints, or dependencies.

• Discrete log-symmetric laws provide competitive fits for skewed and heavy-tailed counts (e.g., computer breakdowns, chronic pain relief durations) (Saulo et al., 2020).
• $b$-distributions from Markov chain generalization enable direct inference and simulation for overdispersed or underdispersed counts relative to classical binomial, Poisson, and negative-binomial baselines (Baker, 2020).
• Graphical multinomial/negative multinomial models facilitate likelihood and Bayes inference in networked, structured settings (e.g., multivariate contingency tables) (Danielewska et al., 2023).
• DS families are germane to both light and heavy-tailed count phenomena, admit efficient simulation via compound Poisson sums, and enable unimodal or multimodal modeling as dictated by dispersion constraints (Townes, 5 Sep 2025).

Numerical methods (e.g., theta-series truncation for discrete normals, recurrence-based algorithms for $b$-distributions) and associated open-source software packages are available for normalization, simulation, and divergence computations (Nielsen, 2021).

7. Comparative Properties and Theoretical Significance

Parametric families achieve flexible interpolation between classical shapes: uniform, triangular, U/inverse-U, J/inverse-J, peaked, or multimodal, often governed by specialized parameters (e.g., $n$ in DTSP, $\alpha$ in DS, $r$ in $\mathrm{PK}^{(r)}(\rho)$) (Chakraborty et al., 2015, Townes, 5 Sep 2025, Ipsen et al., 2018).

Model selection is typically realized by likelihood profiling or dispersion testing, facilitating empirical adaptation. Limitations consist primarily in computational complexity of PMF evaluation (especially via normalization constants or recursions) and the necessity of numerical optimization for parameter inference (Saulo et al., 2020).

Extensions include regression structures for modeling counts conditional on covariates, multivariate generalizations (particularly via graphical models), and discrete-time series analogues for dependent event modeling (Saulo et al., 2020, Danielewska et al., 2023).

In summary, parametric discrete distribution families provide the principal technical foundation for modern count data analysis, multivariate discrete modeling, Bayesian inference in structured models, and stochastic process generalization. Their capacity to encode dispersion, shape, dependence, and tail structure underpins their continuing evolution and central role in both theoretical probability and applied statistics.