Parametric B-Measures in Statistical Modeling

• Parametric B-measures are a class of frameworks that quantify dependence, divergence, and complexity in multivariate and parametric settings.
• They generalize classical nonparametric constructs using parametric families to achieve better estimation accuracy, robustness, and computational efficiency.
• Applications include extreme-value analysis, multivariate spectral estimation, robust statistical inference, and Diophantine approximation with explicit parameterizations.

Parametric B-Measures are a diverse class of quantitative frameworks for modeling dependence, divergence, and complexity in multivariate and parametric statistical settings. The term "B-measure" appears in extremal dependence theory (via spectral or Bernstein–Szegő measures), robust parametric estimation (notably B-exponential divergences), information-theoretic divergence (bounded Bhattacharyya-type measures), and Diophantine quantification (effective irrationality exponents). Each instance deploys parametric families to generalize or regularize classical nonparametric constructs, with concrete implications for representation, estimation, robustness, and signal–noise separation.

1. Parametric Spectral Measures for Extremal Dependence

In the context of multivariate extreme-value theory, a central object is the spectral (or "B-") measure $\Psi$ on the unit simplex $\Delta_d$, which encodes the extremal dependence structure via the tail-dependence function

$$\ell(x_1,\dots,x_d) = \int_{\Delta_d} \max_{1\leq i\leq d}\{w_i x_i\} d\Psi(w)$$

with the standardization $\int w_i d\Psi(w) = 1$ for all $i$ (Beran et al., 2013). The associated extreme-value copula is

$$C(u) = \exp\left[-\ell(-\log u_1, \dots, -\log u_d)\right].$$

Parametric families are constructed as convex hulls of $p$ basis spectral measures: $$\Psi(\cdot;\theta) = \sum_{i=1}^{p-1}\theta_i \Psi_i + \left(1-\sum_{i=1}^{p-1}\theta_i\right)\Psi_p, \quad \theta\in\Theta.$$ Identifiability is ensured if the corresponding Pickands dependence functions $A_i$ are linearly independent. Estimation proceeds via least-squares projection of a nonparametric estimator onto the parametric family: $\widehat{\theta} = S^{-1} r$ where $S$ and $r$ involve integrals of difference basis functions $h_j(w) = A_j(w) - A_p(w)$ against a preliminary estimate.

This parametric projection yields explicit formulas, dimension reduction, and gains in finite-sample accuracy, as demonstrated in simulation studies of bivariate extremes (Beran et al., 2013).

2. Parametric Bernstein–Szegő Measures on the Bi-Circle

Parametric B-measures in the form of Bernstein–Szegő measures support the theory of matrix orthogonal polynomials on the bi-circle $\mathbb{T}^2$, underlying signal processing and multivariate spectral estimation (Geronimo et al., 2011). Measures of the form

$$d\mu(\theta,\phi) = \frac{1}{(2\pi)^2} \frac{d\theta\, d\phi}{|P_{n,m}(e^{i\theta}, e^{i\phi})|^2}$$

where $P_{n,m}(z,w)$ is stable of bidegree $(n,m)$, are parameterized by a finite matrix-valued recurrence array $u_{i,j}$, subject to boundedness ($|u_{i,j}|<1$) and contractivity of associated recurrences. The construction generalizes the classical Verblunsky theorem to bidimensional settings; parameters are encoded as the corner entries/recurrence coefficients in lexicographical and reverse-lex orderings.

This parameterization supports unique reconstruction of $\mu$ from its recurrence string, and enables applications in trigonometric moment problems and filter design (Geronimo et al., 2011).

3. Parametric B-Exponential Divergences in Estimation Theory

The B-exponential divergence (BED) is a parametric member of the Bregman divergence family, defined for densities $g, f$ as

$$d_\alpha(g, f) = \frac{2}{\alpha} \int_{\mathcal X} \left[ e^{\alpha f(x)}\left(f(x) - \frac{1}{\alpha}\right) - e^{\alpha f(x)} g(x) + \frac{1}{\alpha} e^{\alpha g(x)} \right] dx$$

where $\alpha$ is a tuning parameter (Mukherjee et al., 2018). Parametric estimation is carried out via minimization of $d_\alpha(g, f_\theta)$ over $\theta$, yielding generalized minimum divergence estimators (GBEDE). The estimating equation is

$$\frac{1}{n}\sum_{i=1}^n u_\theta(X_i) f_\theta^\beta(X_i) e^{\alpha f_\theta(X_i)} - \int_{\mathcal X} u_\theta(x) f_\theta^{1+\beta}(x) e^{\alpha f_\theta(x)} dx = 0$$

and interpolates between robust and fully efficient estimators by selecting $\alpha, \beta$. The influence function is bounded for $\beta > 0$, conferring strong robustness properties absent in MLE.

Efficiency–robustness tradeoffs are empirically optimized via Warwick–Jones selection and illustrate practical superiority in contaminated data scenarios (Mukherjee et al., 2018).

4. Parametric Bounded Bhattacharyya-Type Divergences

A parametric B-measure formulation arises in bounded Bhattacharyya distance (BBD) measures: $$D_{BBD}^{(\alpha)}(P,Q) = \frac{1}{\alpha (1-\alpha)}[1 - Z_\alpha(P,Q)], \qquad Z_\alpha(P,Q) = \int p(x)^\alpha q(x)^{1-\alpha} dx$$ with $0 < \alpha < 1$ (Jolad et al., 2012). This divergence is bounded, symmetric, and positive semi-definite, and remains finite for mutually singular distributions. As $\alpha \to 0$ or $1$, $D_{BBD}^{(\alpha)}$ coincides with the squared Hellinger distance. The extension to multiple distributions utilizes a weighted affinity integral.

$D_{BBD}^{(\alpha)}$ is a member of the generalized Csiszár $f$-divergence class. In parametric families $p(x|\theta)$, its curvature is proportional to the Fisher–Rao metric: $$D_{BBD}^{(\alpha)}(P_\theta, P_{\theta+\delta}) = \frac{1}{2} C(\alpha) I(\theta) \|\delta\|^2 + o(\|\delta\|^2)$$

where $C(\alpha)$ is an explicit function of $\alpha$. Inequalities connect the measure to Hellinger and Jensen–Shannon divergences, offering interpretable bounds. Bayes error probability can be tightly bounded using $D_{BBD}^{(\alpha)}$, and its application to signal detection (e.g., for monochromatic signals in Gaussian noise) yields analytic detection exponents in terms of SNR (Jolad et al., 2012).

5. Parametric Irregularity Measures in Diophantine Approximation

The concept of effective irrationality "B-measures" arises in bounding approximation exponents for real and $p$-adic roots of rationals near $1$ (Bugeaud, 2016). For $\alpha = (a/b)^{1/n}$ (with $a \approx b$), the effective measure $H$ satisfies

$| \alpha - p/q | > C q^{-H}$

with explicit parametric bounds of the form $$H \leq \max\{820, 10^7 ( \log 2n )/( n \eta \log a ) \}$$ for the real case, where $\eta$ measures the proximity of $a/b$ to $1$. Analogous constructions hold for $p$-adic approximations and involve a related parameterization in terms of local valuation.

These bounds are derived using Baker's method and lower bounds for linear forms in logarithms, and support applications such as parametrically solving Thue–Mahler equations. The presence of $\eta$ as a tuning parameter highlights how proximity to $1$ improves irrationality exponents via the parametric measure (Bugeaud, 2016).

6. Comparative Structure and Connections

B-Measure Family	Domain	Core Parameterization
Spectral (Extreme-Value)	Multivariate extremes, copulas	Convex hull of basis measures
Bernstein–Szegő	Multivariate trigonometric polynomials	Matrix-valued recurrence arrays
B-Exponential Divergence (BED, GBEDE)	Parametric robust estimation	Tuning parameters $(\alpha, \beta)$
Bounded Bhattacharyya (BBD)	Information, hypothesis testing	Symmetry parameter $\alpha$
Effective Irrationality (Diophantine)	Number-theoretic approximation	Proximity parameter $\eta$

All these frameworks employ parametric constructions to regulate model complexity, facilitate inference, and control robustness or sensitivity. The parametric forms allow explicit representation, consistent inference with tractable asymptotics, and controlled trade-offs between bias and efficiency across statistical and computational domains.

7. Applications and Significance

Parametric B-measures figure prominently in multivariate dependence modeling, robust inference, information theory, and computational number theory. Their boundedness properties are critical in regularizing estimation under contaminated or sparse data, their connections to information geometry underpin model selection and efficiency trade-offs, and their parameterizations enable practical computation and theoretical analysis of extremal behavior. In detection theory, they directly encode optimal tradeoffs between error probabilities and separation under signal–noise models. In Diophantine contexts, explicit parametric exponents render previously elusive bounds constructive and applicable to parametric families of equations.

These measures exemplify how parametric generalization confers flexibility, interpretability, and performance guarantees in both classical and modern applications.