Multivariate & Heavy-Tailed Distributions

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Abstractions about specialized probability distributions and their moments, covering multivariate and matrix-valued families (complex normal, matrix t, matrix variate Dirichlet), heavy-tailed and generalized continuous distributions (Weibull variants, variance-gamma, multivariate Pareto), and distributional statistics like cokurtosis and quantization error.

16 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Cokurtosis — Measure fourth-order joint variation by taking standardized expectations of products containing four centered random-variable factors, retaining how extreme deviations co-occur beyond covariance and coskewness.
• Complex normal distribution — A distribution for complex random vectors whose stacked real and imaginary parts are jointly Gaussian, characterized by mean, covariance, and relation (pseudo-covariance) matrices, with circular proper Gaussian as a special case.
• Davis distribution — A three-parameter continuous income distribution on x>μ with a Planck-like exponential denominator and Pareto upper tail, introduced by Harold T. Davis in 1941.
• Exponentiated Weibull distribution — A positive continuous distribution formed by raising the Weibull cumulative distribution function to an additional positive shape parameter.
• Gamma Distribution — A positive continuous distribution family whose shape and scale govern skew and magnitude, with conditional waiting-time, sum, and rate-prior interpretations.
• Generalized chi-squared distribution — The probability distribution of a quadratic function of a multivariate normal vector, equivalently a weighted sum of independent noncentral chi-square variables with an optional normal term.
• Generalized inverse Gaussian distribution — A three-parameter positive continuous distribution with density proportional to x^{p−1}exp[−(ax+b/x)/2], normalized by a modified Bessel K function and containing inverse Gaussian and gamma-related limits.
• Matrix t-distribution — A heavy-tailed probability distribution for random matrices that generalizes the multivariate t distribution with separate row and column scale structure.
• Matrix variate Dirichlet distribution — A probability law on several positive-definite matrices whose sum remains below the identity, generalizing scalar Dirichlet and matrix beta distributions.
• Mean square quantization error — Evaluate a quantizer by averaging the squared difference between each input and its reconstruction value under a declared input distribution, making large reconstruction deviations contribute quadratically.
• Multivariate Pareto distribution — A family of joint heavy-tailed distributions whose margins have Pareto-type behavior and whose dependence construction models simultaneous extremes across variables.
• Multivariate t-distribution — An elliptically contoured heavy-tailed distribution for random vectors, parameterized by location, positive-definite scale matrix and degrees of freedom.
• Normal-exponential-gamma distribution — A heavy-tailed continuous location-scale-shape distribution obtained through a normal variance mixture whose variance follows an exponential-gamma hierarchy.
• Q-Weibull distribution — A positive distribution formed by replacing the Weibull exponential with a q-exponential, recovering Weibull at q=1 and allowing compact-support or heavy-tail behavior according to q.
• Variance-gamma distribution — A continuous heavy-tailed probability family obtained by evaluating Brownian motion with drift at an independent gamma-distributed random time, equivalently a normal variance-mean gamma mixture.
• Wigner semicircle distribution — A compactly supported probability distribution whose density is proportional to a semicircle and which governs limiting eigenvalue spectra of many random symmetric matrices.