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.
Core Idea¶
The q-Weibull distribution is a q-generalization of Weibull whose density is proportional to (x/λ){κ−1}e_q(−(x/λ)κ) on its parameter-dependent support. The q-exponential changes ordinary exponential decay into a deformed law: q→1 recovers Weibull, while other q values alter tail or cutoff behavior. Normalization and alternate q′ parameterizations must be transformed consistently. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Q-Weibull distribution belongs to probability and statistics and is useful where the analyst can specify a nonnegative random variable, q-exponential convention, shape κ, scale λ, normalization restrictions, and density or survival function, then evaluate the q-exponential definition, support, shape and scale domains, normalization, and q=1 limit are stated consistently. The scope is broad within that domain but bounded by the need for the q-exponential definition, support, shape and scale domains, normalization, and q=1 limit are stated consistently. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the q-exponential definition, support, shape and scale domains, normalization, and q=1 limit are stated consistently the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Q-Weibull distribution can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Q-Weibull distribution. Q-Weibull distribution compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: a nonnegative random variable, q-exponential convention, shape κ, scale λ, normalization restrictions, and density or survival function. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the q-exponential definition, support, shape and scale domains, normalization, and q=1 limit are stated consistently independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of probability and statistics because they reuse a nonnegative random variable, q-exponential convention, shape κ, scale λ, normalization restrictions, and density or survival function, The q-exponential changes ordinary exponential decay into a deformed law: q→1 recovers Weibull, while other q values alter tail or cutoff behavior. Normalization and alternate q′ parameterizations must be transformed consistently., and type the carrier, state every parameter and convention in the definition, test that the q-exponential definition, support, shape and scale domains, normalization, and q=1 limit are stated consistently, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Q-Weibull distribution Domain-specific
Parents (1) — more general patterns this builds on
-
Q-Weibull distribution is a kind of Function (Mapping) Prime
The proposed strict upward parent is
prime:function_mapping.
Hierarchy path (1) — routes to 1 parentless root
- Q-Weibull distribution → Function (Mapping)
Neighborhood in Abstraction Space¶
Q-Weibull distribution sits in a sparse region of the domain-specific corpus (60th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Probability Distributions & Quantiles (12 abstractions)
Nearest neighbors
- Exponentiated Weibull distribution — 0.89
- Normal-exponential-gamma distribution — 0.88
- Davis distribution — 0.86
- Generalized inverse Gaussian distribution — 0.86
- Mean square quantization error — 0.86
Computed from structural-signature embeddings · 2026-09-08