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.
Core Idea¶
The Davis family has density proportional to bn(x-μ)(-n-1) divided by exp(b/(x-μ))-1, normalized by Γ(n)ζ(n) under valid parameters. The exponential denominator suppresses density near the lower boundary while its small-argument expansion produces a power-law upper tail, allowing an interior mode and Pareto-like extremes. 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¶
Davis distribution belongs to statistical distributions and is useful where the analyst can specify a continuous random variable above location μ, positive scale b, shape n, a normalized density involving gamma and zeta functions, and income observations, then evaluate parameters satisfy the density's support and normalization conditions and moments are claimed only where the corresponding zeta and power integrals converge. The scope is broad within that domain but bounded by the need for parameters satisfy the density's support and normalization conditions and moments are claimed only where the corresponding zeta and power integrals converge. 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 parameters satisfy the density's support and normalization conditions and moments are claimed only where the corresponding zeta and power integrals converge 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 Davis 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 Davis distribution. Davis 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 continuous random variable above location μ, positive scale b, shape n, a normalized density involving gamma and zeta functions, and income observations. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express parameters satisfy the density's support and normalization conditions and moments are claimed only where the corresponding zeta and power integrals converge independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of statistical distributions because they reuse a continuous random variable above location μ, positive scale b, shape n, a normalized density involving gamma and zeta functions, and income observations, The exponential denominator suppresses density near the lower boundary while its small-argument expansion produces a power-law upper tail, allowing an interior mode and Pareto-like extremes., and type the carrier, state every parameter and convention in the definition, test that parameters satisfy the density's support and normalization conditions and moments are claimed only where the corresponding zeta and power integrals converge, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Davis distribution Domain-specific
Parents (1) — more general patterns this builds on
-
Davis distribution is a kind of Distributional Assumption Prime
The proposed strict upward parent is
prime:distributional_assumption.
Hierarchy paths (7) — routes to 5 parentless roots
- Davis distribution → Distributional Assumption → Assumption → Epistemic Mode Of A Proposition
- Davis distribution → Distributional Assumption → Statistical Inference → Inductive Reasoning
- Davis distribution → Distributional Assumption → Statistical Inference → Uncertainty
- Davis distribution → Distributional Assumption → Probability → Measure → Set and Membership
- Davis distribution → Distributional Assumption → Probability → Measure → Aggregation → Micro Macro Linkage
- Davis distribution → Distributional Assumption → Statistical Inference → Probability → Measure → Set and Membership
- Davis distribution → Distributional Assumption → Statistical Inference → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Davis distribution sits in a moderately populated region (44th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Probability Distributions & Quantiles (12 abstractions)
Nearest neighbors
- Normal-exponential-gamma distribution — 0.91
- Zeta distribution — 0.89
- Location parameter — 0.89
- Folded-t and half-t distributions — 0.89
- Asymptotic theory (statistics) — 0.89
Computed from structural-signature embeddings · 2026-09-08