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.

Version

v2 · 2026-09-19 · History

Domain-specific #

4047

Domain group

Formal Sciences

Origin domain

Experimental Design & Statistics

Subdomains

Statistical Distributions, Income Size Models → Experimental Design & Statistics

Core Idea

The Davis family has density proportional to b^n(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.

How would you explain it like I'm…

The Town Money Shape

Imagine lining up everyone in a town by how much money they earn. Very few earn almost nothing, lots earn a middle amount, and a small number earn giant amounts, many more giant ones than you might guess. The Davis distribution is a math rule some people use to draw that shape.

The Income Curve With a Long Tail

A distribution is a kind of math picture showing how common different amounts are, like how many people earn each amount of money. The Davis distribution is a shape made for describing incomes. It has very few people right near the lowest amount, a hump where most people are, and a long tail on the high side for the few who earn a huge amount. Its formula was borrowed from a famous physics formula about how hot things glow. Because the tail is so long, some averages you might want to calculate don't always work out.

Planck-Style Heavy-Tailed Income Model

The Davis distribution is a family of probability distributions used as a model for the full income distribution, including a heavy upper tail like the Pareto distribution. Its density is proportional to b^n (x − μ)^(−n−1) divided by (exp(b/(x − μ)) − 1), with a normalizing constant that involves the gamma function and the Riemann zeta function. The exponential in the denominator pushes the density toward zero just above the lower boundary μ, while for large x the formula behaves like a power law, creating the heavy tail. Together these give a peak in the interior plus Pareto-like extremes. Its form is derived from Planck's law. Parameters must satisfy conditions for the density to be valid, and moments such as the mean only exist when the relevant integrals converge.

 

The Davis distribution is a Planck-derived probability model for the full income distribution, with a heavy Pareto-like upper tail. Its density is proportional to b^n (x − μ)^(−n−1) divided by exp(b/(x − μ)) − 1, normalized by Γ(n)ζ(n), for x above the location parameter μ under valid parameter values. The structure has two parts that play complementary roles. The exponential term in the denominator becomes very large near the lower boundary, suppressing the density there, while its small-argument expansion at large x makes the density behave like a power law, producing a Pareto-type upper tail. Together these yield an interior mode and Pareto-like extremes in a single formula. Parameters must satisfy the support and normalization conditions, and moments may be claimed only where the corresponding zeta-function and power integrals converge, which fails for higher moments when the tail is heavy. It is narrower than statistical distributions in general: a heavy-tailed income model not built on this Planck-type form is not the Davis distribution.

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

1. 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) — outgoing typed relations

• Davis distribution is a kind of Distributional Assumption Prime

  It posits a particular parametric probability law for observed sizes.

Hierarchy paths (5) — routes to 4 parentless roots

• Davis distribution → is a kind of Distributional Assumption → is a kind of Assumption → is a kind of Epistemic Mode Of A Proposition

Show alternative paths (4)

Neighborhood in Abstraction Space

Davis distribution sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Multivariate & Heavy-Tailed Distributions (16 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 · 2026-10-25