Johnson's SU-distribution¶
An unbounded four-parameter distribution obtained by applying an inverse-hyperbolic-sine transformation to a standard normal variable.
Core Idea¶
Location xi and positive scale lambda transform x, while shape parameters gamma and positive delta make gamma plus delta asinh of the standardized value normally distributed, allowing broad skewness and kurtosis. A normal variate is shifted and rescaled in transformed space, then a hyperbolic-sine inverse map produces real-valued observations with adjustable asymmetry and tail behavior. 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¶
Johnson's SU-distribution belongs to probability distributions and is useful where the analyst can specify the typed probability distributions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit. The scope is broad within that domain but bounded by the need for the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit. 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 four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit 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 Johnson's SU-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 Johnson's SU-distribution. Johnson's SU-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: the typed probability distributions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of probability distributions because they reuse the typed probability distributions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, A normal variate is shifted and rescaled in transformed space, then a hyperbolic-sine inverse map produces real-valued observations with adjustable asymmetry and tail behavior., and type the carrier, state every parameter and convention in the definition, test that the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Johnson's SU-distribution Domain-specific
Parents (1) — more general patterns this builds on
-
Johnson's SU-distribution is a kind of Probability Prime
The proposed strict upward parent is
prime:probability.
Hierarchy paths (2) — routes to 2 parentless roots
- Johnson's SU-distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Johnson's SU-distribution → Probability → Measure → Set and Membership
Neighborhood in Abstraction Space¶
Johnson's SU-distribution sits in a crowded region of the domain-specific corpus (37th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Probability Measures & Random Variables (36 abstractions)
Nearest neighbors
- Modified half-normal distribution — 0.91
- Inverse distribution — 0.90
- Hyperbolic distribution — 0.90
- Exchangeable random variables — 0.89
- Truncated normal distribution — 0.89
Computed from structural-signature embeddings · 2026-09-08