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Student-t Distribution and Heavy Tails

Install and import#

bash
npm install fintech-algorithms
ts
import { studentTDistributionAndHeavyTails } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/student-t-distribution-and-heavy-tails";

Signature#

studentTDistributionAndHeavyTails(input)

Evaluates the Student t density at input.x for df degrees of freedom and reports the distribution's variance, which only exists once df passes two.

Parameters#

NameTypeNotes
inputD00InputOne record carrying x (the point to evaluate) and df (degrees of freedom), plus p and sigma, which the family entry point validates before dispatching.
df: strictly positive · sigma: strictly positive · p: between 0 and 1 inclusive

Returns#

D00Output

An object with pdf (the density at x, using a Lanczos gamma approximation for the normalising constant), degreesOfFreedom (echoes df) and variance, which is df / (df - 2) when df exceeds two and null otherwise.

Errors#

  • When p is missing or outside 0 to 1 — this check runs for every topic in the family — throws RangeError
  • When sigma is zero or negative — this check runs for every topic from A05 onward — throws RangeError
  • When df is zero or negative — throws RangeError

Complexity: time O(1), space O(1).

Worked example#

verified This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

Input#

input
{
  "values": [0.2, 0.5, 0.7, 1, 1.4],
  "x": 1,
  "p": 0.3,
  "n": 5,
  "k": 2,
  "lambda": 2,
  "seed": 42,
  "sampleCount": 8,
  "mu": 0,
  "sigma": 1,
  "df": 5,
  "shape": 2,
  "scale": 1.5,
  "components": [
    {
      "weight": 0.7,
      "mean": 0,
      "sd": 1
    },
    {
      "weight": 0.3,
      "mean": 3,
      "sd": 0.8
    }
  ]
}

Call#

studentTDistributionAndHeavyTails(input)

Returns#

object with 2 fields: pdf, degreesOfFreedom

{
  "pdf": 0.21967979735098053,
  "degreesOfFreedom": 5
}

Diagrams#

Student-t Distribution and Heavy Tails — article hero
Student-t Distribution and Heavy Tails — calculation ledger
Student-t Distribution and Heavy Tails — concept anatomy
Student-t Distribution and Heavy Tails — failure boundary
Student-t Distribution and Heavy Tails — method map
Student-t Distribution and Heavy Tails — scenario contrast

How it works#

This page states the contract — how to call it correctly. The article explains the concept: why it works, and where it breaks.

Read the article →

References#

The rest of the Probability Distributions and Simulation Basics family#