ChiSquare: Create a Chi-Square distribution

In distributions3: Probability Distributions as S3 Objects

Create a Chi-Square distribution

Description

Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.

Usage

ChiSquare(df)

Arguments

df	Degrees of freedom. Must be positive.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a \chi^2 random variable with df = k.

Support: R^+, the set of positive real numbers

Mean: k

Variance: 2k

Probability density function (p.d.f):

f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx

but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard normal is sometimes called the "error function". The notation \Phi(t) also stands for the c.d.f. of a standard normal evaluated at t. Z-tables list the value of \Phi(t) for various t.

Moment generating function (m.g.f):

E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}

Value

A ChiSquare object.

Transformations

A squared standard Normal() distribution is equivalent to a \chi^2_1 distribution with one degree of freedom. The \chi^2 distribution is a special case of the Gamma() distribution with shape (TODO: check this) parameter equal to a half. Sums of \chi^2 distributions are also distributed as \chi^2 distributions, where the degrees of freedom of the contributing distributions get summed. The ratio of two \chi^2 distributions is a FisherF() distribution. The ratio of a Normal() and the square root of a scaled ChiSquare() is a StudentsT() distribution.

See Also

Other continuous distributions: Beta(), Cauchy(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.