Chisq: Chi-Square Distribution

Description

The Chi-Square distribution is a continuous probability distribution commonly used in statistical inference, particularly in hypothesis testing and confidence interval estimation. It is defined by the degrees of freedom parameter $k > 0$.

Usage

Chisq(df = 1)
# S4 method for Chisq,numeric
d(distr, x, log = FALSE)
# S4 method for Chisq,numeric
p(distr, q, lower.tail = TRUE, log.p = FALSE)
# S4 method for Chisq,numeric
qn(distr, p, lower.tail = TRUE, log.p = FALSE)
# S4 method for Chisq,numeric
r(distr, n)
# S4 method for Chisq
mean(x)
# S4 method for Chisq
median(x)
# S4 method for Chisq
mode(x)
# S4 method for Chisq
var(x)
# S4 method for Chisq
sd(x)
# S4 method for Chisq
skew(x)
# S4 method for Chisq
kurt(x)
# S4 method for Chisq
entro(x)
# S4 method for Chisq
finf(x)
llchisq(x, df)
# S4 method for Chisq,numeric
ll(distr, x)
echisq(x, type = "mle", ...)
# S4 method for Chisq,numeric
mle(distr, x, na.rm = FALSE)
# S4 method for Chisq,numeric
me(distr, x, na.rm = FALSE)
vchisq(df, type = "mle")
# S4 method for Chisq
avar_mle(distr)
# S4 method for Chisq
avar_me(distr)

Value

Each type of function returns a different type of object:

Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.
Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.
Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Arguments

df: numeric. The distribution degrees of freedom parameter.
distr: an object of class Chisq.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Chisq. For the log-likelihood and the estimation functions, x is the sample of observations.
log, log.p: logical. Should the logarithm of the probability be returned?
q: numeric. Vector of quantiles.
lower.tail: logical. If TRUE (default), probabilities are $P(X \leq x)$, otherwise $P(X > x)$.
p: numeric. Vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
type: character, case ignored. The estimator type (mle or me).
...: extra arguments.
na.rm: logical. Should the NA values be removed?

Details

The probability density function (PDF) of the Chi-Square distribution is given by: $$ f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \quad x > 0.$$

Examples

Run this code

# -----------------------------------------------------
# Chi-Square Distribution Example
# -----------------------------------------------------

# Create the distribution
df <- 4
D <- Chisq(df)

# ------------------
# dpqr Functions
# ------------------

d(D, c(0.3, 2, 20)) # density function
p(D, c(0.3, 2, 20)) # distribution function
qn(D, c(0.4, 0.8)) # inverse distribution function
x <- r(D, 100) # random generator function

# alternative way to use the function
den <- d(D) ; den(x) # den is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llchisq(x, df)

echisq(x, type = "mle")
echisq(x, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("chisq", x) # the distr argument can be a character

# ------------------
# Estimator Variance
# ------------------

vchisq(df, type = "mle")
vchisq(df, type = "me")

avar_mle(D)
avar_me(D)

v(D, type = "mle")

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