joker (version 0.14.2)

estimation: Parameter Estimation

Description

This set of functions estimates the parameters of a random sample according to a specified family of distributions. See details.

Usage

e(distr, x, type = "mle", ...)
mle(distr, x, ...)
# S4 method for character,ANY
mle(distr, x, ...)
me(distr, x, ...)
# S4 method for character,ANY
me(distr, x, ...)
same(distr, x, ...)
# S4 method for character,ANY
same(distr, x, ...)

Value

list. The estimator 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.

Arguments

distr

A Distribution object or a character. The distribution family assumed.

x

numeric. A sample under estimation.

type

character, case ignored. The estimator type.

...

extra arguments.

Functions

• mle(): Maximum Likelihood Estimator

• me(): Moment Estimator

• same(): Score - Adjusted Moment Estimation

Details

The package covers three major estimation methods: maximum likelihood estimation (MLE), moment estimation (ME), and score-adjusted estimation (SAME).

In order to perform parameter estimation, a new e<name>() member is added to the d(), p(), q(), r() family, following the standard stats name convention. These functions take two arguments, the observations x (an atomic vector for univariate or a matrix for multivariate distributions) and the type of estimation method to use (a character with possible values "mle", "me", and "same".)

Point estimation functions are available in two versions, the distribution specific one, e.g. ebeta(), and the S4 generic ones, namely mle(), me(), and same(). A general function called e() is also implemented, covering all distributions and estimators.

References

General Textbooks

• Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Beta and gamma distribution families

• Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

• Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

• Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

• Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

mle, me, same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- 3
b <- 5
D <- Beta(a, b)

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

d(D, c(0.3, 0.8, 0.5)) # density function
p(D, c(0.3, 0.8, 0.5)) # 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
df <- d(D) ; df(x) # df 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)
llbeta(x, a, b)

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

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

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

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

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

v(D, type = "mle")