Asymptotic Variance

These functions calculate the asymptotic variance (or variance - covariance matrix in the multidimensional case) of an estimator, given a specified family of distributions and the true parameter values.

Usage

avar(distr, type, ...)

avar_mle(distr, ...)

avar_me(distr, ...)

avar_same(distr, ...)

Arguments

distr

A Distribution object.

type

character, case ignored. The estimator type ("mle", "me", or "same").

...

extra arguments.

Value

A named matrix. The asymptotic covariance matrix of the estimator.

Functions

• avar_mle(): Asymptotic Variance of the Maximum Likelihood Estimator

• avar_me(): Asymptotic Variance of the Moment Estimator

• avar_same(): Asymptotic Variance of the Score-Adjusted Moment Estimator

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

avar_mle, avar_me, avar_same

Examples

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

# Simulation
set.seed(1)
a <- 1
b <- 2
D <- Beta(a, b)
x <- r(D)(100)

# Point Estimation - The e Functions

ebeta(x, type = "mle")
#> $shape1
#> [1] 1.066968
#> 
#> $shape2
#> [1] 2.466715
#> 
ebeta(x, type = "me")
#> $shape1
#> [1] 1.074511
#> 
#> $shape2
#> [1] 2.469756
#> 
ebeta(x, type = "same")
#> $shape1
#> [1] 1.067768
#> 
#> $shape2
#> [1] 2.454257
#> 

mle(D, x)
#> $shape1
#> [1] 1.066968
#> 
#> $shape2
#> [1] 2.466715
#> 
me(D, x)
#> $shape1
#> [1] 1.074511
#> 
#> $shape2
#> [1] 2.469756
#> 
same(D, x)
#> $shape1
#> [1] 1.067768
#> 
#> $shape2
#> [1] 2.454257
#> 
e(D, x, type = "mle")
#> $shape1
#> [1] 1.066968
#> 
#> $shape2
#> [1] 2.466715
#> 

mle("beta", x) # the distr argument can be a character
#> $shape1
#> [1] 1.066968
#> 
#> $shape2
#> [1] 2.466715
#> 

# Asymptotic Variance - The v Functions

vbeta(a, b, type = "mle")
#>          shape1   shape2
#> shape1 1.597168 2.523104
#> shape2 2.523104 7.985838
vbeta(a, b, type = "me")
#>        shape1 shape2
#> shape1    2.1    3.3
#> shape2    3.3    9.3
vbeta(a, b, type = "same")
#>          shape1   shape2
#> shape1 1.644934 2.539868
#> shape2 2.539868 8.079736

avar_mle(D)
#>          shape1   shape2
#> shape1 1.597168 2.523104
#> shape2 2.523104 7.985838
avar_me(D)
#>        shape1 shape2
#> shape1    2.1    3.3
#> shape2    3.3    9.3
avar_same(D)
#>          shape1   shape2
#> shape1 1.644934 2.539868
#> shape2 2.539868 8.079736

avar(D, type = "mle")
#>          shape1   shape2
#> shape1 1.597168 2.523104
#> shape2 2.523104 7.985838