Cond.KL.Weib.Gamma: Kullback-Leibler divergence between the (parametrized with...
In RSizeBiased: Hypothesis Testing Based on R-Size Biased Samples

Description Usage Arguments Details Value Author(s) References Examples

View source: R/Cond.KL.Weib.Gamma.R

The function returns the Kullback-Leibler divergence (minus a constant) between the (parametrized with respect to shape and mean or variance) underlying Weibull or gamma distribution and its (assumed) maximum likelihood estimates.

1	Cond.KL.Weib.Gamma(par,nullvalue,hata,hatb,type,dist)

`par`	The (actual) shape parameter α of the distribution.
`nullvalue`	The (actual) distribution mean or variance.
`hata`	Maximum likelihood estimate of the shape parameter of the distribution.
`hatb`	Maximum likelihood estimate of the scale parameter of the distribution.
`type`	Numeric switch, enables the choice of mean or variance: type: 1 for mean, 2 (or any other value != 1) for variance.
`dist`	Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

The Kullback-Leibler divergence between the Weibull(α, β) or the gamma(α, β) and its maximum likelihood estimate Gamma(\hat α, \hat β) is given by

D_{KL} = (\hat α -1)Ψ(\hat α) - \log\hat β - \hat α - \log Γ(\hat α) + \logΓ( α) + α \log β - (α -1)(Ψ(\hat α) + \log \hat β) + \frac{ \hat β \hat α}{λ}.

Since D_{KL} is used to determine the closest distribution - given its mean or variance - to the estimated gamma p.d.f., the first four terms are omitted from the function outcome, i.e. the function returns the result of the following quantity:

\logΓ( α) + α \log β - (α -1)(Ψ(\hat α) + \log \hat β) + \frac{ \hat β \hat α}{λ}.

For the Weibull distribution the corresponding formulas are

D_{KL} = \log \frac{\hat α}{{\hat β}^{\hat α}} - \log \frac{α}{{β}^{α}} + (\hat α - α) ≤ft ( \log \hat β - \frac{γ}{\hat α} \right ) + ≤ft (\frac{\hat β}{β} \right )^α Γ≤ft ( \frac{α}{\hat α} +1 \right ) -1

and since D_{KL} is used to determine the closest distribution - given its mean or variance - to the estimated gamma p.d.f., the first term is omitted from the function outcome, i.e. the function returns the result of the following quantity:

- \log \frac{α}{{β}^{α}} + (\hat α - α) ≤ft ( \log \hat β - \frac{γ}{\hat α} \right ) + ≤ft (\frac{\hat β}{β} \right )^α Γ≤ft ( \frac{α}{\hat α} +1 \right ) -1

A scalar, the value of the Kullback-Leibler divergence (minus a constant).

Polychronis Economou

R implementation and documentation: Polychronis Economou <peconom@upatras.gr>

Economou et. al. (2021). Hypothesis testing for the population mean and variance based on r-size biased samples, under review.

#K-L divergence for the Gamma distribution for shape=2
#and variance=3 and their assumed MLE=(1,1):
 Cond.KL.Weib.Gamma(2,3,1,1,2, "gamma")
#K-L divergence for the Weibull distribution for shape=2
#and variance=3 and their assumed MLE=(1,1):
 Cond.KL.Weib.Gamma(2,3,1,1,2, "weib")