Distributions

The implemented distributions are found in univariateML_models.

library("univariateML")
univariateML_models

##  [1] "beta"       "betapr"     "binom"      "burr"       "cauchy"    
##  [6] "dunif"      "exp"        "fatigue"    "gamma"      "ged"       
## [11] "geom"       "gompertz"   "gumbel"     "invburr"    "invgamma"  
## [16] "invgauss"   "invweibull" "kumar"      "laplace"    "lgamma"    
## [21] "lgser"      "llogis"     "lnorm"      "logis"      "logitnorm" 
## [26] "lomax"      "naka"       "nbinom"     "norm"       "paralogis" 
## [31] "pareto"     "pois"       "power"      "rayleigh"   "sged"      
## [36] "snorm"      "sstd"       "std"        "unif"       "weibull"   
## [41] "zip"        "zipf"

This package follows a naming convention for the ml*** functions. To access the documentation of the distribution associated with an ml*** function, write package::d***. For instance, to find the documentation for the log-gamma distribution write

?actuar::dlgamma

Additional information about the models can found in univariateML_metadata.

univariateML_metadata[["mllgser"]]

## $model
## [1] "Logarithmic series"
## 
## $density
## [1] "extraDistr::dlgser"
## 
## $support
## Object of class Intervals
## 1 interval over Z:
## [1, Inf)
## 
## $names
## [1] "theta"
## 
## $default
## [1] 0.9

From the metadata you can read that

• mllgser estimates the parameters N and s.
• Its a discrete distribution on \(1,2,3,...\),
• Its density function is extraDistr::dlgser.

Problematic Distributions

Some estimation procedures will fail under certain circumstances. Sometimes due to numerical problems, but also because the maximum likelihood estimator fails to exist. Here is a possibly non-exhaustive list of known problematic distributions.

Discrete distributions

• Binomial. The maximum likelihood estimator does not exist for underdispersed data (when \(size\) is estimated). There is an increasing sequence of estimates \(size\), \(p\) so that the binomial likelihood converges to a Poisson, however.
• Negative binomial. The same sort of problem occurs with the negative binomial, which converges to a Poisson for some data sets.
• Lomax. Here we have convergence to an exponential for certain data sets.
• Zipf. The optimal shape parameter may be negative, which still defines a density, but is not supported by extraDistr.
• Logarithic series distribution. When all observations are \(1\) the estimator does not exist, as the “actual” maximum likelihood estimator is the point mass on \(0\).

Continuous distributions

• Gompertz. Here we have a similar problem, with some parameters outside the range of the distribution converging to a density function with a different support. When the b parameter tends towards 0, the Gompertz tends towards an exponential. A failing estimation indicates the exponential has a better fit.
• Lomax. Here we have convergence to an exponential for certain data sets.
• Burr. The Burr distribution tends to the Pareto distribution when shape1*shape2 converges to a constant while shape2 tends to infinity.