# Maximum Likelihood Estimation of Pearson Distributions

### Description

This function tries to find the Maximum Likelihood estimator within the
Pearson distribution system. ML estimation is done for all sub-classes of
the distribution system via numerical optimization
(with `nlminb`

). The sub-class with the
optimal likelihood function value and the corresponding parameters are
returned.

### Usage

1 | ```
pearsonFitML(x, ...)
``` |

### Arguments

`x` |
empirical data (numerical vector) for MLE. |

`...` |
parameters for |

### Details

Starting values for each sub-class are found in a three-step procedure. First, the empirical moments of the input vector are calculated. In the second step, the moments are altered, such that the moment restrictions for the current sub-class are fulfilled (if necessary), and the method of moments estimator is calculated to obtain starting values for the optimizer. In the last step, the starting values are adjusted (if necessary) in order to assure that the whole sample lies in the support of the distribution.

### Value

List of parameters for Pearson distribution. First entry gives type of distribution (0 for type 0, 1 for type I, ..., 7 for type VII), remaining entries give distribution parameters (depending on distribution type).

### Note

The implementation is VERY preliminary (and slow). No analytical results are used, ie. no analytical solutions for ML estimators and no analytical gradients. Most of the distribution types (0, II, III, V, VII) should rather be neglected (for speed reasons), because they will contain the MLE with probability of 0.

### Author(s)

Martin Becker martin.becker@mx.uni-saarland.de

### References

[1] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 1, Wiley Series in Probability and Mathematical Statistics, Wiley

[2] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2, Wiley Series in Probability and Mathematical Statistics, Wiley

### See Also

`PearsonDS-package`

,
`Pearson`

,
`pearsonFitM`

,
`pearsonMSC`

,
`pearsonMoments`

### Examples

1 2 3 4 5 6 7 | ```
## Generate sample
DATA <- rpearson(1000,moments=c(mean=1,variance=2,skewness=1,kurtosis=5))
## find Pearson distribution with these parameters
ppar <- pearsonFitML(DATA)
print(unlist(ppar))
## compare with method of moments estimator
print(unlist(pearsonFitM(moments=empMoments(DATA))))
``` |