# Numerical Optimization

## Use cases

### Maximum-likelihood estimation

Weibull distribution

shape        scale
1.4418009   90.8713787
( 0.1142759) ( 6.5847708)

### $$\chi^{2}$$ test of normality

The $$\chi^{2}$$ normality test that compares observed and expected category frequencies after partitioning the value range into discrete bins (categories) does not work correctly when the sample mean and standard deviation are taken as estimates $$\hat{\mu}$$ and $$\hat{\sigma}$$. The test statistic then does not actually have a $$\chi^{2}$$ distribution.

[1] -0.09821994
[1] 0.9775147

Instead, it is necessary to estimate $$\hat{\mu}$$ and $$\hat{\sigma}$$ taking into account the chosen categories. One possibility to do this is by finding the minimum $$\chi^{2}$$ estimates, i.e., those estimates that minimize the observed $$X^{2}$$ statistic.

Optimization = minimization of objective function

[1] -0.05177565  0.99517343

The other possibility is a grouped maximum likelihood estimate - where the likelihood comes from the multinomial distribution.

[1] -0.05208171  0.99805581