Numerical Optimization

Use cases

Maximum-likelihood estimation

Weibull distribution

     shape       scale  
   1.417318   95.619997 
 ( 0.109165) ( 7.117342)

\(\chi^{2}\) 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] 0.2438403
[1] 0.8985005

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] 0.3431997 1.1345665

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

[1] 0.3541194 1.1531078

Further resources

Useful packages

Get the article source from GitHub

R markdown - markdown - R code - all posts