Generate a data set according to a Weibull distribution with a specified scale and shape parameter.

Backsolve the scale and shape parameters for this data set by maximizing the likelihood function using the Global Optimization Toolbox.

Compare the resulting probability distribution against a histogram of the data set.

$\mathrm{restart}\:$

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$$$\begin{gathered} \mathrm{with}\left(\mathrm{GlobalOptimization}\right)\: \\ \mathrm{with}\left(\mathrm{plots}\right)\: \end{gathered}$$

$n≔10000\:$

$$\begin{gathered} \mathrm{\α}\mathit{≔}73\mathit{\:} \\ \mathrm{\β}\mathit{≔}17\mathit{\:} \end{gathered}$$

$\mathrm{Data}\mathit{:=}\mathrm{Sample}\left(\mathrm{RandomVariable}\left(\mathrm{Weibull}\left(\alpha \mathit{\,}\beta \right)\right)\mathit{\,}n\right)\mathit{\:}$

$P\mathit{≔}\left(t\mathit{\,}\mathrm{\α}\mathit{\,}\mathit{ }\mathrm{\β}\right)\mathit{\to }\frac{\beta t^{\beta \mathit{-}\mathit{1}}{\mathit{\ⅇ}}^{\mathit{-}{\left(\frac{t}{\alpha }\right)}^{\beta }}}{{\mathrm{\α}}^{\beta }}\mathit{\:}$

$\mathrm{maxLike}\mathit{:=}\left(\mathrm{\α}\mathit{\,}\mathrm{\β}\right)\mathit{\→}\mathrm{add}\left(\ln \left(P\left({\mathrm{Data}}_i\mathit{\,}\mathrm{\α}\mathit{\,}\mathrm{\β}\right)\right)\mathit{\,}i\mathit{\=}\mathit{1}\mathit{..}n\right)\mathit{\:}$

$\mathrm{results}\mathit{:=}{\mathrm{GlobalSolve}\left(\mathrm{maxLike}\left(\mathrm{alpha\_ML}\mathit{\,}\mathrm{beta\_ML}\right)\mathit{\,}\mathrm{alpha\_ML}\mathit{\=}1..100\,\mathrm{beta\_ML}\mathit{\=}1..100\mathit{\,}\mathrm{maximize}\right)}_{}\left[2\right]\mathit{\;}$

$\mathrm{results}:=\left[\mathrm{beta\_ML}\=17.1301969320734\,\mathrm{alpha\_ML}\=72.9747976582517\right]$

$$\begin{gathered} \mathrm{\α\_\_GOT}\mathit{≔}\mathrm{rhs}\left(\mathrm{select}\left(\mathrm{has}\mathit{\,}\mathrm{results}\mathit{\,}\mathrm{alpha\_ML}\right)\left[1\right]\right)\mathit{\;} \\ \mathit{ }\mathrm{\β\_\_GOT}\mathit{≔}\mathrm{rhs}\left({\mathrm{select}\left(\mathrm{has}\mathit{\,}\mathrm{results}\mathit{\,}\mathrm{beta\_ML}\right)}_{}\left[1\right]\right)\mathit{\;} \end{gathered}$$

$\mathrm{\α\_\_GOT}:=72.9747976582517$

$\mathrm{\β\_\_GOT}:=17.1301969320734$

$\mathrm{fsolve}\left(\left[\mathrm{Moment}\left(\mathrm{Data}\mathit{\,}1\right)\mathit{\=}\mathrm{Moment}\left(\mathrm{Weibull}\left(a\mathit{\,}b\right)\mathit{\,}1\right)\mathit{\,}\mathrm{Moment}\left(\mathrm{Data}\mathit{\,}2\right)\mathit{\=}\mathrm{Moment}\left(\mathrm{Weibull}\left(a\mathit{\,}b\right)\mathit{\,}2\right)\right]\right)\mathit{ }$

$\left\{a\=72.98206445\,b\=17.00542373\right\}$

$\mathrm{plotopts}:=\mathrm{axesfont}\=\left[\mathrm{Calibri}\right]\,\mathrm{labelfont}\=\left[\mathrm{Calibri}\right]\,\mathrm{labels}\=\left[''t''\,''Probability\; Density''\right]\,\mathrm{labeldirections}\=\left[\mathrm{horizontal}\,\mathrm{vertical}\right]\:$$\mathrm{p1}≔\mathrm{Histogram}\left(\mathrm{Data}\,\mathrm{color}\=''Gainsboro''\,\mathrm{plotopts}\right)\:$$\mathrm{p2}:=\mathrm{plot}\left(P\left(t\,\mathrm{\α}\,\mathrm{\β}\right)\,t\={\mathrm{sort}\left(\mathrm{Data}\right)}_1..{\mathrm{sort}\left(\mathrm{Data}\right)}_n\,\mathrm{color}\=''Black''\,\mathrm{legend}\=''Sample\; Parameters''\,\mathrm{plotopts}\right)\:$$\mathrm{p3}:=\mathrm{plot}\left(P\left(t\,\mathrm{\α\_\_GOT}\,\mathrm{\β\_\_GOT}\right)\,t\={\mathrm{sort}\left(\mathrm{Data}\right)}_1..{\mathrm{sort}\left(\mathrm{Data}\right)}_n\,\mathrm{color}\=''Red''\,\mathrm{legend}\=''Estimated\; Parameters''\,\mathrm{plotopts}\right)\:$$\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\,\mathrm{size}\=\left[800\,''golden''\right]\right)$