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Interpreting Exponentiated Weibull Models

Now I'm going to work on interpreting Exponentiated Weibull models and I'm going to tabulate the frequency of qualitatively different distributions by wiki.

The Exponentiated Weibull has 3 parameters. Two are shape parameters ( $\alpha >0$ and $\gamma >0$ ) and one is a scale parameter ( $\lambda >0$ ). The major qualitative distinctions in interpreting the model are in terms of the shape parameters.

According to this analysis of the Exponentiated Weibull:

If $\alpha =1$ and $\gamma =1$ then we have an exponential distribution with parameter $\lambda$ .
If $\alpha =1$ we have a Weibull distribution.

In this case the failure rate is always increasing (positive ageing) if $\gamma >1$ and always decreasing (negative ageing) if $\gamma <1$ .

If $\gamma =1$ then we have a exponentiated exponential distribution and the failure rate may not be monotonic.

In this case, and if $\alpha >0$ then the failure rate increases when $0<x<\lambda$ .
On the other hand if $\alpha <0$ then the failure rate decreases when $x>\lambda$ .

If $\gamma >1$ and $\alpha >1$ then we have positive ageing (the failure rate is increasing).
If $\gamma <1$ and $\alpha <1$ then we have negative ageing (the failure rate is decreasing).
If the two shape parameters have opposite signs then interpreting the model may require closer inspection of hazard and/or survival curves.

Inconveniently, it looks like almost all of the time we have $\alpha >1$ and $\gamma <1$ .

expweib_tab = table[table.model =='exponweib'].copy().reset_index()
expweib_tab['a'] = expweib_tab.params.apply(lambda r: r[0])
expweib_tab['c'] = expweib_tab.params.apply(lambda r: r[1])
expweib_tab['scale'] = expweib_tab.params.apply(lambda r: r[3])
expweib_tab['a_ge_1'] = expweib_tab.a >= 1 
expweib_tab['c_ge_1'] = expweib_tab.c >= 1 

expweib_tab = expweib_tab.drop(['level_0','index'],1)
pd.crosstab(expweib_tab['a_ge_1'],expweib_tab['c_ge_1'])

Note that in the code a = $\alpha$ and c = $\gamma$ .

$\gamma \geq 1$	False	True
$a\geq 1$
False	0	1
True	241	0

So, inconveniently, I don't know what we can say qualitatively about reading times just from looking at the parameter estimates.

Next I'm going to plot hazard functions for a handful of wikis to see if there is anything we can say in general about these parameters from the distributions.

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Lab Notebook

Interpreting Exponentiated Weibull Models