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When describing the Weibull distribution on Wikipedia, it is claimed, among other things, that

"A value of $ k > 1 $ indicates that the failure rate increases with time."

However, for $ k > 3 $ the failure rate increases with time, but then decreases after a while.

For example for $ k = 4 $ the failure rate $ h(x;\lambda,4) $ is as follows:

$$ h \left(x \right) = \frac{4~x ^{3}~e^{\frac{x ^{3}~\left(\lambda -x \right)}{\lambda ^{4}}}}{\lambda ^{4}} $$

Is this correct?

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The failure rate is $h(x)=f(x)/(1-F(x))$, and when substituted it is $$h(x)=\frac{\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}}{1-(1-e^{-(x/\lambda)^k})}=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}$$ which increases as $x$ increases if $k>1$.

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  • $\begingroup$ I am very, very sorry for making an error in the derivation of $h(x)$. $\endgroup$ – Ad van der Ven Nov 2 '19 at 12:11
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    $\begingroup$ @AdvanderVen there is nothing to be sorry. This is the purpose of asking. And, if you think the answer is ok, can you please accept and/or upvote it? $\endgroup$ – gunes Nov 2 '19 at 12:44

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