# Can I fit a Weibull distribution on both positive and negative values (observations)?

I have dataset with observations having both positive and negative values. I would like to know if I can check if my dataset follows a Weibull distribution.

• Obviously you do not mean the usual Weibull distribution, which applies only to positive values by definition. What generalization do you have in mind?
– whuber
May 9, 2016 at 21:22

Classic two-parameter Weibull pdf is

$$f(x;\lambda,k) = \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0 ,\\ 0 & x<0. \end{cases}$$

so it's support is non-negative. The story behind it is that $x$ is time-to-failure and $k$ is change of failure rate over time, so $x$ cannot be negative.

However it can be parametrized also by an additional location parameter $\mu$, then it's pdf becomes

$$f(x;\lambda,k,\mu) = \begin{cases} \frac{k}{\lambda}\left(\frac{x-\mu}{\lambda}\right)^{k-1}e^{-(\frac{x-\mu}{\lambda})^{k}} & x\geq \mu ,\\ 0 & x < \mu. \end{cases}$$

In such case negative values (for $\mu<0$) are possible, but you need to define the lower limit of the distribution. Because of that it is not the best choice for values that can be negative.

• It is a bit tricky, I have a dataset with continuous values that are both positive and negative but they do not however follow a normal distribution for according to the qq plot. What are the options I could have for my case study? as i know that it can not be lognormal or exponential or gamma as they are only for positive observations. May 9, 2016 at 21:35
• @user3841581 You seem to be asking a new question (and should probably post it as one). You'll need to give more information for it to be answerable, though -- things like: What are you measuring? What does your qq-plot look like? What made you ask about the Weibull? Why do you need to identify a distribution at all -- what are you trying to use it for? May 10, 2016 at 1:15

The Weibull distribution is only defined for x ≥ 0. So you would have so transform your data before trying to fit a Weibull distribution. But it depends on your data if that is feasible. It might be better to look at other distributions.