Weibull with unequal observation periods

[Travis Folk]

All,
I am using the Weibull distribution to model daily movement probabiliites for radio-marked animals. Accordingly, most of my data consists of daily locations, but there are some periods where I missed a day or two. My question concerns accomodating for this in a maximum likelihood context. Could the scale parameter be divided by the interval length? Does anybody have experience with adjustment of shape or scale parameters?

[tarik]

My knowledge about the Weibull distribution concentrates more on its application in modeling life data (failure data), ie. probability of failure given a certain time. I know that Weibull analysis (and Weibull++ software) can be used in other.

One of the data types that can be analyzed in Weibull++ is interval censored data which reflects uncertainty as to the exact times the units failed within an interval. This type of data frequently comes from tests or situations where the objects of interest are not constantly monitored. This sounds somewhat similar to your case.

You could read the following material and hopefully you will find a way to model your type of data
http://www.weibull.com/LifeDataWeb/data_classification.htm

[Dr. Dave Olwell]

Hi, Travis,

With the normal distribution, the sum of two days' readings is also a normal random variable, so a scale adjustment such as you suggest might work for normal data.

Unfortunately, the sum of two days Weibull readings is in turn NOT distributed with the Weibull distribution, so what you suggest would not work exactly. It might work as an approximation. To test that, I suggest that you run a simulation experiment where you generate two days worth of data, sum them, and then try to model the sum as a weibull distribution.

For example, I generated 500 readings twice from a W(2,100) distribution, and summed them, and then fit them using Weibull++, and got a W(3.14, 200) back using RRX. This would give a reasonable approximation. Note that the scale parameters appeared to add, but that the shape parameter did not. Some experimentation might allow you to construct a reasonable rule of thumb for the shape parameter.

As an aside, the central limit theorem suggests that the sum of several W(beta,eta) would eventually follow the normal distribution, so we'd expect a Weibull with a shape parameter neer 3.5 to be a good fit for the sum, since is pretty closely matches the symmtery of a normal distribution. That is why, for example, the shape parameter doesn't remain as 2 in the example in the preceding paragraph.

This is no help, howeere, in figuring out how to reverse engineer the original data, since two draws from W(4,100) will also tend to have a shape parameter close to 3.5.

If you know apriori what the shapre parameter usaully is, then it might work as a hueristic to use that known value and just adjust the scale parameter.

Hope this helps! Yours was a great question.

Best wishes
Dr. Dave