Need to calculate tolerance Intervals with a set of non-normal data and 3-Parameter Weibull distribution

Hi,

I need to calculate tolerance Intervals with a set of non-normal data and 3-Parameter Weibull distribution (only distribution that fits my data), but the "3-parameter Weibull" is not listed as an option, what I can do, what do you recommend to use instead, Weibull or Non-Parametric option?.

I appreciate any help you can provide.

Hi. Have your same problem. Did you manage to solve your problem ? I appreciate any help you can provide.

I would recommend the following:

1. First, verify that your data comes from a stable process and that there are no mixtures of different process streams.
2. If the data truly comes from a stable process, contains no mixtures of different process streams, and meets a 3-parameter Weibull distribution, find the minimum value and subtract that from all of the values. The distribution should now meet that of a Weibull distribution. Find your tolerance intervals then add that minimum value back to both tolerance intervals.
3. If you have sufficient data, use the non-parametric option and compare results with the previous option. They should be fairly close.

Thank you for your suggestion.

I do believe that Miner's recommendation yields rather good estimates, because the threshold parameter is usually rather well know -- if the dataset is sufficiently large. Nevertheless, Miner's method completely omits the uncertainty in the threshold parameter. Depending on the usage of the analysis this could raise questions.
Therefore, you could consider using cross-validation to obtain an estimate for the uncertainty of the threshold parameter. Next use Miner's suggested method, but substituting "best estimate" of the threshold parameter by its confidence limits. This will not be perfect either, because we are omitting the correlation between the parameters. Nevertheless, we take the uncertainty of every parameter into account.

That is certainly true (pun intended). I try to refrain from over complicating my explanations to ensure the main point gets across. Realistically, every statistic that we calculate has uncertainty associated with it (e.g., mean, standard deviation, Cp/Cpk, R&R, control limits, tolerance limits, etc.), but we typically and conveniently ignore this uncertainty. Again, you are 100% correct. My goal is to promote understanding of the main concept.

I have found over the years here that there is often a critical un-asked question behind the question posted. When it comes to sophisticated statistical questions my first thoughts are: why are you trying this sophisticated analysis? What are you actually trying to do? If a person doesn’t understand how to perform a sophisticated mathematical calculation I wonder if it is really necessary. In my last organization I saw this a lot. A scientist would read some journal article that used a sophisticated technique and they would then have to find a reason to use it themselves (like a hypochondriac who watches medical shows and doom-scrolls medical web sites gets what ever the latest scary disease is being discussed)

So here goes to the OP @randwick : I think it’s great that you came here to ask your math question. But now can you tell us why you think you need this particular analysis? What are you ultimately trying to do with this calculation?

Hi. Thanks for your comments. I was asked by a customer of mine how to deal with this problem. I was searching on the Web for an analytical calulation of the tolerance interval with a Weibull 3-parameter. I couldn't find anything and I came across this link which I submitted to my customer who works in the medical device field.
My customer was doing the following after assessing that the data were best fitted by Weibull 3.:
1) Subtract Weibull3_Threshold from original data
2) Check that new data follow Weibull 2 (that was OK)
3) Determine tolerance intervals of Weibull 2
4) Add Weibull3_Threshold to the these tolerance limits.

@randwick That is a good approach.

Alarm bells. Why would your Customer ask you? And you found a link on the interwebs?
You have answered a mathematical question but how about the practical question that led to the math question? What are they (you) going to do with the mathematical answer? While you may be able to properly manipulate this mathematical formula - hwo do we, you adn they know it is the right formula to be calculating? (I am reminded of one of Dr. Wheeler’s article titles: the exact answer to the wrong question)