Properties of Probability Models, Part 2

What they forgot to tell you about the Gammas

Story update 9/8/2015: There was an error in the data set for columns "Skew" and "Kurt" in figure 7. The error has been corrected.

Clear thinking and simplicity of analysis require concise, clear, and correct notions about probability models and how to use them. Last month in part one we looked at the properties of the Weibull probability models and discovered that some ideas about skewed distributions are incorrect. Here we shall examine the basic properties of the family of Gamma models.

How would you characterize a skewed distribution? When asked this question most will answer, "A skewed distribution is one that has a heavy, elongated tail." This idea is expressed by saying that a distribution becomes more heavy-tailed as its skewness and kurtosis increase. Last month, for Weibull models at least, we discovered that as the tail was elongated it grew lighter, not heavier. Does this happen with other families of probability models? Here we consider the Gamma models.

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Comments

Maximum likelihood is better

Re: "The value for the alpha parameter may be estimated from the average and standard deviation statistics, and this estimate will, in turn, determine the shape of the specific Gamma model you fit to your data. Since these statistics will be more dependent upon the middle 95 percent of the data than the outer one percent or less, you will end up primarily using the middle portion of the data to choose a Gamma model."

This is true because the average and standard deviation are functions of alpha and gamma, so we have 2 equations with 2 variables. The problem with this approach is, however, exactly the one Dr. Wheeler points out. The maximum likelihood method is much better, although it requires iterative methods to determine alpha and gamma. The estimates from the average and standard deviation are good starting guesses, but they rarely equal the (best) final results.

Another problem with using the average and standard deviation to find the parameters is that you cannot then find the threshold parameter for a 3-parameter gamma model. This has to be done with a double-iterative process that optimizes the threshold parameter (delta), which in turn requires maximum likelihood estimation of alpha and gamma for each delta.

It is not about the method of estimation

The point is not about how we find an estimate of the parameters for a probability model. But rather that regardless of how we estimate our parameters, the whole process is filled with uncertainty, and that these uncertainties will have the greatest impact upon the extreme critical values. The statistical approach and Shewhart's approach are diametrically opposite, and until someone understands this, they cannot begin to understand how Shewhart's distribution free approach can work.

More than logic needed

A great paper as always but most will never grasp it. More importantly, those in power making decisions will never even bother to understand it. It is safer for their careers to simply follow the herd.

I've always seen parallels with the $1.5T pa global warming scam. There is zero evidence of any kind to support the idea that man's CO2 is causing a global warming that stopped 2 decades ago, yet incredibly, people still believe. Science and logic is not enough to sway the masses. Unlike quality, the next ice age will show just how gullible the believers have been. In quality, no amount of logic alone will bring the majority back to Shewhart.

The cap and trade scam

http://www.telegraph.co.uk/news/earth/copenhagen-climate-change-confe/6… shows just how much the people who are pushing carbon emission regulations really believe in man-made global warming.

"The airport says it is expecting up to 140 extra private jets during the peak period alone, so far over its capacity that the planes will have to fly off to regional airports – or to Sweden – to park, returning to Copenhagen to pick up their VIP passengers."

Sort of like the CEO who talks about the importance of quality, and then responds to a nonconformance report by telling the people to ship it to the customer anyway.

Which sigma are we using to bound 98% ?

Are we wrapping up 98% or more of the data within 3 standard deviations computed the typical way (sample standard deviation calculation with squared differences from the mean, summed, n minus one divisor, square root) or through a moving range estimate? I mention moving range since the article invokes Shewhart. I've seen plenty of datasets in which 98% or more of the data are within 3 standard deviations of the mean using the typical calculation but more than 2% of the results are beyond 3-sigma (MR-based) limits.

Thanks.

Rational subgroup issue

If the two estimates are not roughly equal, the rational subgroup has not been defined correctly. This is when Cpk >> Ppk.

I'm interested in individual-x/mr results

I'm trying to understand whether Dr. Wheeler expects this 98% notion to hold up when running individuals charts. He's obviously a big fan of such charts, so it seems like a reasonable question.

My point is that a moving-range estimate of the standard deviation is likely to be less than the (global) estimate of standard deviation for skewed data. Try it in your favorite software package -- generate some lognormal data and compare the (global) standard deviation with the moving-range estimate. Isn't d2 (1.128 for MR of 2) based on a normal population?

Note from Dr. Wheeler's column dated February 2010: There are only two measures of dispersion to use when creating a chart for individual values: The average moving range and the median moving range ... The use of any other measure of dispersion is wrong.

Thanks.