SUMM: control charts, heteroscedasticity, heterogeneity, predicted distribution/moments

[josef-pkt]

(some vague random thoughts)

standard control charts use a estimate of the standard deviation to specify the control limits (3 sigma)

traditional Xbar charts seem to use withing group variation (e.g. based on within group range statistic)
The following article argues in favor of using a between group range statistic to capture between group variation
https://www.smartersolutions.com/orl/30000-ft-level-performance-metric-reporting_6sigmaforum.pdf

minitab handles excess dispersion in Poisson and Binomial charts by estimating the excess dispersion based on the between group variation.

more general: we want to know the spread or quantiles for the distribution of a new observation, i.e. we want to predict a new observation and some of it's quantiles.
(a bit similar to GARCH, or predicting several moments of the prediction distribution)

• unobserved heterogeneity or unconditional distribution: If we don't observe the variables driving heteroscedasticity/heterogeneity, then we observe only the unconditional distribution which is a mixture distribution. The appropriate standard deviation is the total variation, or between variation if we use groups. The within variance will not correctly estimate this.
• conditional distribution or observed heterogeneity: If we can observe the variables affecting the variance or distributional properties, then we can improve on the unconditional prediction using a model for other moments than mean. In this case we use group specific standard deviation and group specific prediction intervals.

We don't have anything GARCH like, that would predict the variance or excess dispersion.

Similar to the discussion on excess dispersion in GLM, we could have different sources that results in overall excess dispersion or heteroscedasticity, e.g. random effects or withing group correlation.

Our current QMLE estimates the mean function being robust to misspecified higher moments (heteroscedasticity or correlation).

To get prediction intervals for new observations, we also need the corresponding models for variance and higher moments, e.g. QMLE for variance that is robust to higher moment misspecification.
(maybe GLM-Gamma with robust cov_type)

We essentially need to apply what we have for mean to variance.
e.g. do we want population averaged effect, and if yes for which population, or do we want conditional effects under stronger parametric assumptions.

[josef-pkt]

related newer issue #8533