Heteroskedasticity and Non-normality together; few solutions exist

In linear regression, most of the methods that combat the two common problems of heteroskedasticity and non-normality are designed to counter the effect of one-problem only (either hetero or non-normality; not both in the same time).
One of the rare methods that combat both problems is one type of wild bootstrapping (it has an R-package for that).
On my searches for solution of the two problems, I found an R-package designed to do White-test (for heteroskedasticity) but conducted via bootstrapping (I guess package name is “whitestrap”). In many times when I try this package to obtain a bootstrap test of heteroskedasticity, I found good results (the bootstrap p-value of white test is generally larger than the conventional White test; this is under replications = 2000 or 1000). Once I increase the replications (B = 5000), the P-value goes towards non-significance, and usually once you try B = 10,000, non-signifiance is surely obtained (which is what we hope).
After such experiments on bootstrap white test, can one say that doing a bootstrap regression (of a model contaminated with both non-normality and hetero) with high replications (say 10000) would solve both problems, non-normality and hetero?

I’m not sold on that overall approach for several reasons.

  • The Huber-White robust covariance matrix can give you the right answer to the wrong question. It can’t fix having less meaningful parameters due to model misspecification especially getting the transformation wrong.
  • Even in simple situations it is not clear how Y should be transformed before doing parametric modeling.
  • In general the bootstrap tells you how bad something is but not how to fix it.
  • Semiparametric ordinal regression models cut through much of this:
    • They are Y-transformation invariant
    • The are completely robust to Y-outliers
    • The can provide all needed readouts, including effect ratios, cumulative probabilities, individual cell probabilities, quantiles, and the mean
    • They handle lower and upper limits of detection, floor and ceiling effects, bimodal distributions, heteroscedasticy (to some extent), excessive ties anyone in the range of Y

A resource for semiparametic models is here and here is a detailed case study of ordinary regression for continuous Y. It uses the R rms packge orm function which can handle more than 6,000 distinct Y values, i.e., it can do calculations efficiently with > 6000 intercepts in the model. A Bayesian counterpart to this is here.

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