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Mathematical principles of the construction and characterization of a parameterized family of Gaussian mixture distributions suitable to serve as models for the probability distributions of measurement errors in nonlinear quality control
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    Some of the nonGaussian distribution functions used to model observational error in nonlinear quality control are known to lead to undesirable multiple-minima in the cost function of the variational assimilation, which can potentially degrade the degree of convergence, and hence degrade the quality of the analysis itself in the most adverse cases. One remedy is to construct a system of plausible probability functions which, while possessing broad tails, still retain the Gaussian's property of convexity of the negative-logarithm of the density. Another remedy (to be dealt with in a sequel to this note) is to attempt to simulate, in the assimilation, the effect of including a loss model, which will tend to regularize the minimization problem even when the probabilities themselves do not all have convex negative-logarithms. This article employs methods of integral transforms to construct a system of probability models guaranteed to be expressible in the form of positive Gaussian mixtures, which we argue to be plausible in cases where the actual effective error is (as is so often the case) dominated by representation error. The simplest two-parameter form guarantees that the aforementioned convexity condition is met, while a further generalization extends the family beyond this limitation, but in a way in which the degree of violation is under the control of the third shape parameter. The archetypal example of the proposed new family is just the classical logistic or sech-squared distribution. It therefore seems appropriate to refer to the proposed system of distributions as the 'Superlogistic' family.
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