I feel uncomfortable working with truncated distributions at their truncation point. A truncated probability distribution can be very useful in modelling populations that are known to be finite for working with the bulk of the model. If there is very little data beyond a point, truncation is a simple and effective way to deal with the "finiteness" of the distribution. But a jump discontinuity of a density function to zero does not sound right. In fact, why would a jump discontinuity (jd) to zero be more justifiable than any other jd in a density function.
I know the basics of entropy, and something tells me that (sharp) truncation in a density function has a low entropy when selecting a model. I don't know if the Akaike criterion would favour a smooth, yet nonanalytic function above a truncated distribution or not? I don't know if discontinuities has any effect on the Akaike information criterion (it does add parameters).
Also, from a formal point of view, a truncated distribution does consist of two separate analytic functions: the "support", and the part above or below which it is zero.
I am stating what I think the answer is, but is a truncated distribution not a bad choice for a model where the region of truncation is of importance? And is my suspicion correct that entropy maximization does not favor truncation?
"The only real, valuable thing is intuition" A. Einstein
"...trust... your gut instincts. If you feel something is wrong, it usually is" Anonymous
And I am sure that even Popper would agree that you should pay attention to your intuition is it relates to something being WRONG.
I think that a sharp truncation is probably more artificial and a scientific simplification for many phenomena. The characteristic of the entire object, process or region should determine a natural sharp truncation. For example, the length of a crack in a window pain is limited by the geometrical size of the window (besides further influencing factors).
Besides, the estimation of the truncation point by a maximum likelihood method does not fulfil the regularity condition, which must be fulfilled for many properties and applications of the maximum likelihood method. One result of this item: the estimation error of classical maximum likelihood estimation is (asymptotically) normally distributed. The estimation of a truncation point with the corrected maximum likelihood estimator result in an exponential distributed error. The Akaike information criterion (AIC) should not be applied in such cases. I only applied the AIC to validate alternatives with the same estimation for the truncation point (https://www.researchgate.net/publication/262568654_Modeling_of_magnitude_distributions_by_the_generalized_truncated_exponential_distribution).
I expect that a mixture of sharp truncated distributions better describes many phenomena. However, formulation and estimation of such a (more complex) model is more difficult and needs a large sample size. Therefore, simplifications are applied.
I apologize for my poor English.
I see your point clearly. Yet, it is a singular case in that it does not conform to several of the properties of a distribution without discontinuity, specifically in an open interval around the truncation point. As our model of of the frequency-magnitude distribution, I have an intuitive argument that the PDF of the freq-mag distribution should decrease (I would like to believe continuously, to zero). My argument is that faults, modeled as large cracks, needs increasingly more energy to show movement until, eventually, it reaches the point where the fault actually has to propagate (at a very high energy). But there is no boundary in a nice hypothetical fault that stops the crack from propagating. Applying, without justification, my pragmatic belief in an upper limit, the likelihood of having increasingly larger earthquakes decreases continuously to zero at the upper bound of magnitude. I would appreciate more thoughts on this.
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