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Information and Entropy [⋆]
Published in Tucker S. McElroy, Dimitris N. Politis, Time Series, 2019
Tucker S. McElroy, Dimitris N. Politis
Example 8.3.13. Transformation to Uniform as Entropy-Increasing Among all continuous distributions supported on [0,1], the Uniform distribution has maximum entropy (see Example 8.3.4); see Exercise 8.17, which indicates the entropy of such a Uniform r.v. is zero. For a general r.v. X with continuous CDF F, the probability integral transform is defined via Y=F(X). By Exercise 8.19, Y has a Uniform distribution on [0,1], and therefore this is an entropy-increasing transformation. The probability integral transform can be generalized to a random vector X_, using the so-called Rosenblatt transformation mapping X_ to a random vector Y_ whose entries are i.i.d. Uniform on [0,1]; see Ch. 8.6 of Politis (2015) for more details.
Approximate multivariate distribution of key performance indicators through ordered block model and pair-copula construction
Published in IISE Transactions, 2019
With these pseudo-observations, the OBM-PCC model yields the log-likelihood function as where is a parameter vector for the pair-copula parameter of the corresponding edge in the OBM, with is n i.i.d. samples of the random variables . The order of the parent can be specified based Kendall’s tau between parent nodes and their shared child node. A higher Kendall’s tau between a parent and a specific child node will give higher priority to the parent in the order (Hanea et al., 2015). To correctly specify the Maximum Likelihood Estimate (MLE) of Equation (16), we are concerned about not only the parameter vector , but the specific type of the pair-copula for each edge in the OBM. To complete the comprehensive inference for the OBM-PCC model, we give algorithm 1 to specify the type of each copula and the initial value for the MLE of Equation (16), where we choose the Akaike’s Information Criterion (AIC) (Akaike, 1974) to select the pair-copula. The use of the AIC in copula selection is comprehensively studied in Brechmann (2010), where a large scale simulation study shows that copula selection using the AIC is more reliable than that using goodness-of-fit tests (based on a probability integral transform). In general, AIC can give stable result with a quick response, which makes it a widely used criterion in selecting copulas (Aas et al., 2009).