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J. Risk Financial Manag. 2021, 14, 440 6 of 15 and robust measure of volatility/risk than variance for non Gaussian phenomena. It is indeed unique for any distribution, much as the characteristic function is, both representing all the moments of a distribution, which could be merely the mean and variance in the case of a Normal variable. Asset returns are not Gaussian! Given a prior or competing distribution q, defined on R, the cross-entropy (Kullback and Leibler 1951) measure is M I(p; q) = ∑ pilog(pi/qi), (4) i=1 where a uniform q reduces I(p; q) to H(p). This measure reflects the gain in information with respect to R resulting from the additional knowledge in p relative to q. Like with H(p), I(p; q) is an information theoretic distance of p from q. It can be symmetrized by averaging I(p; q) and I(q; p). Facing the fundamental question of drawing inferences from limited and insuffi- cient data, Jaynes proposed the maximum entropy (ME) principle, which he viewed as a generalization of Bernoulli and Laplace’s Principle of Insufficient Reason. Given T constraints, perhaps in the form of moments, Jaynes proposed the ME method, which is to maximize H(p) subject to the T structural constraints. Thus, given moment conditions, Xt (t = 1, 2, ..., T), where T < M, the ME principle prescribes choosing the p(ai) that maximizes H(p) subject to the given constraints (moments) of the problem. The solution to this underdetermined problem is ˆ p(ai) ∝ exp{− ∑λtXt(ai)}, (5) t where λ are the T Lagrange multipliers, and λˆ are the values of the optimal solution (estimated values) of λ. Naturally, if no constraints are imposed, H(p) reaches its maximum value and the p are distributed uniformly. Building on Shannon’s work, a number of generalized entropies and information measures were developed. Starting with the idea of describing the gain of information, (Renyi 1970) developed the entropy of order α for incomplete random variables. The relevant generalized entropy measure of a proper probability distribution is HαR(p)= 1 log∑pαk. (6) 1−α k Shannon measure is a special case of this measure where α → 1. Similarly, the Renyi cross-entropy of order α is R R 1 pαk Iα (x|y) = Iα (p, q) = 1 − α log ∑ qα−1 , (7) kk which is equal to the traditional cross-entropy measure as α → 1. Only one member of these “divergence” measures is a metric, which we define below. Entropy has been actively considered in finance theory since at least 1999. According to (Gulko 1999), “entropy pricing theory” suggests that in information efficient markets, perfectly uncertain market beliefs must prevail. Using entropy to measure market uncer- tainty, entropy-maximizing market beliefs must prevail. One can derive (entropy) optimal asset pricing models that are similar to the Black–Scholes model (with the log-normal distribution replaced by other probability distributions). 3.2. Using Entropy to Test Equality of Univariate Densities Maasoumi and Racine (2002) considered a metric entropy that is useful for testing for equality of densities for two univariate random variables X and Y. The functionPDF Image | Contrasting Cryptocurrencies with Other Assets
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