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Urquhart (2016) investigated the market efficiency of Bitcoin by means of five different tests on Bitcoin returns. It was concluded that the Bitcoin returns do not satisfy the efficient market hypothesis. We show here that a simple power transformation of the Bitcoin returns do satisfy the hypothesis through the use of eight different tests. The transformation used does not lead to any loss of information.

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Undoubtedly, the normal distribution is the most popular distribution in statistics. In this paper, we introduce a natural generalization of the normal distribution and provide a comprehensive treatment of its mathematical properties. We derive expressions for the nth moment, the nth central moment, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon entropy, and the asymptotic distribution of the extreme order statistics. We also discuss estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix.Estimation, entropy, generalized normal distribution, moments, normal distribution, order statistics,

Gupta et al. [Commun. Stat., Theory Methods 27, 887-904, 1998] introduced the exponentiated exponential distribution as a generalization of the standard exponential distribution. In this paper, we introduce four more exponentiated type distributions that generalize the standard gamma, standard Weibull, standard Gumbel and the standard Fréchet distributions in the same way the exponentiated exponential distribution generalizes the standard exponential distribution. A treatment of the mathematical properties is provided for each distribution.

A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions.A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.

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