Power kurtosis transformations: Definition, properties and ordering

Summary Heavy tail distributions can be generated by applying specific non-linear transformations to a Gaussian random variable. Within this work we introduce power kurtosis transformations which are essentially determined by their generator function. Examples are theH-transformation of Tukey (1960), theK-transformation of MacGillivray and Cannon (1997) and theJ-transformation of Fischer and Klein (2004).Furthermore, we derive a general condition on the generator function which guarantees that the corresponding transformation is actually tail-increasing. In this case the exponent of the power kurtosis transformation can be interpreted as a kurtosis parameter. We also prove that the transformed distributions can be ordered with respect to the partial ordering of van Zwet (1964) for symmetric distributions.     

Gamblers’ Habits: Empirical Evidence on the Behavior of Regulars, Newcomers and Dropouts

Electronic gambling offers the opportunity to analyze huge and unbiased data sets of automatically recorded actual gambling behavior. This study refers to data on 2,127,887 poker playing identities from the Online Poker Database of the University of Hamburg (OPD-UHH) to analyze three subgroups of gamblers: regulars, newcomers, and dropouts. Their gambling habits over 6 months are analyzed in total, as well as over time. Regulars show a much higher involvement than non-regulars and increase their playing volume slightly over the observation period. Newcomers have a lower involvement than non-newcomers and most of them decrease their playing volume over time. Still, there is a small group of newcomers which increases their playing volume sharply and is, hence, very interesting for the industry as well as for the early prevention of pathological gambling. Dropouts have a higher gambling involvement than newcomers but play less than players who have not stopped stop gambling. Most dropouts also show a decreasing playing volume before dropping out. An analysis of the correlations between different variables of gambling habits shows that most of them reinforce each other, for example: gamblers with a higher total playing time tend to play at more tables simultaneously. Only playing frequency is a moderating variable of gambling involvement.     

Skewness by Splitting the Scale Parameter

This article describes a recursive nonparametric estimation for the local partial first derivative of an arbitrary function satisfied some regularity conditions and establishes its consistency and asymptotic normality under the assumption of strong mixing sequence. The proposed estimator is a variable window width version of the Watson-Nadaraya type of derivative estimator. The window width varied as more data points become available enables a recursive algorithm that reduce computational complexity from order N ³ normally required by batch methods for kernel regression to order N ². This approach is computationally simple and attractive from practical viewpoint especially when the situation call for frequent updating of first derivative estimates. For example, maintaining a delta-hedged position of a portfolio of equities with index options is one of many applications of such estimation.