Discrete approximations of continuous probability distributions obtained by minimizing Cramér-von Mises-type distances

We consider the problem of approximating a continuous random variable, characterized by a cumulative distribution function (cdf) F(x), by means of k points, x1< x2< ⋯ < xk, with probabilities pi, i= 1 , ⋯ , k. For a given k, a criterion for determining the xi and pi of the approximating k-point discrete distribution can be the minimization of some distance to the original distribution. Here we consider the weighted Cramér-von Mises distance between the original cdf F(x) and the step-wise cdf F^ (x) of the approximating discrete distribution, characterized by a non-negative weighting function w(x). This problem has been already solved analytically when w(x) corresponds to the probability density function of the continuous random variable, w(x) = F′(x) , and when w(x) is a piece-wise constant function, through a numerical iterative procedure based on a homotopy continuation approach. In this paper, we propose and implement a solution to the problem for different choices of the weighting function w(x), highlighting how the results are affected by w(x) itself and by the number of approximating points k, in addition to F(x); although an analytic solution is not usually available, yet the problem can be numerically solved through an iterative method, which alternately updates the two sub-sets of k unknowns, the xi’s (or a transformation thereof) and the pi’s, till convergence. The main apparent advantage of these discrete approximations is their universality, since they can be applied to most continuous distributions, whether they possess or not the first moments. In order to shed some light on the proposed approaches, applications to several well-known continuous distributions (among them, the normal and the exponential) and to a practical problem where discretization is a useful tool are also illustrated

Goodman and Kruskal\u2019s Gamma Coefficient for Ordinalized Bivariate Normal Distributions

We consider a bivariate normal distribution with linear correlation [Formula: see text] whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal's gamma coefficient, [Formula: see text], which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson's [Formula: see text] and Kendall's rank correlation [Formula: see text] for the bivariate normal distribution, and since in the continuous case, Kendall's [Formula: see text] coincides with Goodman and Kruskal's [Formula: see text], the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall's rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal's [Formula: see text] by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided

Dissecting hedge funds' strategies

This paper dissects the dynamics of the hedge fund industry with four financial markets, including the equity market, commodities, currencies, and debt market by employing a large number of assets from these markets. We employ four main representative hedge fund strategy indices, and a cap-weighted global index to estimate an asymmetric dynamic conditional correlation (ADCC) GJR-GARCH model using daily data from April 2003 to May 2021. We break down the performance, riskiness, investing style, volatility, dynamic correlations, and shock transmissions of each hedge fund strategy thoroughly. Further, the impact of commodity futures basis on hedge funds’ return is analyzed. Comparing the dynamic correlations during the 2008 global financial crisis (GFC) with COVID-19 pandemic reveals changing patterns in hedge funds’ investing styles. There are strong and pervasive shock spillovers from hedge fund industry to other financial markets, especially to futures commodities. An increase in the futures basis of several commodities drives up hedge funds’ performance. While hedge fund industry underperforms compared to equity market and commodities, the risk-reward measures show that hedge funds are superior to other markets, and safer than the bond market

Discrete half-logistic distributions with applications in reliability and risk analysis

In the statistical literature, several discrete distributions have been developed so far for modeling non-negative integer-valued phenomena, yet there is still room for new counting models that adequately capture the diversity of real data sets. Here, we first discuss a count distribution derived as a discrete analogue of the continuous half-logistic distribution, which is obtained by preserving the expression of its survival function at each non-negative integer support point. The proposed discrete distribution has a mode at zero and allows for overdispersion; these two features make it suitable for modeling purposes in many fields (e.g., insurance and ecology), when these conditions are satisfied by the data. In order to widen its spectrum of applications, a discrete analogue is also presented of the type I generalized half-logistic distribution (obtained by adding a shape parameter to the simple one-parameter half-logistic), which allows us to model count data whose mode is not necessarily zero. For these new count distributions, the main statistical properties are outlined, and parameter estimation along with related issues is discussed. Their feasibility is proved on two real data sets taken from the literature, which have already been fitted by other well-established count distributions. Finally, a possible application is illustrated in the insurance field, related to the exact/approximate determination of the distribution of the total claims amount through the well-known Panjer’s recursive formula, within the framework of collective risk models

Portfolio selection with independent component analysis

We analyze a methodology for portfolio selection based on the independent component analysis. In this paper parametric and non-parametric approaches are used for capturing the behavior of independent components that generate the distribution of asset returns. Although the setup is quite general, we focus mainly on the numerical issues encountered for parametric models and suggest the inclusion of a penalty function in the optimization problem

Constructing a class of stochastic volatility models: empirical investigation with VIX data

We propose a class of discrete-time stochastic volatility models that, in a parsimonious way, captures the time-varying higher moments observed in financial series. Three desirable results are obtained. First, we have a recursive procedure for the log-price characteristic function which allows a semi-analytical formula for option prices as in Heston and Nandi [2000]. Second, we reproduce some features of the VIX Index. Finally, we derive a simple formula for the VIX index and use it for option pricing

Smart network based portfolios

In this article we deal with the problem of portfolio allocation by enhancing network theory tools. We propose the use of the correlation network dependence structure in constructing some well-known risk-based models in which the estimation of the correlation matrix is a building block in the portfolio optimization. We formulate and solve all these portfolio allocation problems using both the standard approach and the network-based approach. Moreover, in constructing the network-based portfolios we propose the use of three different estimators for the covariance matrix: the sample, the shrinkage toward constant correlation and the depth-based estimators . All the strategies under analysis are implemented on three high-dimensional portfolios having different characteristics. We find that the network-based portfolio consistently performs better and has lower risk compared to the corresponding standard portfolio in an out-of-sample perspective