Reliable estimation and approximation of probability density functions is fundamental for their further processing. However, their specific properties, i.e. scale invariance and relative scale, prevent the use of standard methods of spline approximation and have to be considered when building a suitable spline basis. Bayes Hilbert space methodology allows to account for these properties of densities and enables their conversion to a standard Lebesgue space of square integrable functions using the centered log-ratio transformation. As the transformed densities fulfill a zero integral constraint, the constraint should likewise be respected by any spline basis used. Bayes Hilbert space methodology also allows to decompose bivariate densities into their interactive and independent parts with univariate geometric marginals. As this yields a useful framework for studying the dependence structure between random variables, a spline basis ideally should admit a corresponding decomposition. This paper proposes a new spline basis for (transformed) bivariate densities respecting the desired zero integral property. We show that there is a one-to-one correspondence of this basis to a corresponding novel basis in the Bayes Hilbert space of bivariate densities using tools of this methodology. Furthermore, the spline representation and the resulting decomposition into interactive and independent parts are derived. Finally, this novel spline representation is evaluated in a simulation study and we illustrate its usefulness in an application to a regression analysis of bivariate soil contamination distributions.

Abstract The simplest multistate model is named “illness-death” because subjects can move from the initial state to the final state (death) possibly transiting to an intermediate state (illness). This framework applies also when the intermediate state is a therapeutic intervention (treatment), administered after waiting some time since the initial event. Within such context, it could be of interest to compare the hazards of mortality without vs after treatment administration. In the Mantel-Byar test, an extension of the log-rank test accounting for the time-dependent nature of treatment indicator, all patients initially belong to the “untreated” group but some are shifted to the “treated” group at the time of treatment administration. Using this approach, the hazard of death is always defined as function of time since origin, also after treatment (“clock-forward” time scale). Using theoretical arguments and simulations, we show that this method is valid only under the Markov assumption but it is not suitable when the process is not markovian. This happens when the hazard of death after treatment depends: (i) on time since treatment (semi-Markov process) or (ii) on time since treatment and also on the waiting time to treatment (extended semi-Markov process). We propose two modifications of the Mantel-Byar test suitable for semi-Markov and extended semi- Markov scenarios, respectively. These procedures are both based on the adoption of the “clock-reset” scale that allows the hazard of death for treated patients to depend on time since treatment administration.

Preference-approval structures combine preference rankings and approval voting for declaring opinions over a set of alternatives. In this paper, we propose a new procedure for clustering alternatives in order to reduce the complexity of the preference-approval space and provide a more accessible interpretation of data. To that end, we present a new family of pseudometrics on the set of alternatives that take into account voters’ preferences via preference-approvals. To obtain clusters, we use the Ranked k-medoids (RKM) partitioning algorithm, which takes as input the similarities between pairs of alternatives based on the proposed pseudometrics. Finally, using non-metric multidimensional scaling, clusters are represented in 2-dimensional space.