Halfspace depth for general measures: The ray basis theorem and its consequences

The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under minimal assumptions on the underlying measure. The proof applies not only to the median, but also to other trimmed regions. Motivated by the technical development of the amended ray basis theorem, we specify connections between the trimmed regions, floating bodies, and additional equi-affine convex sets related to the depth. As a consequence, minimal conditions for the strict monotonicity of the depth are obtained. Applications to the computation of the depth and robust estimation are outlined

Another look at halfspace depth: Flag halfspaces with applications

The halfspace depth is a well studied tool of nonparametric statistics in multivariate spaces, naturally inducing a multivariate generalisation of quantiles. The halfspace depth of a point with respect to a measure is defined as the infimum mass of closed halfspaces that contain the given point. In general, a closed halfspace that attains that infimum does not have to exist. We introduce a flag halfspace - an intermediary between a closed halfspace and its interior. We demonstrate that the halfspace depth can be equivalently formulated also in terms of flag halfspaces, and that there always exists a flag halfspace whose boundary passes through any given point $x$, and has mass exactly equal to the halfspace depth of $x$. Flag halfspaces allow us to derive theoretical results regarding the halfspace depth without the need to differentiate absolutely continuous measures from measures containing atoms, as was frequently done previously. The notion of flag halfspaces is used to state results on the dimensionality of the halfspace median set for random samples. We prove that under mild conditions, the dimension of the sample halfspace median set of $d$-variate data cannot be $d-1$, and that for $d=2$ the sample halfspace median set must be either a two-dimensional convex polygon, or a data point. The latter result guarantees that the computational algorithm for the sample halfspace median form the R package TukeyRegion is exact also in the case when the median set is less-than-full-dimensional in dimension $d=2$

Nenegativni celobrojni autoregresivni procesi u slučajnoj sredini generisani geometrijskim brojačkim nizovima

Here are analyzed integer autoregressive (INAR) processes in the random environment generated by geometric counting series. Firstly, the first order random environment INAR model is introduced. Later, random environment INAR models of higher order, as well as their general form, are defined. Finally, the bivariate model based on the bivariate random process is defined. The properties of all introduced models are analyzed. Estimation of unknown parameters is given and validate on the simulated data. Model quality is confirmed by application on the real-life data, comparing results with the competitive models

Forecasting with two generalized integer-valued autoregressive processes of order one in the mutual random environment

In this article, we consider two univariate random environment integer-valued autoregressive processes driven by the same hidden process. A model of this kind is capable of describing two correlated non-stationary counting time series using its marginal variable parameter values. The properties of the model are presented. Some parameter estimators are described and implemented on the simulated time series. The introduction of this bivariate integer-valued autoregressive model with a random environment is justified at the end of the paper, where its real-life data-fitting performance was checked and compared to some other appropriate models. The forecasting properties of the model are tested on a few data sets, and forecasting errors are discussed through the residual analysis of the components that comprise the model.Peer Reviewe

CONDITIONAL LEAST SQUARES ESTIMATION OF THE PARAMETERS OF HIGHER ORDER RANDOM ENVIRONMENT INAR MODElS

Two different random environment INAR models of higher order, precisely RrNGINARmax(p) and RrNGINAR1(p), are presented as a new approach to modeling non-stationary nonnegative integer-valued autoregressive processes. The interpretation of these models is given in order to better understand the circumstances of their application to random environment counting processes. The estimation statistics, defined using the Conditional Least Squares (CLS) method, is introduced and the properties are tested on the replicated simulated data obtained by RrNGINAR models with different parameter values. The obtained CLS estimates are presented and discussed

Nenegativni celobrojni autoregresivni procesi u slučajnoj sredini generisani geometrijskim brojačkim nizovima

Here are analyzed integer autoregressive (INAR) processes in the random environment generated by geometric counting series. Firstly, the first order random environment INAR model is introduced. Later, random environment INAR models of higher order, as well as their general form, are defined. Finally, the bivariate model based on the bivariate random process is defined. The properties of all introduced models are analyzed. Estimation of unknown parameters is given and validate on the simulated data. Model quality is confirmed by application on the real-life data, comparing results with the competitive models

Forecasting with two generalized integer-valued autoregressive processes of order one in the mutual random environment

In this article, we consider two univariate random environment integer-valued autoregressive processes driven by the same hidden process. A model of this kind is capable of describing two correlated non-stationary counting time series using its marginal variable parameter values. The properties of the model are presented. Some parameter estimators are described and implemented on the simulated time series. The introduction of this bivariate integer-valued autoregressive model with a random environment is justified at the end of the paper, where its real-life data-fitting performance was checked and compared to some other appropriate models. The forecasting properties of the model are tested on a few data sets, and forecasting errors are discussed through the residual analysis of the components that comprise the model

Forecasting with two generalized integer-valued autoregressive processes of order one in the mutual random environment

In this article, we consider two univariate random environment integer-valued autoregressive processes driven by the same hidden process. A model of this kind is capable of describing two correlated non-stationary counting time series using its marginal variable parameter values. The properties of the model are presented. Some parameter estimators are described and implemented on the simulated time series. The introduction of this bivariate integer-valued autoregressive model with a random environment is justified at the end of the paper, where its real-life data-fitting performance was checked and compared to some other appropriate models. The forecasting properties of the model are tested on a few data sets, and forecasting errors are discussed through the residual analysis of the components that comprise the model