A weighted $$\chi ^2$$ test to detect the presence of a major change point in non-stationary Markov chains

Abstract

The problem of detecting a major change point in a stochastic process is often of interest in applications, in particular when the effects of modifications of some external variables, on the process itself, must be identified. We here propose a modification of the classical Pearson χ 2 test to detect the presence of such major change point in the transition probabilities of an inhomogeneous discrete time Markov Chain, taking values in a finite space. The test can be applied also in presence of big identically distributed samples of the Markov Chain under study, which might not be necessarily independent. The test is based on the maximum likelihood estimate of the size of the 'right' experimental unit, i.e. the units that must be aggregated to filter out the small scale variability of the transition probabilities. We here apply our test both to simulated data and to a real dataset, to study the impact, on farmland uses, of the new Common Agricultural Policy, which entered into force in EU in 2015.

Generalized Rescaled Pólya urn and its statistical application

“…The following statistical application of the GRP urn was inspired by [1,15,42]. However, those papers only deal with the case when the statistics (1.1) is asymptotically distributed as χ 2 (k − 1)λ, with λ > 1, while we also face the case when the statistics (1.1) is asymptotically distributed as χ 2 (k − 1)N 1−2υ λ, illustrating a suitable estimation procedure for the fundamental parameters η = 1 − 2υ and λ.…”

Generalized Rescaled Polya urn and its statistical applications

“…In a big sample the units typically can not be assumed independent and identically distributed, but they exhibit a structure in clusters, with independence between clusters and with correlation inside each cluster [11,15,32,39,51,52]. The model and the related results presented in [1] and in the present paper may be useful in the situation when inside each cluster the probability that a certain unit chooses the value i is affected by the number of units in the same cluster that have already chosen the value i, hence according to a reinforcement rule.…”

“…for each bounded continuous real function f on S. In [21] the notion of H-stable convergence is firstly generalized in a natural way replacing in (39) the single sub-σ-field H by a collection G = (G n ) (called conditioning system) of sub-σ-fields of A and then it is strengthened by substituting the convergence in σ(L 1 , L ∞ ) by the one in probability (i.e. in L 1 , since f is bounded).…”

“…Section 5 contains the proofs and some intermediate results. In Section 6 we describe a possible statistical application in the context of a big sample, inspired by [1,11,39]. Section 7 provides some complements and, finally, the paper ends with an appendix containing some technical lemmas, some recalls about stochastic approximation theory and about stable convergence.…”

Modeling Provincial Covid-19 Epidemic Data Using an Adjusted Time-Dependent SIRD Model

“…This variation in the parameters makes our model able to predict the short term evolution of the epidemics with good reliability, particularly in the absence of sudden oversized changes in population behavior or health policy. We leave the detection of significant change points in the parameters behavior to subsequent papers, using time series or stochastic processes techniques like the one described in [ 18 ]. For the reasons described above about the need for modeling the virus spread locally, we also implement a spatial model to allow for correlation between contiguous provinces.…”