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BACKGROUND Current prognosis in oncology is reduced to the tumour stage and performance status, leaving out many other factors that may impact the patient´s management. Prognostic stratification of early stage non-small-cell lung cancer (NSCLC) patients with poor prognosis after surgery is of considerable clinical relevance. The objective of this study was to identify clinical factors associated with long-term overall survival in a real-life cohort of patients with stage I-II NSCLC and develop a prognostic model that identifies features associated with poor prognosis and stratifies patients by risk. METHODS This is a cohort study including 505 patients, diagnosed with stage I-II NSCLC, who underwent curative surgical procedures at a tertiary hospital in Madrid, Spain. RESULTS Median OS (in months) was 63.7 (95% CI, 58.7–68.7) for the whole cohort, 62.4 in patients submitted to surgery and 65 in patients submitted to surgery and adjuvant treatment. The univariate analysis estimated that a female diagnosed with NSCLC has a 0.967 (95% CI 0.936–0.999) probability of survival one year after diagnosis and a 0.784 (95% CI 0.712–0.863) five years after diagnosis. For males, these probabilities drop to 0.904 (95% CI 0.875–0.934) and 0.613 (95% CI 0.566–0.665), respectively. Multivariable analysis shows that sex, age at diagnosis, type of treatment, ECOG-PS, and stage are statistically significant variables (p 1) while adjuvant chemotherapy is a good prognostic variable (HR

Funding Information: Funding: For the second author, this work was done under partial financial support of RFBR (Grant n. 19-01-00451). For the first and third author this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020 financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was funded by the insurance company Fidelidade. We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times. publishersversion published

Motor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cramér–Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems, we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work, the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that, we used a real commercial scale of an insurer operating in the Portuguese market, and we also work on various well-known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussed.

In this paper, we study, by means of randomized sampling, the long-run stability of some open Markov population fed with time-dependent Poisson inputs. We show that state probabilities within transient states converge—even when the overall expected population dimension increases without bound—under general conditions on the transition matrix and input intensities.Following the convergence results, we obtain ML estimators for a particular sequence of input intensities, where the sequence of new arrivals is modeled by a sigmoidal function. These estimators allow for the forecast, by confidence intervals, of the evolution of the relative population structure in the transient states.Applying these results to the study of a consumption credit portfolio, we estimate the implicit default rate.

In this paper, following an open portfolio approach, we show how to estimate a Bonus-malus system evolution.Considering a model for the number of new annual policies, we obtain ML estimators, asymptotic distributions and confidence regions for the expected number of new policies entering the portfolio in each year, as well as for the expected number and proportion of insureds in each bonus class, by year of enrollment. Confidence regions for the distribution of policyholders result in confidence regions for optimal bonus scales.Our treatment is illustrated by an example with numerical results.

Using a Markov model, we study the evolution of open populations subject to periodical reclassifications considering a random number of entrances. In this paper, we focus on the estimation of the subpopulations' relative sizes, obtaining maximum likelihood (ML) estimators, asymptotic distributions, and confidence regions. We show, under general conditions, that the stable distribution is dependent on the rate of entrances to the population. We illustrate the model by estimating the evolution of a population of a pension fund beneficiaries.

Populations with periodical re-classifications occur in many fields, such as Insurance Companies, Bank Institutions, Pension Funds, Epidemiology and Health/Disease studies, where population elements are periodically reclassified. p]We show how to carry out inference for such populations, under general assumptions. Our treatment differs from previous ones since we consider open populations and focus on the sizes of sub-populations, availing ourselves of the Stochastic Vortices theory, see Guerreiro and Mexia (2004) and Guerreiro and Mexia (2008).

We propose a model for determining the spread of a credit portfolio based on the actuarial principle. In this model the spread is a function of the recovery rate and of the probability of default. In an application to data, from a consumer credit portfolio of a Cape Verde bank, we estimate the recovery rate by a beta regression and the probability of default by a logistic regression, both regressions using as independent variables sociodemographic information and consumer credit contract variables in the data set. We show that the data support the possibility of defining a spread for each client - the borrower - that is coherent with the portfolio spread given by the model.

This work was partially supported by Financiamento Base 2009 ISFL-1-297 from FCT/MCTES/PT. Considering open portfolios, we analyze bonus–malus systems (BMS) under a realistic approach, as we already did in Guerreiro and Mexia (Discuss. Math. Probab. Stat. 24(2):197–213, 2004). Using stochastic vortices model we are now able to predict long-run distribution through confidence intervals. authorsversion published

Our paper considers open populations with arrivals and departures whose elements are subject to periodic reclassications. These populations will be divided into a nite number of sub-populations. Assuming that: a) entries, reclassications and departures occur at the beginning of the time units; b) elements are reallocated at equally spaced times; c) numbers of new elements entering at the beginning of the time units are realizations of independent Poisson distributed random variables; we use Markov chains to obtain limit results for the relative sizes of the sub-populations corresponding to the states of the chain. Namely we will obtain conditions for stability of the relative sizes for transient and recurrent states as well as for all states. The existence of such stability corresponds to the existence of a stochastic structure based either on the transient or on the recurrent states or even on all states. We call these structures stochastic vortices because the structure is maintained despite entrances, departures and reallocations.

Under the assumptions of an open portfolio, i.e., considering that a policyholder can transfer his policy to another insurance company and the continuous arrival of new policyholders into a portfolio which can be placed into any of the bonus classes and not only in the "starting class", we developed a model (Stochastic Vortices Model) to estimate the Long Run Distribution for a Bonus Malus System. These hypothesis render the model quite representative of the reality. With the obtained Long Run Distribution, a few optimal bonus scales were calculated, such as Norberg’s (1979), Borgan, Hoem’s and Norberg’s (1981), Gilde and Sundt’s (1989) and Andrade e Silva’s (1991). To compare our results, since this was the rst application of the model, we used the Classic Model for Bonus Malus and the Open Model developed by Centeno and Andrade e Silva (2001). The results of the Stochastic Vortices and the Open Model are highly similar and quite different from those of the Classic Model. Besides this the distribution of policyholders in the various bonus classes was derived assuming that the entrances followed adequate stochastic models.