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Mathematics
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Multivariate Sarmanov Distributions and Applications
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Multivariate Sarmanov Distributions and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: closed (31 March 2021) | Viewed by 15653

Special Issue Editors

Prof. Dr. Vernic Raluca

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Guest Editor
Faculty of Mathematics and Informatics, Ovidius University of Constanta, 900527 Constanța, Romania
Interests: actuarial and financial mathematics; probability theory; statistics; mathematical optimization; mathematical software
Prof. Dr. Catalina Bolancé

E-Mail Website
Guest Editor
Department of Econometrics, Statistics and Applied Economics, Riskcenter, University of Barcelona, 08007 Barcelona, Spain
Interests: nonparametric statistics; risk quantification; micro econometrics; data analysis
Dr. Elena Pelican

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, Ovidius University of Constanta, 900527 Constanța, Romania
Interests: numerical analysis; optimization techniques; algorithms and programming; face and digit recognition; machine learning

Special Issue Information

Dear Colleagues,

In many fields, there is increasing interest in developing systems of multivariate distributions in connection with capturing the dependence observed from practical data. In this sense, the multivariate Sarmanov distribution is a good candidate because of its specific form, which allows for a flexible dependence structure between the given marginals. As these marginals can be of various types, the resulting multivariate Sarmanov distribution can be continuous; discrete; or even of a mixed type, joining both continuous and discrete marginals. In particular, the Farlie–Gumbel–Morgenstern distribution is probably the best-known member of the Sarmanov family.

Regarding practical applications, Sarmanov’s distribution has already been used to model real data from fields like medicine, economy, insurance and finance, marketing, physics, or biology. The purpose of this Special Issue is to increase the interest in this class of distributions, and to enlarge its applicability by reviewing the recent developments and by extending it; more precisely, we are concerned with extending this class of distributions, such that the correlation range is enlarged, as for the classical Sarmanov distribution, it proved to be limited.

We invite you to submit papers conducting probabilistic studies on this distribution and on its extensions (e.g., concerning the relations among the subsets of the marginal variables), regarding copula representation, proposing estimation methods and applications in different fields, including comparative studies with other distributions on real data.

Prof. Vernic Raluca
Prof. Dr. Catalina Bolancé
Dr. Elena Pelican
Guest Editors

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Keywords

  • continuous and discrete distributions
  • correlation and dependence
  • estimation methods and algorithms
  • numerical methods
  • copula representation

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Published Papers (5 papers)

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Research

13 pages, 326 KiB  
Article
Multivariate INAR(1) Regression Models Based on the Sarmanov Distribution
by Lluís Bermúdez and Dimitris Karlis
Mathematics 2021, 9(5), 505; https://doi.org/10.3390/math9050505 - 1 Mar 2021
Cited by 14 | Viewed by 2494
Abstract
A multivariate INAR(1) regression model based on the Sarmanov distribution is proposed for modelling claim counts from an automobile insurance contract with different types of coverage. The correlation between claims from different coverage types is considered jointly with the serial correlation between the [...] Read more.
A multivariate INAR(1) regression model based on the Sarmanov distribution is proposed for modelling claim counts from an automobile insurance contract with different types of coverage. The correlation between claims from different coverage types is considered jointly with the serial correlation between the observations of the same policyholder observed over time. Several models based on the multivariate Sarmanov distribution are analyzed. The new models offer some advantages since they have all the advantages of the MINAR(1) regression model but allow for a more flexible dependence structure by using the Sarmanov distribution. Driven by a real panel data set, these models are considered and fitted to the data to discuss their goodness of fit and computational efficiency. Full article
(This article belongs to the Special Issue Multivariate Sarmanov Distributions and Applications)
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18 pages, 351 KiB  
Article
Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity
by Ramon Alemany, Catalina Bolancé, Roberto Rodrigo and Raluca Vernic
Mathematics 2021, 9(1), 73; https://doi.org/10.3390/math9010073 - 31 Dec 2020
Cited by 6 | Viewed by 3312
Abstract
The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus [...] Read more.
The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables. Full article
(This article belongs to the Special Issue Multivariate Sarmanov Distributions and Applications)
17 pages, 773 KiB  
Article
Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being
by José María Sarabia and Vanesa Jorda
Mathematics 2020, 8(11), 2095; https://doi.org/10.3390/math8112095 - 23 Nov 2020
Cited by 2 | Viewed by 2544
Abstract
The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can [...] Read more.
The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries. Full article
(This article belongs to the Special Issue Multivariate Sarmanov Distributions and Applications)
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11 pages, 351 KiB  
Article
A Sarmanov Distribution with Beta Marginals: An Application to Motor Insurance Pricing
by Catalina Bolancé, Montserrat Guillen and Albert Pitarque
Mathematics 2020, 8(11), 2020; https://doi.org/10.3390/math8112020 - 13 Nov 2020
Cited by 16 | Viewed by 2645
Abstract
Background: The Beta distribution is useful for fitting variables that measure a probability or a relative frequency. Methods: We propose a Sarmanov distribution with Beta marginals specified as generalised linear models. We analyse its theoretical properties and its dependence limits. Results: We use [...] Read more.
Background: The Beta distribution is useful for fitting variables that measure a probability or a relative frequency. Methods: We propose a Sarmanov distribution with Beta marginals specified as generalised linear models. We analyse its theoretical properties and its dependence limits. Results: We use a real motor insurance sample of drivers and analyse the percentage of kilometres driven above the posted speed limit and the percentage of kilometres driven at night, together with some additional covariates. We fit a Beta model for the marginals of the bivariate Sarmanov distribution. Conclusions: We find negative dependence in the high quantiles indicating that excess speed and night-time driving are not uniformly correlated. Full article
(This article belongs to the Special Issue Multivariate Sarmanov Distributions and Applications)
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17 pages, 814 KiB  
Article
Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution
by Catalina Bolancé and Raluca Vernic
Mathematics 2020, 8(9), 1400; https://doi.org/10.3390/math8091400 - 21 Aug 2020
Cited by 3 | Viewed by 3553
Abstract
In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is [...] Read more.
In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio. Full article
(This article belongs to the Special Issue Multivariate Sarmanov Distributions and Applications)
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