Multivariate INAR(1) Regression Models Based on the Sarmanov Distribution

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 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.

Multivariate mixed Poisson Generalized Inverse Gaussian INAR(1) regression

“…At this point it is worth noting that modelling positively correlated claims has been explored by many articles. See for example, Bermúdez and Karlis (2011), Bermúdez and Karlis (2012), Shi and Valdez (2014a, b), Abdallah et al (2016), Bermúdez and Karlis (2017), Bermúdez et al (2018), Bermúdez et al (2018), Pechon et al (2018), Pechon et al (2019), Bolancé and Vernic (2019), Fung et al (2019, Bolancé et al (2020), Pechon et al (2021), Jeong and Dey (2021), Gómez-Déniz and Calderín-Ojeda (2021), Tzougas and di Cerchiara (2021a, b) and Bermúdez and Karlis (2021). Finally, the proportion of zeros and kurtosis show that the marginal distributions of X 1,t , X 2,t are positively skewed and exhibit a fat-tailed structure which indicates the appropriateness of adopting a positive skewed and fat-tailed distribution (GIG distribution).…”

“…Many articles have been devoted to this topic, see for example, Bermúdez and Karlis (2011), Bermúdez and Karlis (2012), Shi and Valdez (2014a, b), Abdallah et al (2016), Bermúdez and Karlis (2017), Pechon et al (2018), Pechon et al (2019), Vernic (2019), Denuit et al (2019), Fung et al (2019), Bolancé et al (2020), Pechon et al (2021), Jeong and Dey (2021), Gómez-Déniz and Calderín-Ojeda (2021), Tzougas and di Cerchiara (2021a, b). However, with the exception of very few articles, such as Bermúdez et al (2018) and Bermúdez and Karlis (2021), the construction of bivariate INAR(1) models which can capture the serial correlation between the observations of the same policyholder over time and the correlation between different claim types remains a largely uncharted territory. This is an additional contribution of this study.…”

Zero-and-One Integer-Valued AR(1) Time Series with Power Series Innovations and Probability Generating Function Estimation Approach

“…This approach started from the famous work by Al-Osh and Alzaid [6], which first introduced the so-called INAR(1) process, and since then many results related to these models have been obtained (cf. [7][8][9][10][11][12][13][14][15][16][17]). One of the recently frequent problems in count data modeling is the presence of inflated zero-and-one values in the data, which can appear in various areas of human activity (e.g., the number of requests for issuing policies, breakdowns in the production process, injury in traffic accidents, etc.).…”

Bonus-Malus Scale models: creating artificial past claims history

“…With detailed data about each contract for each insured which has become available for insurers, new longitudinal models have been proposed to improve and generalize credibility models. Complex approaches have been proposed to model claim counts: models based on series of correlated random effects (Bolancé et al (2007), Abdallah et al (2016) and Pechon et al (2019b)), jitter models (Shi and Valdez (2014)), models with pair copulas or multiple hierarchical copulas (Shi and Yang (2018) and Shi et al (2016)), time series for count data (Gourieroux and Jasiak (2004), Bermúdez et al (2018) or Bermúdez and Karlis (2021)), etc. However, like using credibility models in a practical context, these longitudinal models are often difficult for insurers to implement.…”

“…Indeed, with detailed data about each contract for each insured which has become available for insurers, new longitudinal models have been proposed to improve and generalise credibility models. Complex approaches have been proposed to model claim counts: models based on series of correlated random effects (Bolancé et al 2007), jitter models (Shi & Valdez 2014), models with pair copulas or multiple hierarchical copulas Yang 2018 andShi et al 2016), time series for count data (Gourieroux & Jasiak 2004;Bermúdez et al 2018;Bermúdez & Karlis 2021or Pinquet 2020, etc. Longitudinal models allowing for complex dependence structures between many type of claims (Abdallah et al 2016;Pechon et al 2019Pechon et al , 2021Gómez-Déniz & Calderín-Ojeda 2018) or between claim frequency and claim severity (Shi et al 2016;Oh et al 2020) were also proposed recently.…”