Orthogonal series estimation of the pair correlation function of a spatial point process

The pair correlation function is a fundamental spatial point process characteristic that, given the intensity function, determines second order moments of the point process. Non-parametric estimation of the pair correlation function is a typical initial step of a statistical analysis of a spatial point pattern. Kernel estimators are popular but especially for clustered point patterns suffer from bias for small spatial lags. In this paper we introduce a new orthogonal series estimator. The new estimator is consistent and asymptotically normal according to our theoretical and simulation results. Our simulations further show that the new estimator can outperform the kernel estimators in particular for Poisson and clustered point processes

Quasi-likelihood for Spatial Point Processes

Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering not accounted for by the available covariates, likelihood based inference becomes computationally cumbersome due to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practice is solved numerically. The approximate solution is equivalent to a quasi-likelihood for binary spatial data and we therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient

A hierarchical spatio-temporal model to analyze relative risk variations of COVID-19: a focus on Spain, Italy and Germany

The novel coronavirus disease (COVID-19) has spread rapidly across the world in a short period of time and with a heterogeneous pattern. Understanding the underlying temporal and spatial dynamics in the spread of COVID-19 can result in informed and timely public health policies. In this paper, we use a spatio-temporal stochastic model to explain the temporal and spatial variations in the daily number of new confirmed cases in Spain, Italy and Germany from late February to mid September 2020. Using a hierarchical Bayesian framework, we found that the temporal trend of the epidemic in the three countries rapidly reached their peaks and slowly started to decline at the beginning of April and then increased and reached their second maximum in August. However decline and increase of the temporal trend seems to be sharper in Spain and smoother in Germany. The spatial heterogeneity of the relative risk of COVID-19 in Spain is also more pronounced than Italy and Germany.Comment: 20 pages, 4 figure

ETAS: An R Package for Fitting the Space-Time ETAS Model to Earthquake Data

The epidemic-type aftershock sequence (ETAS) model is the most widely used statistical model to describe earthquake catalogs. ETAS is an R package for fitting the space-time ETAS model to an earthquake catalog using the stochastic declustering approach introduced by Zhuang, Ogata, and Vere-Jones (2002). The package provides two classes and several functions to facilitate data preparation, model fitting and some simple diagnostic checks. The present paper is a description of the package and illustrates modeling earthquake data using the space-time ETAS model

Decomposition of variance for spatial Cox processes

Spatial Cox point processes is a natural framework for quantifying the various sources of variation governing the spatial distribution of rain forest trees. We introduce a general criterion for variance decomposition for spatial Cox processes and apply it to specific Cox process models with additive or log linear random intensity functions. We moreover consider a new and flexible class of pair correlation function models given in terms of normal variance mixture covariance functions. The proposed methodology is applied to point pattern data sets of locations of tropical rain forest trees

A central limit theorem for a sequence of conditionally centered and α\alpha-mixing random fields

A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain $\alpha$-mixing conditions in space or/and time. The limiting normal distribution is obtained for increasing spatial domain, increasing length of the sequence or a combination of these. The applicability of the theorem is demonstrated by examples regarding estimating functions for a space-time point process and a space-time Markov process

Fitting Skew Distributions to Iranian Auto Insurance Claim Data

In actuary, the derivation of loss distributions from insurance data is of great interest. Fitting an adequate distribution to real insurance data is not an easy task, mainly due to the nature of the data, which shows several features to be accounted for. Although, because of its stochastic and numerical simplicity, it is often assumed that the involved financial risk factors are normally distributed, but empirical studies indicate that most of financial risk factors have distributions with high peaks and heavy tails. Thus, it is important in the actuarial science to model insurance risks with skewed distributions. Claims size data in non-life insurance policies are very skewed and exhibit high kurtosis and extreme tails. Skew distributions are reasonable models for describing claims in property-liability insurance. We fit several well-known skew distributions (skew-normal, skew-Laplace, generalized logistic, generalized hyperbolic, variance gamma, normal inverse Gaussian, Marshal-Olkin Log-Logistic and Kumaraswamy Marshal-Olkin Log-Logistic distributions) to the amount of automobile accident claims for property damage to a third party. The data are from financial records of a state-owned major general insurance company in Iran. The fitted models are compared using AIC (Akaike information criterion), BIC (Bayesian information criterion) and Kolmogorov-Smirnov goodness-of-fit test statistics. We find that the Kumarasamy Marshal-Olkin Log-Logistic distribution is better than other considered distributions in describing the features of the observed data. This distribution is a very perfect distribution to describe the skew data. The value at risk and conditional tail expectation, as most common risk measures in insurance, are estimated for the data under consideration

Suicide Mortality Risk in Kermanshah Province, Iran: A County-level Spatial Analysis

Background: Kermanshah province has one of the highest suicide rates in Iran. The aim of this study is to explore spatial variations in the relative risk of suicide across the counties of Kermanshah province. Methods: This is an applied ecological study in which county-level counts of suicide deaths recorded by the forensic medicine organization of Kermanshah province during the period March 21, 2006 to March 20, 2013 have been used. Following a Bayesian approach, Besag, York and Mollie's (BYM) model was fitted to the number of suicide deaths of males, females and all persons to make inference about the relative risk of suicide across the counties of the province. Results: Over the study period and based on 95% credible intervals, Kangavar, Harsin and Sonqor counties had significantly lower relative risks of suicide for both males and females, Slas-Babajani, Paveh, Javanrud and Ravansar counties had significantly lower relative risks of suicide only for males and Kermanshah county had a significantly higher relative risk of suicide only for males. The relative risk of suicide for the other counties were not significantly different from the province’s overall risk neither for males nor females. Conclusion: The counties of Kermanshah province can be classified into four categories by the level of relative risk of suicide: low relative risk for both males and females, low relative risk only for males, high relative risk only for males and average relative risk. Findings from this study could be used to specify priority counties for suicide prevention initiatives

Spatial heterogeneity in gender and age of fatal suicide in Iran

Background: The suicide incident has had an increasing trend in Iran over the past years. This study mainly aimed to investigate and visualize the spatial variations of registered suicide cases at the province level. A two-step modeling approach was employed in order to estimate the relative risks (RRs) and model the age of fatal suicide across provinces in Iran. Study design: An applied ecological study. Methods: This study used the suicide death data recorded by the Iranian forensic medicine organization from March 21, 2016, to March 20, 2018. Furthermore, a Bayesian spatial approach - Besag, York, and Mollie (BYM) model- was applied to estimate the RR of suicide across provinces in Iran. Results: This risk was found to be significantly higher than the average in both men and women in the west of Iran. For women, higher population density (mean: 0.003; 95% CrI: 0.001-0.005) and lower urbanization rate of provinces (mean: -0.025; 95% CrI: -0.038, -0.012) were associated with increased RR of suicide. Based on the log-normal model fitted to the data, the overall mean age of the fatal suicide at the national level was 34 years. Conclusions: The magnitude of gender and age differences was quantified, and many spatial variations were identified in suicide mortality across provinces in Iran. Given the heterogeneity in suicide mortality risk among different subgroups of age and gender, our findings point to the urgent need in developing gender- and age-specific suicide prevention strategies. Moreover, efficient allocation of healthcare resources for suicide prevention can be attained by targeting provinces with higher risk