Trend Analysis of Temperature and Rainfall of Rajasthan, India

Increasing temperature and declining and erratic rainfall is one of the greatest global challenges. This study presents the trend analysis of temperature and rainfall in five divisional headquarters of Rajasthan, namely, Bikaner, Jaipur, Jodhpur, Kota, and Udaipur. The historic data of minimum and maximum temperature and rainfall for a period of 49 years from 1971 to 2019 were collected from Climate Research and Services, India Meteorological Department, Pune. Detection of trends and change in magnitude was done using the Mann–Kendall (MK) test and Sen’s slope, respectively. The results of the study indicated a significant increase in both minimum and maximum temperature over time for all the five stations. However, rainfall showed a nonsignificant increasing trend for Kota and Udaipur district, whereas Bikaner, Jaipur, and Jodhpur detected a negative trend.

Reliability Estimation for the Remained Stress-Strength Model under the Generalized Exponential Lifetime Distribution

A stress-strength reliability model compares the strength and stresses on a certain system; it is used not only primarily in reliability engineering and quality control but also in economics, psychology, and medicine. In this paper, a novel extension of stress-strength models is presented. The mew model is applied under the generalized exponential distribution. The maximum likelihood estimator, asymptotic distribution, and Bayesian estimation are obtained. A comprehensive simulation study along with real data analysis is performed for illustrating the importance of the new stress-strength model.

Permutation Invariant Strong Law of Large Numbers for Exchangeable Sequences

We provide a permutation invariant version of the strong law of large numbers for exchangeable sequences of random variables. The proof consists of a combination of the Komlós–Berkes theorem, the usual strong law of large numbers for exchangeable sequences, and de Finetti’s theorem.

Modeling of the COVID-19 Cases in Gulf Cooperation Council Countries Using ARIMA and MA-ARIMA Models

Coronavirus disease 2019 (COVID-19) is still a great pandemic presently spreading all around the world. In Gulf Cooperation Council (GCC) countries, there were 1015269 COVID-19 confirmed cases, 969424 recovery cases, and 9328 deaths as of 30 Nov. 2020. This paper, therefore, subjected the daily reported COVID-19 cases of these three variables to some statistical models including classical ARIMA, kth SMA-ARIMA, kth WMA-ARIMA, and kth EWMA-ARIMA to study the trend and to provide the long-term forecasting of the confirmed, recovery, and death cases of the novel COVID-19 pandemic in the GCC countries. The data analyzed in this study covered the period starting from the first case of coronavirus reported in each GCC country to Jan 31, 2021. To compute the best parameter estimates, each model was fitted for 90% of the available data in each country, which is called the in-sample forecast or training data, and the remaining 10% was used for the out-of-sample forecast or testing data. The AIC was applied to the training data as a criterion method to select the best model. Furthermore, the statistical measure RMSE and MAPE were utilized for testing data, and the model with the minimum RMSE and MAPE was selected for future forecasting. The main finding, in general, is that the two models WMA-ARIMA and EWMA-ARIMA, besides the cubic and 4th degree polynomial regression, have given better results for in-sample and out-of-sample forecasts than the classical ARIMA models in fitting the confirmed and recovery cases while SMA-ARIMA and WMA-ARIMA were suitable to model the recovery and death cases in the GCC countries.

Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata

The goal of this paper is to expand the explicit formula for the solutions of the Extended Skorokhod Problem developed earlier for a special class of constraining domains in ℝ n with orthogonal reflection fields. We examine how affine transformations convert solutions of the Extended Skorokhod Problem into solutions of the new problem for the transformed constraining system. We obtain an explicit formula for the solutions of the Extended Skorokhod Problem for any ℝ n - valued càdlàg function with the constraining set that changes in time and the reflection field naturally defined by any basis. The evolving constraining set is a region sandwiched between two graphs in the coordinate system generating the reflection field. We discuss the Lipschitz properties of the extended Skorokhod map and derive Lipschitz constants in special cases of constraining sets of this type.

Assessing the Performance of the Discrete Generalised Pareto Distribution in Modelling Non-Life Insurance Claims

The generalised Pareto distribution (GPD) offers a family of probability spaces which support threshold exceedances and is thus suitable for modelling high-end actuarial risks. Nonetheless, its distributional continuity presents a critical limitation in characterising data of discrete forms. Discretising the GPD, therefore, yields a derived distribution which accommodates the count data while maintaining the essential tail modelling properties of the GPD. In this paper, we model non-life insurance claims under the three-parameter discrete generalised Pareto (DGP) distribution. Data for the study on reported and settled claims, spanning the period 2012–2016, were obtained from the National Insurance Commission, Ghana. The maximum likelihood estimation (MLE) principle was adopted in fitting the DGP to yearly and aggregated data. The estimation involved two steps. First, we propose a modification to the μ and μ + 1 frequency method in the literature. The proposal provides an alternative routine for generating initial estimators for MLE, in cases of varied count intervals, as is a characteristic of the claim data under study. Second, a bootstrap algorithm is implemented to obtain standard errors of estimators of the DGP parameters. The performance of the DGP is compared to the negative binomial distribution in modelling the claim data using the Akaike and Bayesian information criteria. The results show that the DGP is appropriate for modelling the count of non-life insurance claims and provides a better fit to the regulatory claim data considered.

Hidden Geometry of Bidirectional Grid-Constrained Stochastic Processes

Bidirectional Grid Constrained (BGC) stochastic processes (BGCSPs) are constrained Itô diffusions with the property that the further they drift away from the origin, the more the resistance to movement in that direction they undergo. The underlying characteristics of the BGC parameter Ψ X t , t are investigated by examining its geometric properties. The most appropriate convex form for Ψ , that is, the parabolic cylinder is identified after extensive simulation of various possible forms. The formula for the resulting hidden reflective barrier(s) is determined by comparing it with the simpler Ornstein–Uhlenbeck process (OUP). Applications of BGCSP arise when a series of semipermeable barriers are present, such as regulating interest rates and chemical reactions under concentration gradients, which gives rise to two hidden reflective barriers.

Evaluation of Four Multiple Imputation Methods for Handling Missing Binary Outcome Data in the Presence of an Interaction between a Dummy and a Continuous Variable

Multiple imputation by chained equations (MICE) is the most common method for imputing missing data. In the MICE algorithm, imputation can be performed using a variety of parametric and nonparametric methods. The default setting in the implementation of MICE is for imputation models to include variables as linear terms only with no interactions, but omission of interaction terms may lead to biased results. It is investigated, using simulated and real datasets, whether recursive partitioning creates appropriate variability between imputations and unbiased parameter estimates with appropriate confidence intervals. We compared four multiple imputation (MI) methods on a real and a simulated dataset. MI methods included using predictive mean matching with an interaction term in the imputation model in MICE (MICE-interaction), classification and regression tree (CART) for specifying the imputation model in MICE (MICE-CART), the implementation of random forest (RF) in MICE (MICE-RF), and MICE-Stratified method. We first selected secondary data and devised an experimental design that consisted of 40 scenarios (2 × 5 × 4), which differed by the rate of simulated missing data (10%, 20%, 30%, 40%, and 50%), the missing mechanism (MAR and MCAR), and imputation method (MICE-Interaction, MICE-CART, MICE-RF, and MICE-Stratified). First, we randomly drew 700 observations with replacement 300 times, and then the missing data were created. The evaluation was based on raw bias (RB) as well as five other measurements that were averaged over the repetitions. Next, in a simulation study, we generated data 1000 times with a sample size of 700. Then, we created missing data for each dataset once. For all scenarios, the same criteria were used as for real data to evaluate the performance of methods in the simulation study. It is concluded that, when there is an interaction effect between a dummy and a continuous predictor, substantial gains are possible by using recursive partitioning for imputation compared to parametric methods, and also, the MICE-Interaction method is always more efficient and convenient to preserve interaction effects than the other methods.

Two-Stage Joint Model for Multivariate Longitudinal and Multistate Processes, with Application to Renal Transplantation Data

In longitudinal studies, clinicians usually collect longitudinal biomarkers’ measurements over time until an event such as recovery, disease relapse, or death occurs. Joint modeling approaches are increasingly used to study the association between one longitudinal and one survival outcome. However, in practice, a patient may experience multiple disease progression events successively. So instead of modeling of a single event, progression of the disease as a multistate process should be modeled. On the other hand, in such studies, multivariate longitudinal outcomes may be collected and their association with the survival process is of interest. In the present study, we applied a joint model of various longitudinal biomarkers and transitions between different health statuses in patients who underwent renal transplantation. The full joint likelihood approaches are faced with the complexities in computation of the likelihood. So, here, we have proposed two-stage modeling of multivariate longitudinal outcomes and multistate conditions to avoid these complexities. The proposed model showed reliable results compared to the joint model in case of joint modeling of univariate longitudinal biomarker and the multistate process.

Adjusted Extreme Conditional Quantile Autoregression with Application to Risk Measurement

In this paper, we propose an extreme conditional quantile estimator. Derivation of the estimator is based on extreme quantile autoregression. A noncrossing restriction is added during estimation to avert possible quantile crossing. Consistency of the estimator is derived, and simulation results to support its validity are also presented. Using Average Root Mean Squared Error (ARMSE), we compare the performance of our estimator with the performances of two existing extreme conditional quantile estimators. Backtest results of the one-day-ahead conditional Value at Risk forecasts are also given.