Pakistan Journal of Statistics and Operation Research
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Characterizations of Certain (2023-2024) Introduced Univariate Continuous Distributions II
No full textThis paper is a continuation of our previous work with the same title, which deals with various characterizations of certain univariate continuous distributions proposed in (2023-2024) after the publication of our first paper in (2024). These characterizations are based on: (i) a simple relationship between two truncated moments; (ii) the hazard function; (iii) reverse hazard function and (iv) conditional expectation of a single function of the random variable. It should be mentioned that for the characterization (i) the cumulative distribution function need not have a closed form and depends on the solution of a first order differential equation, which provides a bridge between probability and differential equation
A Flexible Discrete Rayleigh-G Family for Engineering and Reliability Modeling: Properties, Characterizations, Bayesian and Non-Bayesian Inference
No full textA novel flexible probability tool for modeling extreme and zero-inflated count data with various hazard rate shapes is introduced in this work. Numerous pertinent statistical and mathematical features are developed and examined. Some important mathematical features are obtained, including, ordinary moments, central moment, dispersion index, L-moments, cumulant generating function and moment generating function. A specific example is investigated numerically and visually examined. The new class of hazard rate function offers a broad range of flexibility, including "monotonically decreasing," "upside down," "monotonically increasing," "constant," "decreasing-constant," and "decreasing-constant-increasing (U-hazard rate function)". Furthermore, the new mass function accommodates many useful forms in the field of modeling, including the "right skewed with one peak", "right skewed with two peaks (right skewed and bimodal)", "symmetric mass function" "left skewed with one peak". The conditional expectation of a certain function of the random variable as well as the hazard function are used to provide relevant characterization results. For the estimation process, evaluating and comparing inferential effectiveness, Bayesian and non-Bayesian estimation approaches are taken into consideration. We propose and explain the Bayesian estimation method for the squared error loss function. For comparing non-Bayesian versus Bayesian estimates, Markov chain Monte Carlo simulation experiments are carried out using the Metropolis Hastings algorithm and the Gibbs sampler. The Bayesian and non-Bayesian approaches are compared using four real-life applications of count data sets. By using four additional real count data applications, the significance and adaptability of the new discrete class are demonstrated
Addressing the Autocorrelation Problem in the Poisson Regression Model: Theory and Numerical Illustrations
No full textThe Poisson regression model (PRM) is usually applied in the situations where the dependent variable is in the form of count data. The purpose of this study is to compare methods of estimation for the Poisson Regression Model's first-order autocorrelation (AR(1)). The Kibria and Lukman Estimator Method (KL), Generalized Least Square Estimator Method (GLS), the Liu Estimator Method (LE), and the Reduction Liu Estimator Method (RLE) were employed. Monte Carlo simulations are used to compare these methods. The data generated follows Poisson Regression Model, however because of sample size and autocorrelation levels among other things, to create first-order autocorrelation among random errors. The Mean square Error (MSE) criterion was used for comparison. The methods are also evaluated on actual data, Moreover, the findings demonstrated that the KL approach is superior to the other estimation techniques in terms of its performance
The Statistical Distributional Validation under a Novel Generalized Gamma Distribution with Value-at-Risk Analysis for the Historical Claims, Censored and Uncensored Real-life Applications
No full textThis study introduces and examines a new probability distribution, presenting various characterizations. Key financial risk measures, including the value-at-risk (VaR), tail-value-at-risk (TVaR), also referred to as conditional tail expectation or conditional value-at-risk (CVaR), tail variance (TV), tail mean-variance (TMV), and mean excess loss (MExL) function are evaluated using maximum likelihood estimation, ordinary least squares, weighted least squares, and the Anderson-Darling estimation methods. These methods are applied for actuarial analysis in both a simulation study and an insurance claims data application. For validation of the distribution using complete data, the widely recognized Nikulin-Rao-Robson statistic is utilized and assessed through simulations and three real data sets. Two uncensored real-life data sets for failure times and remission times are used in uncensored validation. Additionally, for censored data validation, a modified version of the Nikulin-Rao-Robson statistic is proposed and evaluated through extensive simulations and three censored real data sets
A New Version of the Compound Quasi-Lomax Model: Properties, Characterizations and Risk Analysis under the U.K. Motor Insurance Claims Data
No full textThis paper introduces a new lifetime distribution, the Compound Quasi-Lomax (CQLx) model, designed to enhance the modeling of heavy-tailed data in actuarial and financial risk analysis. The CQLx distribution is developed through a novel extension of the Lomax family, offering increased flexibility in capturing extreme values and complex data behaviors. Key mathematical properties are derived. Characterization of the model is achieved via truncated moments and the reverse hazard function. Several estimation methods are employed including the Maximum Likelihood Estimation (MLE), Cramér–von Mises (CVM), Anderson–Darling Estimation (ADE), Right-Tail Anderson-Darling Estimation (RTADE), and Left-Tail Anderson-Darling Estimation (LTADE). A comprehensive simulation study evaluates the performance of these methods in terms of bias and root mean square error (RMSE) across various sample sizes. Risk measures such as Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Tail Variance (TV), Tail Mean Variance (TMV), and Expected Loss (EL) are computed under artificial and real financial insurance claims data. The results demonstrate that MLE generally provides the most accurate and stable estimates, particularly for larger samples, while CVM and ADE tend to overestimate risk, especially at higher quantiles. The CQLx model shows superior performance in fitting extreme claim-size data, making it a robust tool for risk management
The New Topp-Leone-Heavy-Tailed Type II Exponentiated Half Logistic-G Family of Distributions: Properties, Actuarial Measures, with Applications to Censored Data
No full textThe Topp-Leone heavy-tailed type II exponentiated half logistic-G (TL-HT-TIIEHL-G) is the newly proposed familyof distributions (FoDs) introduced in this research. The study thoroughly investigates the statistical properties ofthis FoDs, as well as its relevance in actuarial risk assessment. The estimation of the unknown model parameters is done using the method of maximum likelihood estimation, and the consistency of these estimates is assessed through the implementation of Monte Carlo simulations. Additionally, numerical simulations are conducted to analyze the risk measures associated with the TL-HT-TIIEHL-G FoDs. The Topp-Leone heavy-tailed type II exponentiated half logistic-Weibull (TL-HT-TIIEHL-W) distribution, a particular case of the TL-HT-TIIEHL-G FoDs is compared with other contending distributions including heavy-tailed distributions to evaluate its performance. The model’s capacity, adaptability, and practicality are convincingly showcased through its application to real data
Enhancing Food Security Analysis in South Sulawesi Using Robust Mixed Geographically and Temporally Weighted Regression with M-Estimator
No full textMGTWR (Mixed Geographically and Temporally Weighted Regression) combines a global linear regression model with GTWR by incorporating spatial and temporal dimensions. However, it remains sensitive to outliers, which can reduce accuracy. To address this limitation, a robust regression approach with the M-Estimator was applied to model the food security index in South Sulawesi Province from 2018 to 2022. The resulting Robust MGTWR (RMGTWR) model demonstrated improved performance, with a lower AIC ( ) and a high explanatory power ( ). Key factors influencing food security include the ratio of normative consumption per capita to net production, the percentage of households with a proportion of expenditure on food more significant than 65% of total spending, the percentage of households without access to electricity, the percentage of households without access to clean water, and the percentage of stunting toddlers. These findings highlight the effectiveness of RMGTWR with M-Estimator in addressing data irregularities and provide valuable insights for policymakers in designing targeted strategies to strengthen food security in South Sulawesi Province
L−moment Based Regional Frequency Analysis of Annual Maximum Relative Humidity Across Pakistan
No full textClimate change has significantly impacted regional weather patterns in Pakistan, leading to an increase in the frequency and intensity of extreme weather events, such as heatwaves, droughts, and floods. This study aimed to investigate the regional frequency analysis of extreme relative humidity across Pakistan, identify potential geographically related trends, and estimate possible frequencies associated with extreme relative humidity in distinct regions. The data were collected from 24 meteorological stations. Discordance measures were used to evaluate distinct sites in the region, with GAWADAR and KARACHI exhibiting the highest discordance values. The study region was divided into three homogeneous regions based on geographic and statistical measures to verify the heterogeneity statistics. The L−moment ratio diagram and goodness−of−fit statistic identified suitable distributions for each region, with the generalized extreme value and Pearson type III distributions proving to be most effective in the first region. The second region was best represented by the generalized extreme value and Weibull distributions, whereas the third region was most accurately characterized by the generalized log-normal and polynomial density−quantile III distributions. These results offer valuable insights into the spatial patterns of humidity extremes, potentially supporting efforts in climate change adaptation and flood risk management
The Flexible Nadarajah–Haghighi Distribution: Properties, Inference, and Applications
No full textThis article explores a flexible extension of the Nadarajah–Haghighi (NH) model, referred to as the odd inverse Pareto Nadarajah–Haghighi (OIPNH) distribution. We derive the mathematical properties of the probability density function of the OIPNH distribution, which exhibits a variety of behavior shapes, including decreasing, increasing, J-shaped, reversed J-shaped, bathtub, upside-down bathtub, and decreasing-increasing-decreasing hazard rates. Additionally, the distribution can display right-skewed, symmetrical, and concave-down densities. The parameters of the OIPNH distribution are examined using eight classical estimation approaches. We present extensive simulation results to evaluate the performance of these methods for both small and large sample sizes. Furthermore, we analyze three real-life datasets from engineering, medicine and agricultural sciences, demonstrating the flexibility of the OIPNH distribution compared to existing NH distributions
The Negative Binomial-Bilal Distribution: Regression Model and Applications to Health Care Data
No full textIn health care research, overdispersion often arises in count data. The Poisson distribution is a traditional distribution for modeling count data. However, it cannot handle overdispersed count data. This article introduces a new count distribution for overdispersed data. Statistical properties and a multivariate version of the proposed distribution are derived. Two parameter estimation methods are discussed by the maximum likelihood method and Bayesian approach. A simulation study is conducted to assess the performance of the estimators. A regression model based on the proposed distribution is constructed. Finally, two health care applications are analyzed to show the potential of the proposed distribution and its associated regression model