Search Results (11)

Insurance loss data are usually in the form of left-truncation and right-censoring due to deductibles and policy limits, respectively. This paper investigates the model uncertainty and selection procedure when various parametric models are constructed to accommodate such left-truncated and right-censored data. The joint asymptotic properties of the estimators have been established using the Delta method along with Maximum Likelihood Estimation when the model is specified. We conduct the simulation studies using Fisk, Lognormal, Lomax, Paralogistic, and Weibull distributions with various proportions of loss data below deductibles and above policy limits. A variety of graphic tools, hypothesis tests, and penalized likelihood criteria are employed to validate the models, and their performances on the model selection are evaluated through the probability of each parent distribution being correctly selected. The effectiveness of each tool on model selection is also illustrated using well-studied data that represent Wisconsin property losses in the United States from 2007 to 2010. Full article

The coefficient of quartile variation is a valuable measure used to assess data dispersion when it deviates from a normal distribution or displays skewness. In this study, we focus specifically on the delta-lognormal distribution. The lognormal distribution is characterized by its asymmetrical nature and comprises exclusively positive values. However, when these values undergo a logarithmic transformation, they conform to a symmetrical (or normal) distribution. Consequently, this research aims to establish confidence intervals for the difference between coefficients of quartile variation within lognormal distributions incorporating zero values. We employ the Bayesian, generalized confidence interval, and fiducial generalized confidence interval methods to construct these intervals, involving data simulation using RStudio software. We evaluate the performance of these methods based on coverage probabilities and average lengths. Our findings indicate that the Bayesian method, employing Jeffreys’ prior, performs well in low variability, while the generalized confidence interval method is more suitable for higher variability. Therefore, we recommend using both approaches to construct confidence intervals for the difference between the coefficients of the quartile variation in lognormal distributions that include zero values. Furthermore, we apply these methods to rainfall data in Thailand to illustrate their alignment with actual and simulated data. Full article

A delta-lognormal distribution consists of zero and positive values. The positive values follow a lognormal distribution, which is an asymmetric distribution. It is well known that the logarithm of these values follows a normal distribution, which is a symmetric distribution. The delta-lognormal distribution is used in medical and environmental sciences. This study considers the challenges of constructing confidence intervals for the mean and difference between means of delta-lognormal distributions containing left-censored data and applies them to compare two daily rainfall average areas in Thailand. Three different approaches for constructing confidence intervals for the mean of the delta-lognormal distribution containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, and the parametric bootstrap approach, are developed. Moreover, four different approaches for constructing confidence intervals for the difference between means of delta-lognormal distributions containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, the parametric bootstrap approach, and the method of variance estimates recovery approach, are considered. The performance of the proposed confidence intervals is evaluated by Monte Carlo simulation. The simulation studies indicate that the Bayesian approach can be considered as an alternative to construct a credible interval for the mean of the delta-lognormal distribution containing left-censored data. Additionally, the generalized confidence interval and Bayesian approaches can be recommended as alternatives to estimate the confidence interval for the difference between means of delta-lognormal distributions containing left-censored data. All approaches are illustrated using the daily rainfall data from Chiang Mai and Lampang provinces in Thailand. Full article

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently observed in the fields of environmental studies, biology, and medicine. The number of zero observations follows a binomial distribution. However, problems can arise in the analysis of data containing zero observations along with log-normally distributed data, for which the delta-lognormal distribution is often referred to for using the analysis of the data. In statistics, the percentile provides the relative standing of a numerical data point when compared to all of the others in a distribution with reference to the observations at or below it. In this study, estimates for the confidence interval for the ratio of the percentiles of two delta-lognormal distributions are constructed using fiducial generalized confidence interval approaches based on the fiducial quantity and the optimal generalized fiducial quantity, the Bayesian approach, and the parametric bootstrap method. As assessed by Monte Carlo simulations using the RStudio programming in terms of the coverage probability and the average length, the Bayesian approach performed quite well by providing adequate coverage probabilities along with the shortest average lengths in all of the scenarios tested. Daily rainfall data contain both zero and positive values. The daily rainfall data can usually be fitted to the delta-lognormal distribution. Their application to rainfall data is also provided to illustrate their efficacies with real data. The efficacy of the approach is used to compare two rainfall dispersion populations. Full article

Information on the relationship between rainfall intensity, duration and accumulation frequency or return period (IDF) is commonly utilized in the design and management of urban drainage systems. Can Tho City, located in the Vietnamese Mekong Delta, is a city which has recently invested heavily in upgrading its stormwater drainage systems in the hope of preventing reoccurring flood events. Yet, much of these works were designed based on obsolete and outdated IDF rainfall curves. This paper presents an updated IDF curve for design rainfall for Can Tho City. For each duration and designated return period, a cumulative distribution function (CDF) was developed using the Pearson III, Log-Pearson III, and Log-Normal distribution functions. In order to choose the best IDF rainfall curve for Can Tho City, the CDF rainfall curve and empirical formulas used in Vietnam and Asia (Vietnamese standard 7957:2008, Department of Hydrology, Ministry of Transportation, Talbot, Kimijima, and Bermard) were compared. The goodness of fit between the IDF relationship generated by the frequency analysis (CDF curve), and that predicted by the IDF empirical formulas was assessed using the efficiency index (EI), and the root mean squared error (RMSE). The IDF built from Vietnam’s standard TCVN 7957:2008 with new parameters (A = 9594, C = 0.5, b = 26, n = 0.96) showed the best performance, with the highest values of EI (0.84 $\le$EI$\le$ 0.93) and the lowest values of RMSE (2.5 $\le$RMSE$\le$ 3.2), when compared to the other remnants. Full article

Epidemiological distributions of the coronavirus disease 2019 (COVID-19), including the intervals from symptom onset to diagnosis, reporting, or death, are important for developing effective disease-control strategies. COVID-19 case data (from 19 January 2020 to 10 January 2022) from a national database maintained by the Korea Disease Control and Prevention Agency and the Central Disease Control Headquarters were analyzed. A joint Bayesian subnational model with partial pooling was used and yielded probability distribution models of key epidemiological distributions in Korea. Serial intervals from before and during the Delta variant’s predominance were estimated. Although the mean symptom-onset-to-report interval was 3.2 days at the national level, it varied across different regions (2.9–4.0 days). Gamma distribution showed the best fit for the onset-to-death interval (with heterogeneity in age, sex, and comorbidities) and the reporting-to-death interval. Log-normal distribution was optimal for ascertaining the onset-to-diagnosis and onset-to-report intervals. Serial interval (days) was shorter before the Delta variant-induced outbreaks than during the Delta variant’s predominance (4.4 vs. 5.2 days), indicating the higher transmission potential of the Delta variant. The identified heterogeneity in region-, age-, sex-, and period-based distributions of the transmission dynamics of COVID-19 will facilitate the development of effective interventions and disease-control strategies. Full article

We present the Generalized Gamma (GG) distribution as a possible risk neutral distribution (RND) for modeling European options prices under Heston’s stochastic volatility (SV) model. We demonstrate that under a particular reparametrization, this distribution, which is a member of the scale-parameter family of distributions with the mean being the forward spot price, satisfies Heston’s solution and hence could be used for the direct risk-neutral valuation of the option price under Heston’s SV model. Indeed, this distribution is especially useful in situations in which the spot’s price follows a negatively skewed distribution for which Black–Scholes-based (i.e., the log-normal distribution) modeling is largely inapt. We illustrate the applicability of the GG distribution as an RND by modeling market option data on three large market-index exchange-traded funds (ETF), namely the SPY, IWM and QQQ as well as on the TLT (an ETF that tracks an index of long-term US Treasury bonds). As of the writing of this paper (August 2021), the option chain of each of the three market-index ETFs shows a pronounced skew of their volatility ‘smile’, which indicates a likely distortion in the Black–Scholes modeling of such option data. Reflective of entirely different market expectations, this distortion in the volatility ‘smile’ appears not to exist in the TLT option data. We provide a thorough modeling of the option data we have on each ETF (with the 15 October 2021 expiration) based on the GG distribution and compare it to the option pricing and RND modeling obtained directly from a well-calibrated Heston’s SV model (both theoretically and also empirically, using Monte Carlo simulations of the spot’s price). All three market-index ETFs exhibited negatively skewed distributions, which are well-matched with those derived under the GG distribution as RND. The inadequacy of the Black–Scholes modeling in such instances, which involves negatively skewed distribution, is further illustrated by its impact on the hedging factor, delta, and the immediate implications to the retail trader. Similarly, the closely related Inverse Generalized Gamma distribution (IGG) is also proposed as a possible RND for Heston’s SV model in situations involving positively skewed distribution. In all, utilizing the Generalized Gamma distributions as possible RNDs for direct option valuations under the Heston’s SV is seen as particularly useful to the retail traders who do not have the numerical tools or the know-how to fine-calibrate this SV model. Full article

We aimed to elucidate the range of the incubation period in patients infected with the SARS-CoV-2 Omicron variant in comparison with the Alpha variant. Contact tracing data from three Japanese public health centers (total residents, 1.06 million) collected following the guidelines of the Infectious Diseases Control Law were reviewed for 1589 PCR-confirmed COVID-19 cases diagnosed in January 2022. We identified 77 eligible symptomatic patients for whom the date and setting of transmission were known, in the absence of any other probable routes of transmission. The observed incubation period was 3.03 ± 1.35 days (mean ± SDM). In the log-normal distribution, 5th, 50th and 95th percentile values were 1.3 days (95% CI: 1.0–1.6), 2.8 days (2.5–3.1) and 5.8 days (4.8–7.5), significantly shorter than among the 51 patients with the Alpha variant diagnosed in April and May in 2021 (4.94 days ± 2.19, 2.1 days (1.5–2.7), 4.5 days (4.0–5.1) and 9.6 days (7.4–13.0), p < 0.001). As this incubation period, mainly of sublineage BA.1, is even shorter than that in the Delta variant, it is thought to partially explain the variant replacement occurring in late 2021 to early 2022 in many countries. Full article

Few studies have assessed incubation periods of the severe acute respiratory syndrome coronavirus 2 Delta variant. This study aimed to elucidate the transmission dynamics, especially the incubation period, for the Delta variant compared with non-Delta strains. We studied unvaccinated coronavirus disease 2019 patients with definite single exposure date from August 2020 to September 2021 in Japan. The incubation periods were calculated and compared by Mann–Whitney U test for Delta (with L452R mutation) and non-Delta cases. We estimated mean and percentiles of incubation period by fitting parametric distribution to data in the Bayesian statistical framework. We enrolled 214 patients (121 Delta and 103 non-Delta cases) with one specific date of exposure to the virus. The mean incubation period was 3.7 days and 4.9 days for Delta and non-Delta cases, respectively (p-value = 0.000). When lognormal distributions were fitted, the estimated mean incubation periods were 3.7 (95% credible interval (CI) 3.4–4.0) and 5.0 (95% CI 4.5–5.6) days for Delta and non-Delta cases, respectively. The estimated 97.5th percentile of incubation period was 6.9 (95% CI 5.9–8.0) days and 10.4 (95% CI 8.6–12.7) days for Delta and non–Delta cases, respectively. Unvaccinated Delta variant cases had shorter incubation periods than non–Delta variant cases. Full article

Industrial wastewater and sewage are both important sources of heavy metals and metalloids in urban wastewater treatment plants (WWTPs). China has made great efforts to control heavy metal and metalloid pollution by setting discharge limits for WWTPs. There is, however, limited discharge data and no systematic methodology for the derivation of discharge limits. In this study, 14 heavy metals and metalloids (Hg, alkyl mercury, As, Cd, Cr, Cr⁶⁺, Pb, Ni, Be, Ag, Cu, Zn, Mn, Se) that are listed in the Discharge Standard of Pollutants for Municipal Wastewater Treatment Plant (GB 18918-2002) were selected for the analysis of discharge characteristics while using the supervised monitoring data from more than 800 WWTPs located in nine provinces in China. Of the 14 heavy metals and metalloids, all but alkyl mercury were detected in the discharge water. There was a high rate of detection of As, Cu, Mn, Se, and there were some samples that exceeded the standard concentrations of Cr, Cr⁶⁺, Pb, and Ni. Removal rates of Hg, As, Cd, Cr, Cr⁶⁺, Pb, Ni, Cu, Zn, Mn, and Se were higher than 40%, comparable to values from other countries. Hg and As were selected to analyze the influencing factors of effluent and derive discharge limits of WWTPs using a statistical method, because these two metals had more detected data than other metals. The study used supervised monitoring data from Zhejiang WWTPs with 99 for Hg and 112 for As. Based on the delta-lognormal distribution, the results showed that geographic location was significantly closely correlated with Hg (P = 0.027 < 0.05) and As (P ≈ 0 < 0.05) discharge concentrations, while size (for Hg P = 0.695 > 0.05, for As P = 0.088 > 0.05) and influent concentration (R² < 0.5) were not. Derived Hg and As discharge limits suggest that it is necessary to establish stricter discharge limits for WWTPs, which is more consistent with the real-world situation in China. The study here comprehensively researches the discharge characteristics of heavy metals and metalloids in effluent of WWTPs in China, and developed for the first time in China heavy metals and metalloids discharge limits based on statistical methods. The results may inform special discharge limit settings for WWTPs in China. Full article

Trace elements pollution has attracted a lot of attention worldwide. However, it is difficult to identify and apportion the sources of multiple element pollutants over large areas because of the considerable spatial complexity and variability in the distribution of trace elements in soil. In this study, we collected total of 2051 topsoil (0–20 cm) samples, and analyzed the general pollution status of soils from the Yangtze River Delta, Southeast China. We applied principal component analysis (PCA), a finite mixture distribution model (FMDM), and geostatistical tools to identify and quantitatively apportion the sources of seven kinds of trace elements (chromium (Cr), cadmium (Cd), mercury (Hg), copper (Cu), zinc (Zn), nickel (Ni), and arsenic (As)) in soil. The PCA results indicated that the trace elements in soil in the study area were mainly from natural, multi-pollutant and industrial sources. The FMDM also fitted three sub log-normal distributions. The results from the two models were quite similar: Cr, As, and Ni were mainly from natural sources caused by parent material weathering; Cd, Cu, and Zu were mainly from mixed sources, with a considerable portion from anthropogenic activities such as traffic pollutants, domestic garbage, and agricultural inputs, and Hg was mainly from industrial wastes and pollutants. Full article