Model Comparison and Parameter Estimation for Gompertz Distributions Under Constant Stress Accelerated Lifetime Tests

Aim of study: The aim of this work was to model diameter distributions of Quercus suber stands. The ultimate goal was to construct models enabling the development of more affordable forest inventory methods. This is the first study of this type on cork oak forests in the area.Area of study: The area of study is “Los Alcornocales” Natural Park (Cádiz-Málaga, Spain).Material and methods: The diameter distributions of 100 permanent plots were modelled with the two-parameter Weibull function. Distribution parameters were fitted with the non-linear regression, maximum likelihood, moment and percentile-based methods. Goodness of fit with the different methods was compared in terms of number of plots rejected by the Kolmogorov-Smirnov test, bias, mean square error and mean absolute error. The scale and shape parameters in the Weibull function were related to the stand variables by using the parameter prediction model.Main results: The best fitting was obtained with the non-linear regression approach, using as initial values those obtained by maximum likelihood method, the percentage of rejections by the Kolmogorov-Smirnov test was 2% of the total number of cases. The scale parameter (b) was successfully modelled in terms of the quadratic mean diameter under cork (R2 adj = 0.99). The shape parameter (c) was modelled by using maximum diameter, minimum diameter and plot elevation (R2 adj = 0.40).Research highlights: The proposed model diameter distribution can be a highly useful tool for the inventorying and management of cork oak forests.Key words: maximum likelihood method; moment method; non linear regression approach; parameter prediction model; percentile method; scale parameter; shape parameter.

How the subjective value of an outcome changes as a function of time, probability, or effort has been an active area of psychological and economic research for decades. The exact functional form of how a commodity is discounted has been debated, and there have been numerous forms proposed. One of the challenges when trying to determine the functional form of discounting data is how models are compared, what modeling methods are used, how many data points are used, and what comparison metrics were used. Thus, we sought to replicate and extend previous research comparing discounting model selection methods by simulating discounting data from five functional forms: the Mazur hyperbolic model (Mazur, 1987), Rachlin hyperboloid (Rachlin, 2006), Myerson-Green hyperboloid (Myerson & Green, 1995), Samuelson exponential model (Samuelson, 1937), and beta-delta model (Laibson, 1997). With each of these models we manipulated the number (i.e., density) of data points, used two forms of modeling, and assessed the degree to which each model generalizes to data it has not used in the fitting process. Model comparisons were conducted using the Akaike information criterion (AIC), Bayesian information criterion (BIC), and leave-one-out cross validation (LOOCV). In general, AIC, BIC, and LOOCV selected the correct model, whereas the Rachlin model had the lowest error across folds of LOOCV when relying on multilevel modeling.

The age and growth information of fish is important for assessment of the dynamics, planning, and management in fisheries. In this study, the mean size at age data of 10 fish species collected from literature were fitted by the five growth models and the growth parameters were estimated by the maximum likelihood method. The best model was selected using the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The growth of most species can be best described by the von Bertalanffy and allometric growth models. The results showed that both AIC and BIC have their advantages in the testing of significance of the difference between the functions of models. As for the demersal fishes (snappers and groupers), BIC is better than AIC in selecting the best growth model.

Introduction and objectives Cough frequency is known to correlate well with perceived cough severity and is commonly used as an endpoint to evaluate the efficacy of new anti-tussive agents. However, little is known about the distribution of cough and its impact on cough severity. The Weibull distribution is a probability distribution model often used to predict time between events, as applied in COPD exacerbations. The shape parameter describes if coughs are more or less likely to occur over time. We investigated if cough temporal distribution is modulated by an effective anti-tussive agent (morphine) and how this distribution influences perceived cough severity. Methods Cough recordings, made using the VITALOJAK™ cough monitoring system were analysed from 22 chronic cough patients (18 female, mean age 61.7 years, mean cough duration 14 years) who had participated in a previously reported randomised controlled trial of low dose morphine sulphate. Coughs were identified and the cough intervals recorded and fit to an exponential (random) and Weibull distribution. The goodness of fit was checked with the Akaike information criterion (AIC). The shape parameter was calculated and using generalised estimating equations compared for morphine versus placebo. Relationships between cough severity visual analogue scale (VAS), cough frequency and the shape parameter were also explored. Results The AIC indicated that cough distribution better fitted the Weibull equation than the exponential in 38/41 (93%) recordings. The Weibull shape parameter was significantly reduced by morphine compared with placebo (geometric mean 0.418 versus 0.548, p=0.04), suggesting more coughs occurred early, with a fall in the rate of subsequent coughs over time with morphine. Furthermore, cough severity VAS was significantly predicted by a combination of cough frequency (p Conclusions In chronic cough patients, low dose morphine, in addition to reducing cough frequency may also reduce the temporal clustering of cough events. This may be important as temporal clustering appears to be a component of perceived cough severity, independent of cough frequency.

Modeling the distribution of DCT coefficients for JPEG reconstruction

The ranked set sampling (RSS) is a cost-effective method of sampling that can be used in a wide range of statistical problems. In this paper, the shape and the scale parameters of Nadarajah-Haghighi extension of the exponential distribution are estimated based on a simple random sample (SRS) and RSS. Three cases are considered: 1) the scale parameter is known; 2) the shape parameter is known; 3) both shape and scale parameters are unknown. Observations are done when the ranking mechanism in the ranked set sample is perfect and when it is not. Method of moments, the maximum likelihood method, and a modification of the maximum likelihood method are used. The obtained estimators are compared in terms of their biases and mean square errors (MSE). The results revealed that estimators based on RSS tend to show better properties (smaller bias and MSE) relative to their SRS counterparts, regardless of the quality of the ranking.

Kenya is among the countries that are continuously investing in wind energy to meet her electricity demand. Kenya is working towards its vision 2030 of achieving a total of 2GW of energy from wind industry. To achieve this, there is a need that all the relevant data on wind characteristics must be available. The purpose of this study is, therefore, to find the most efficient two-parameter model for fitting wind speed distribution for Narok County in Kenya, using the maximum likelihood method. Hourly wind speed data collected for a period of three years (2016 to 2018) from five sites within Narok County was used. Each of the distribution’s parameters was estimated and then a suitability test of the parameters was conducted using the goodness of fit test statistics, Kolmogorov-Smirnov, and Anderson-Darling. An efficiency test was determined using the Akaike’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC) values, with the best decision taken based on the distribution having a smaller value of AIC and BIC. The results showed that the best distributions were the gamma distribution with the shape parameter of 2.47634 and scale parameter of 1.25991, implying that gamma distribution was the best distribution for modeling Narok County wind speed data.

Medication adherence after myocardial infarction remains low. Pharmacy claims have typically been used to measure medication adherence, but electronic pill bottles may offer additional information. The main objectives of this study were to compare the association of adherence measured by prescription claims and remote monitoring technologies with cardiovascular events. This study was a secondary analysis of a remote monitoring intervention to increase medication adherence in myocardial infarction patients. In total, 682 myocardial infarction patients were randomized to the intervention group with both medical and pharmacy benefits. Pharmacy claims adherence was measured using proportion of days covered (PDC) and GlowCap adherence (GC) was measured as the proportion of days the pill bottle was opened. We compared the association of PDC and GC adherence for statins with time to first vascular readmission or death and assessed model fit using Akaike information criterion and Bayesian information criterion and the likelihood ratio test. Higher PDC was significantly associated with a lower hazard rate for vascular readmissions or death (hazard ratio=0.435; P=0.009). There was also an association between GC adherence and vascular readmissions or death (hazard ratio=0.313; P≤0.001). Adding the GC adherence variable to the model using only PDC improved the model fit (likelihood ratio test, P=0.001), as well as vice versa (P=0.050). Pharmacy claims data provide useful but not complete data for medication adherence monitoring. New wireless technologies have the potential to provide additional data about clinical outcomes.

Accelerated life testing is a method for estimating the reliability of products at normal use conditions from the failure data obtained at the severe conditions. In this paper, constant-stress accelerated life testing is considered when the lifetime of a product follows a two- parameter Birnbaum-Saunders distribution. Based on type-II censoring. A certain conditions (stress) are assumed to affect the scale parameter through the inverse power law model. Maximum likelihood estimators (MLEs) of an accelerated life tests model are derived. Prediction of the scale parameter under usual conditions is obtained. Moreover, the reliability function at a certain mission time under the same conditions is also predicted. Simulation results are carried out to study the precision of the MLEs for the parameters involved.

Current estimates of the European windstorm climate and their associated losses are often hampered by either relatively short, coarse resolution or inhomogeneous datasets. This study tries to overcome some of these shortcomings by estimating the European windstorm climate using dynamical seasonal-to-decadal (s2d) climate forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF). The current s2d models have limited predictive skill of European storminess, making the ensemble forecasts ergodic samples on which to build pseudoclimates of 310–396 yr in length. Extended winter (October–April) windstorm climatologies are created using scalar extreme wind indices considering only data above a high threshold. The method identifies up to 2363 windstorms in s2d data and up to 380 windstorms in the 40-yr ECMWF Re-Analysis (ERA-40). Classical extreme value analysis (EVA) techniques are used to determine the windstorm climatologies. Differences between the ERA-40 and s2d windstorm climatologies require the application of calibration techniques to result in meaningful comparisons. Using a combined dynamical–statistical sampling technique, the largest influence on ERA-40 return period (RP) uncertainties is the sampling variability associated with only 45 seasons of storms. However, both maximum likelihood (ML) and L-moments (LM) methods of fitting a generalized Pareto distribution result in biased parameters and biased RP at sample sizes typically obtained from 45 seasons of reanalysis data. The authors correct the bias in the ML and LM methods and find that the ML-based ERA-40 climatology overestimates the RP of windstorms with RPs between 10 and 300 yr and underestimates the RP of windstorms with RPs greater than 300 yr. A 50-yr event in ERA-40 is approximately a 40-yr event after bias correction. Biases in the LM method result in higher RPs after bias correction although they are small when compared with those of the ML method. The climatologies are linked to the Swiss Reinsurance Company (Swiss Re) European windstorm loss model. New estimates of the risk of loss are compared with those from historical and stochastically generated windstorm fields used by Swiss Re. The resulting loss-frequency relationship matches well with the two independently modeled estimates and clearly demonstrates the added value by using alternative data and methods, as proposed in this study, to estimate the RP of high RP losses.

Recently, in order to get closer agreement at the extremes, skewed distributions are playing an important role in various research studies. The generalized logistic distribution (GLD) of Type II, which is indexed by one shape parameter, is introduced here to extend the scope of this distribution in some asymmetrical studies. Several properties of this distribution in relation to other probability distributions are stated. Furthermore, the maximum-likelihood (ML) method and an approximate ML method are used to derive the point estimators of the parameters based on progressive Type II censoring. A wide range of sample sizes and progressive-censoring schemes are considered in a simulation study to see the performance of estimates of location and scale parameters of the Type II GLD. The coverages probability of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. To improve the coverage probabilities, we suggest the use of unconditional simulated percentage points for the construction of confidence intervals. Two numerical examples are presented to illustrate the methods of estimation discussed here.

This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape parameters. First, the Newton-Raphson method is used to derive the maximum likelihood estimators (MLEs), and the existence and uniqueness of the estimators are also demonstrated. We used the stochastic expectation maximization (SEM) method to construct MLEs for unknown parameters, which simplified and facilitated computation. Based on the asymptotic normality of the MLEs and SEM methods, we create the corresponding confidence intervals for unknown parameters, and the delta approach is utilized to obtain the interval estimation of the reliability function. Additionally, using two bootstrap techniques, the approximative interval estimators for all unknowns are created. Furthermore, we computed the Bayes estimates of unknown parameters as well as the survival function using the Markov chain Monte Carlo (MCMC) method in the presence of square error and LINEX loss functions. Finally, we look into two real data sets and create a simulation study to evaluate the efficacy of the established approaches.

Item response theory (IRT) models provide an appropriate alternative to the classical ordinal confirmatory factor analysis (CFA) during the development of patient-reported outcome measures (PROMs). Current literature has identified the assessment of IRT model fit as both challenging and underdeveloped (Sinharay & Johnson, 2003; Sinharay, Johnson, & Stern, 2006). This study evaluates the performance of Ordinal Bayesian Instrument Development (OBID), a Bayesian IRT model with a probit link function approach, through applications in two breast cancer-related instrument development studies. The primary focus is to investigate an appropriate method for comparing Bayesian IRT models in PROMs development. An exact Bayesian leave-one-out cross-validation (LOO-CV) approach (Vehtari & Lampinen, 2002) is implemented to assess prior selection for the item discrimination parameter in the IRT model and subject content experts’ bias (in a statistical sense and not to be confused with psychometric bias as in differential item functioning) toward the estimation of item-to-domain correlations. Results support the utilization of content subject experts’ information in establishing evidence for construct validity when sample size is small. However, the incorporation of subject experts’ content information in the OBID approach can be sensitive to the level of expertise of the recruited experts. More stringent efforts need to be invested in the appropriate selection of subject experts to efficiently use the OBID approach and reduce potential bias during PROMs development.

Quantifying inequality in the leaf area distribution within a single module is critical for elucidating plant resource allocation strategies, but the accuracy of theoretical Gini coefficients derived from statistical distributions remains poorly validated against observed values. To resolve this gap, we analyzed 9,242 leaves from 121 culms of the bamboo Semiarundinaria densiflora, a model system with minimal ontogenetic noise and moderate leaf counts (36–187 leaves per culm) that enables robust Lorenz curve construction. Four candidate distributions were tested: the normal, log-normal, two-parameter Gamma, and two-parameter Weibull distributions. The parameters of the normal and log-normal distributions were estimated directly from sample statistics, whereas the parameters of the Gamma and Weibull distributions were estimated using the maximum likelihood method. Goodness of fit was assessed using the Kolmogorov-Smirnov (K-S) test for distributional validity, and the Akaike’s information criterion (AIC) for model selection. Although the Gamma distribution passed the K-S test for a slightly higher percentage of culms (99%) than the Weibull distribution (97.5%), the Weibull distribution was selected as the superior model because it yielded significantly lower AIC values. Crucially, the theoretical Gini coefficients of the Gamma and Weibull distributions (denoted as GG and GW, respectively) were tested against the observed Gini coefficients (GP) calculated nonparametrically using the polygon method. Linear regression demonstrated that GW​ predicted GP with near isometric accuracy: the intercept’s 95% confidence interval included zero (−0.006 to 0.017) and the slope’s 95% confidence interval included unity (0.929 to 1.039). In contrast, GG exhibited significant bias. Notably, pooling leaves across culms violated all distributions due to microhabitat driven multimodality, confirming that intra-culm inequality assessments require organism level analysis. This work provides an empirical validation that the Weibull shape parameter reliably quantifies intra-culm leaf area inequality. By bridging theoretical distribution models with field-derived inequality metrics, our approach provides insights into canopy efficiency, photosynthetic optimization, and hydraulic trade-offs. Future work should test this approach in other grass species and assess its generalizability in plants with contrasting canopy architectures.

This paper deals with the problem of uniformly minimum variance unbiased estimation for the parameter of the Gompertz distribution based on progressively Type II censored data with binomial removals. We have obtained the uniformly minimum variance unbiased estimator (UMVUE) for powers of the shape parameter and its functions. The UMVUE of the variance of these estimators is also given. The UMVUE of (i) pdf, (ii) cdf, (iii) reliability function, and (iv) hazard function of the Gompertz distribution is derived. Further, an exact % confidence interval for the th quantile is obtained. The UMVUE of pdf is utilized to obtain the UMVUE of . An illustrative numerical example is presented.