Log-logistic Distribution Research Articles

Fertility model evolution: A survey on mathematical models of age-specific fertility with application to Nepalese and Malaysian data

Fertility pattern analysis and modeling to smooth age-specific fertility rates (ASFRs) form a well-established research field that holds particular importance for Asian countries. In developed nations, ASFRs typically display a bimodal skewed fertility curve, whereas, in developing countries, they usually exhibit a unimodal skewed fertility curve that diverges from the normal one. For decades, demographic experts worldwide have been interested in creating models using deterministic and stochastic approaches to represent these fertility curves. In this regard, parametric and non-parametric models have been created, with the latter providing a better fit for ASFR data. This research investigates the evolution of fertility models aimed at smoothing ASFRs. It explores suitable alternative models for countries with fast-declining, unimodal, and skewed fertility curves of ASFRs, such as Nepal and Malaysia. Nepal’s fertility rate is transitioning from a high level toward the replacement rate (2.1) at the year 2021; meanwhile, Malaysia’s fertility rate (1.7) in the year 2021 has dropped below the replacement rate. Given the lack of a universally applicable model for ASFR pattern variation, this study proposes the Kumaraswamy log-logistic distribution as a promising model to represent the ASFRs of Nepal and Malaysia accurately. Various approaches, including the Akaike information criterion, and Bayesian information criterion, are employed to validate the fitting of the proposed model.

Likelihood-Based Inference of Log-Logistic Accelerated Hazards Regression Models for Cross Hazards Data

ABSTRACT The proportional hazards model and accelerated failure time model are two important classes for analyzing survival data. However, the two models cannot capture time-scaled effects such as crossing hazard or the gradual effect of treatment over time. For the purpose of this detection, accelerated hazards model has been studied. In this paper, we propose the use of log-logistic distribution for the baseline hazard function of the accelerated hazards model which allows for time-dependent group effect. In particular, the log-logistic distribution gives various forms (e.g. non-monotone) of hazard function and also an easy implementation for likelihood-based model fitting due to an explicit form of hazard and survival functions. The likelihood-based estimation procedures of the accelerated hazards models are derived. Our method is demonstrated with two practical data sets, via the procedures of systematic model checking.

Prognostic factors for premature cardiovascular disease mortality in Malaysia: a modelling approach using semi-parametric and parametric survival analysis with national health and morbidity survey linked mortality data.

Cardiovascular disease (CVD) is the leading cause of premature mortality worldwide. Despite existing research on CVD risk factors, the study of premature CVD mortality in Malaysia remains limited. This study employs survival analysis to model modifiable risk factors associated with premature CVD mortality among Malaysian adults. We utilised data from Malaysia's National Health and Morbidity Survey (NHMS) conducted in 2006, 2011, and 2015, linked with mortality records. The cohort comprised individuals aged 18 to 70 during the NHMS interview. Follow-up extended to 2021, focusing on CVD-related premature mortality between ages 30 and 70. We employed six survival models: a semi-parametric Cox proportional hazard (PH) and five parametric survival models, which were Exponential, Weibull, Gompertz, log-normal and log-logistic distributions using R software. The age standardized incidence rate (ASIR) of premature CVD mortality was calculated per 1000 person-years. Among 63,722 participants, 886 (1.4%) experienced premature CVD mortality, with an ASIR of 1.80 per 1000 person-years. The best-fit models (based on AIC value) were the stratified Cox model by age (semi-parametric) and the log-normal accelerated failure time (AFT) model (parametric). Males had higher risk (Hazard Ratio, HR = 2.68) and experienced 49% shorter survival time (Event Time Ratio, ETR = 0.51) compared to females. Compared to Chinese ethnicity, Indians, Malays, and other Bumiputera had higher HR and lower survival times. Rural residents and those with lower education also faced increased HRs and reduced survival times. Diabetes (diagnosed: HR = 3.26, ETR = 0.37; undiagnosed: HR = 1.63, ETR = 0.63), hypertension (diagnosed: HR = 1.84, ETR = 0.53; undiagnosed: HR = 1.46, ETR = 0.68), and undiagnosed hypercholesterolemia (HR = 1.31, ETR = 0.80) increased risk and decreased survival times. Additionally, current smoking and abdominal obesity elevated risk (HR = 1.38, 1.60) and shortened survival (ETR = 0.81, 0.71). The semi-parametric and parametric survival models both highlight the considerable impact of socioeconomic status and modifiable risk factors on premature CVD mortality, underscoring the imperative for targeted interventions to effectively mitigate these effects.

NON-HOMOGENEOUS POISSON PROCESS OF A QUADRATIC TRANSMUTED LOG-LOGISTIC MODEL

In recent years, software reliability engineering has become a major topic due to rapid technological advancements, as researchers work to interpret and analyze software reliability data. Therefore, establishing new software reliability models is essential to possess reliable tools for diagnosing software systems. This paper proposes a new NHPP model based on the quadratic transmuted log-logistic (QTLL) distribution. The characteristics of the model were obtained and graphically illustrated. Three real software data sets were applied to evaluate the proposed model based on different criteria. The parameters of the NHPP QTLL were estimated using maximum likelihood estimation (MLE), and nonlinear least squares estimation (NLSE), and a comparative analysis of the two estimation methods has been performed to estimate the model parameters. Additionally, a study was carried out to compare the suggested model with other well-known NHPP models. The results indicate that NLSE outperforms MLE in parameter estimation. Furthermore, the NHPP QTLL model demonstrated superior performance across all evaluation assessments compared to other models. The graphical representation also shows that the proposed model is more effective in fitting fault data than existing models.

Distribution of radon in large workplaces: an analysis performed on radon levels measured in UK schools

Radon is a radioactive, carcinogenic gas formed by the radioactive decay of uranium and radium that occur naturally in small amounts in all rocks and soils. It is the largest single source of radiation exposure to the UK population, contributing to more than 1 100 lung cancer deaths each year according to an analysis conducted in 2005. Regulations exist to protect employees (and other persons) where radon concentrations exceed the reference level of 300 Bq m-3. Once the reference level is exceeded, annual doses of more than the public dose limit of 1 mSv a-1are considered to be excessive. A radon measurement campaign for schools, which started in 2009, generated a large dataset, including those with high numbers of simultaneous radon measurements. Radon data between buildings (e.g. homes) have been shown to correspond broadly to the lognormal distribution, after the additive contribution of outside air has been removed. However, there are fewer studies of the distribution of radon levels within a single, large property. Radon data collected from 533 UK schools with at least 20 valid, simultaneous results were analysed against several statistical models. In approximately 50% of schools the radon levels could be represented by the lognormal distribution and in 60% by the loglogistic lognormal distribution, the latter being a better fit probably owing to its lower sensitivity to the tails of the distribution. Qualitatively, the lognormal and the loglogistic probability plots appeared to be indistinguishable. These findings indicate that the lognormal and loglogistic might be appropriate models to characterise the distribution of radon in most large workplaces. For each statistical model, the two distribution parameters can be used to provide a better estimate of the average dose to the occupants. However, caution is required when assessing doses, since the average estimator of the radon concentration does not predict the highest value and may significantly underestimate or overestimate the dose in specific areas.

Bayesian Approach for Estimating the Parameters of the Time-To-Failure Distribution and Its Percentiles Under Power Degradation Model

The degradation models are often applied on the degradation data for studying time-to-failure distribution. In this study, the Bayesian approach is applied on the power degradation model for estimating the parameters of the time-to-failure distribution and its percentiles. Two different distributions are assumed for the degradation parameter of the model. The degradation parameter is firstly assumed to follow the skew-normal distribution with three jointly independently distributed parameters such that the gamma prior is assumed for the shape parameter, while the scale and the location parameters are assumed uniform. The second distribution assumed for the degradation parameter is the log-logistic distribution with two jointly independent random parameters where the shape parameter is assumed gamma, while the scale parameter is assumed uniform. Based on the Gibbs sampling method carried out under the JAGS platform, the models considered are applied on the simulated data and the NASA turbofan Jet engine dataset and the results found are compared. In modeling the time-to-failure distribution, it is shown that based on the simulated data and real data, the Bayesian approach for the power degradation model with the skew-normal degradation parameter outperformed the Bayesian approach for the power degradation model with the log-logistic degradation parameter.

A lifetime economic model of mortality and secondary care use for patients discharged from hospital following acute stroke.

The long-term health-economic consequences of acute stroke are typically extrapolated from short-term outcomes observed in different studies, using models based on assumptions about longer-term morbidity and mortality. Inconsistency in these assumptions and the methods of extrapolation can create difficulties when comparing estimates of life-time cost-effectiveness of stroke care interventions. To develop a long-term model consisting of a set of equations to estimate the life-time effects of stroke care interventions to promote consistency in extrapolation of short-term outcomes. Data about further admissions and mortality was provided for acute stroke patients discharged between 2013 and 2014 from a large English service. This was combined with data from UK life tables to create a set of parametric equations in a model that use age, sex, and modified Rankin Scores to predict the life-time risk of mortality and secondary care resource utilisation including ED attendances, non-elective admissions, and elective admissions. A cohort of 1,509 (male 51%; mean age 74) stroke patients had median follow-up of seven years and represented 7,111 post-discharge patient years. A logistic model estimated mortality within twelve months of discharge and a Gompertz model was used over the remainder of the lifetime. Hospital attendances were modelled using a Weibull distribution. Non-elective and elective bed days were both modelled using a log-logistic distribution. Mortality risk increased with age, dependency, and male sex. Although the overall pattern was similar for resource utilisation, there were different variations according to dependency and gender for ED attendances and non-elective/elective admissions. For example, 65-year-old women with a discharge mRS of 1 would gain an extra 6.75 life years compared to 65-year-old women with a discharge mRS of 3. Over their lifetime, 65-year-old women with a discharge mRS of 1 would experience 0.09 less ED attendances, 2.12 less non-elective bed days and 1.28 additional elective bed days than 65-year-old women with a discharge mRS of 3. Using long-term follow-up publicly available data from a large clinical cohort, this new model promotes standardised extrapolation of key outcomes over the life course, and potentially can improve the real-world accuracy and comparison of long-term cost-effectiveness estimates for stroke care interventions. Data is available upon reasonable request from third parties.

Regression-based model predicting cost contingencies for road network projects

Cost contingencies represent a significant markup percentage for construction projects. Many construction projects experience cost overruns due to unexpected events before or during the construction phase. Determining the cost contingency for construction projects is one of the most frequent and widespread financial problems many countries face. Since the amount included in the estimates is always insufficient to absorb the actual cost overruns of the project. The main objective of this paper is to develop a simple realistic regression model for predicting the appropriate contingency amount for road network projects before the bidding stage. The regression model has been developed based on data from real case studies for 75 road network projects to adequately consider all potential risks may face the project. Good-of-fit tests were used to select the best-fitting statistical distribution for a given dataset. Based on the analysis of selected data, the log-logistic probability distribution function is the best-fit curve representing cost overrun behavior, and the cumulative distribution function has been used to determine the realistic probability of incurred cost overrun. Based on the analysis of real case studies, it is suggested that an average of 59.722% be used as a cost overrun contingency amount for road network projects. The study outcomes will help practitioners and researchers improve future accuracy/calculations of contingency amount for road construction projects and improve the decision-making process.

Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point

AbstractFor a sample $$X_1, X_2,\ldots X_N$$ X 1 , X 2 , … X N of independent identically distributed copies of a log-logistically distributed random variable X the maximum likelihood estimation is analysed in detail if a left-truncation point $$x_L>0$$ x L > 0 is introduced. Due to scaling properties it is sufficient to investigate the case $$x_L=1$$ x L = 1 . Here the corresponding maximum likelihood equations for a normalised sample (i.e. a sample divided by $$x_L$$ x L ) do not always possess a solution. A simple criterion guarantees the existence of a solution: Let $$\mathbb {E}(\cdot )$$ E ( · ) denote the expectation induced by the normalised sample and denote by $$\beta _0=\mathbb {E}(\ln {X})^{-1}$$ β 0 = E ( ln X ) - 1 , the inverse value of expectation of the logarithm of the sampled random variable X (which is greater than $$x_L=1$$ x L = 1 ). If this value $$\beta _0$$ β 0 is bigger than a certain positive number $$\beta _C$$ β C then a solution of the maximum likelihood equation exists. Here the number $$\beta _C$$ β C is the unique solution of a moment equation,$$\mathbb {E}(X^{-\beta _C})=\frac{1}{2}$$ E ( X - β C ) = 1 2 . In the case of existence a profile likelihood function can be constructed and the optimisation problem is reduced to one dimension leading to a robust numerical algorithm. When the maximum likelihood equations do not admit a solution for certain data samples, it is shown that the Pareto distribution is the $$L^1$$ L 1 -limit of the degenerated left-truncated log-logistic distribution, where $$L^1(\mathbb {R}^+)$$ L 1 ( R + ) is the usual Banach space of functions whose absolute value is Lebesgue-integrable. A large sample analysis showing consistency and asymptotic normality complements our analysis. Finally, two applications to real world data are presented.

Properties, estimation, and applications of the extended log-logistic distribution

This paper presents the exponentiated alpha-power log-logistic (EAPLL) distribution, which extends the log-logistic distribution. The EAPLL distribution emphasizes its suitability for survival data modeling by providing analytical simplicity and accommodating both monotone and non-monotone failure rates. We derive some of its mathematical properties and test eight estimation methods using an extensive simulation study. To determine the best estimation approach, we rank mean estimates, mean square errors, and average absolute biases on a partial and overall ranking. Furthermore, we use the EAPLL distribution to examine three real-life survival data sets, demonstrating its superior performance over competing log-logistic distributions. This study adds vital insights to survival analysis methodology and provides a solid framework for modeling various survival data scenarios.

Statistical analysis of the ground deformation of Vulcanian explosions at Sakurajima volcano, Japan

The forecast of pulsatory explosions during volcanic unrest periods is an essential issue for the assessment and mitigation of volcanic hazards. Although various precursors are detectable through geophysical and geochemical monitoring, difficulties remain in precisely constraining possible scenarios. A probabilistic approach is effective in assessing risk while considering various uncertainties. Sakurajima volcano characterized by frequent Vulcanian activity is one of the suitable fields for the probabilistic forecast of pulsatory explosions. Their inflation-deflation patterns of ground deformation related to Vulcanian explosions are useful for evaluating the imminence and size of the next event. The large database obtained from its vigorous activity can contribute to statistical analysis. In this study, aiming the probabilistic forecast of the timing and size of explosions, we investigated the duration of inflation and volume changes at the pressure source using strain records of over 5000 events of Sakurajima volcano. Then, a stochastic model was estimated to explain the distribution of these events. The log-logistic distribution was found to be an appropriate model for data distribution, indicating the presence of competing processes, such as pressurization and depressurization, in the conduit. The model parameters of the log-logistic distribution temporally fluctuated reflecting the volcanic activity, especially increasing the magma supply from a deep region. We also suggested a methodology to constrain the probabilities of the likely timing and size of an imminent explosion using real-time strain monitoring and an estimated model distribution. Although some improvements would be needed for practical forecasting, our approach could be useful in predicting possible ash hazards.

Time-Variant Chloride Ingress Models for Probabilistic Assessment of Concrete Bridge Decks with De-Icing Salt Attack

This study is to propose time-variant probabilistic models of surface chloride and diffusion coefficient based on the survey data of 16 concrete bridge decks with the attack of de-icing salts. These models are developed, because there is no study that simultaneously considers both time-variance and probabilistic descriptors in the model for concrete bridge decks. From the study, it can be found that long-term surface chloride and its time-variant development are fitted with Log-normal and Weibull distributions, respectively. In addition, the 28-day diffusion coefficient and age factor are fitted with Log-logistic and Triangular distributions, respectively. Considering only the mean value in the models, the corrosion-free residual life of concrete bridge decks is equal to 18.3 years based on the target value of critical chloride of 1.2 kg/m3, whereas their cracking-free residual life is equal to 29.5 years based on the target value of critical chloride of 2.0 kg/m3. In comparison with the probabilistic analysis, it was nevertheless found that at year 18.3, there are 38% and 20% probabilities to have rebar corrosion and concrete cracking, respectively. However, at year 29.5, there are 63% and 42% probabilities to have rebar corrosion and concrete. Specifically, there are 6 and almost 7 out of 16 bridge locations having rebar corrosion in the year 18.3 and concrete cracking in the year 29.5, respectively.

Study on ring-road incident duration based on latent class accelerated hazard model.

Accurately estimating the duration of freeway incidents can enhance emergency management practices and reduce the likelihood of secondary incidents. To investigate the mechanisms through which key factors influence incident duration, this study sorted out the characteristics and variables of the incident duration on a special freeway in Zhejiang Province, that is, the ring road, and developed a latent class accelerated hazard model. Heterogeneity was incorporated into the model. Three distributions (Weibull, Log-normal, and Log-logistic) were compared, and the Log-logistic distribution exhibited superior performance. The analysis revealed two distinct latent classes: Latent Class 1 and Class 2, had class membership probability of 0.53 and 0.47, respectively, with a total of 11 variables being statistically significant at the 0.05 significance level. It is worth noting that, some neglected explanatory variables are discussed in depth in this study. For example, the mechanism of which specific lane is closed has an impact on the incident duration, rather than a general discussion of the number of lane closures. Furthermore, the way in which the driver involved in the incident reports to the police has a significant impact on the duration of incidents. Notably, potential heterogeneity and its influencing mechanism are captured in the model. Additionally, by predicting class membership using posterior probabilities, it was determined that most data points were more likely to belong to Class 1, and the incident duration primarily ranged between 0 and 60 minutes. These findings are helpful to reduce the duration of incidents on ring-roads and freeways in China, and provide theoretical support for the formulation of freeway incident management and treatment policies.

A method for assessing the operational status of a microseismic monitoring system using energy distribution changes of observed data

Abstract The operational status of geophones plays a pivotal role in ensuring the accuracy and reliability of microseismic monitoring systems. However, conventional techniques used to evaluate the operational status of geophones require human intervention or significant time delays. To address this issue, we propose a method for online monitoring of geophone status using observed data obtained from a microseismic system. Firstly, the energy features of the preprocessed observation data are extracted via wavelet packet decomposition. Subsequently, the distribution parameters of energy features are obtained through loglogistic distribution fitting. These parameters are then applied to a change-point detection model, enabling the online monitoring of seismic geophones. In addition, we selected long short-term memory (LSTM) network to classify the operational status of the geophones, which is trained using the obtained energy distribution data and the time-frequency characteristics of the observed data. The experimental results indicate that the model achieves an accuracy of 98.33%, surpassing the 89.58% accuracy of the support vector machine (SVM). The proposed method not only contributes to online monitoring and precise determination of the operating status of detectors, but also has enormous application potential in other fields that require monitoring and evaluating the operating status of instruments.

MEDi-SOL: Multi Ensemble Distribution Model for Estimating Sleep Onset Latency.

Sleep onset latency (SOL) is an important factor relating to the sleep quality of a subject. Therefore, accurate prediction of SOL is useful to identify individuals at risk of sleep disorders and to improve sleep quality. In this study, we estimate SOL distribution and falling asleep function using an electroencephalogram (EEG), which can measure the electric field of brain activity. We proposed a Multi Ensemble Distribution model for estimating Sleep Onset Latency (MEDi-SOL), consisting of a temporal encoder and a time distribution decoder. We evaluated the performance of the proposed model using a public dataset from the Sleep Heart Health Study. We considered four distributions, Normal, log-Normal, Weibull, and log-Logistic, and compared them with a survival model and a regression model. The temporal encoder with the ensemble log-Logistic and log-Normal distribution showed the best and second-best scores in the concordance index (C-index) and mean absolute error (MAE). Our MEDi-SOL, multi ensemble distribution with combining log-Logistic and log-Normal distribution, shows the best score in C-index and MAE, with a fast training time. Furthermore, our model can visualize the process of falling asleep for individual subjects. As a result, a distribution-based ensemble approach with appropriate distribution is more useful than point estimation.

Different Estimation Methods for A New Extension Three Parameter Log-Logistic Distribution with Applications

This article presents a new extension of the three-parameter log-logistic distribution, the so-called heavy-tailed log-logistic (HTLL) distribution. Some important mathematical properties of the new HTLL distribution are calculated. In addition, some numerical results of moments for the HTLL distribution are calculated. Extensive simulations were performed to investigate the estimation of the model parameters using many established approaches, including maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM), Anderson-Darling (AD), and right-tail Anderson-Darling (RTAD). The simulation results show that the AD approach has the highest efficiency among these approaches. The usefulness of the newly proposed model is demonstrated by analyzing two real data sets.

Statistical characterization of reliability indices in medium voltage networks using a Monte Carlo-based method

Distribution companies are compelled to provide high-quality services to their customers in compliance with existing regulations. Key performance indicators such as system average interruption duration index and system average interruption frequency index are commonly employed to assess the quality of service provided by these utilities. In certain countries, distribution system operators may face penalties or receive bonuses based on their performance within specified thresholds in their reliability indices. This paper introduces a novel, integrated algorithm, based on Monte Carlo simulations, for conducting a risk assessment in real MV networks using local data from the distribution system operator. The proposed technique is tested in a real MV network, and the obtained results are analyzed both at feeder level and for the entire network. The feeder-level results might be used to identify the sections of the network with the worst quality of power supply, based on the estimated KPIs. The overall results of the study offer valuable insights into the likelihood of specific KPI values using probability density function. The Weibull distribution and log-logistic distribution are found to be those with the best-fitting probability density functions for the two indices evaluated in this work. Furthermore, a penalty-based rate approach is presented as an application of the proposed method to assess the economic risk of the MV network. The findings reveal that the test system is expected to incur penalties amounting to 7.55 million dollars per year.

Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model

Accelerated life tests play a very critical role in reliability analysis because highly reliable products are being produced with recent advanced technologies to sustain the market demand and competition. A progressive-stress accelerated life test is one of the kinds of accelerated life tests that allows applied stress to change continuously. Considering the importance of progressive-stress accelerated life tests, this paper deals with progressive-stress accelerated life tests when the lifetimes of units follow the log-logistic distribution and are progressively type-II censored where the associated scale parameter conforms to the inverse power law. Different estimators of model parameters are derived using maximum likelihood and Bayesian methods. Interval estimation is also considered. In the sequel, approximate confidence, bootstrap, and Bayes credible intervals are constructed. Bayes estimators are obtained under squared error, LINEX, and entropy loss functions using proper and improper prior distributions. A simulation study is conducted based on various censoring schemes. The coverage percentages and interval widths are computed via Monte Carlo simulations. Bayes predictive estimates and intervals are also obtained. Finally, two different accelerated life test data are analyzed for illustration purposes.

Advancing Solar Energetic Particle Event Prediction through Survival Analysis and Cloud Computing. I. Kaplan–Meier Estimation and Cox Proportional Hazards Modeling

Solar energetic particles (SEPs) pose significant challenges to technology, astronaut health, and space missions. This initial paper in our two-part series undertakes a comprehensive analysis of the time to detection for SEPs, applying advanced statistical techniques and cloud-computing resources to deepen our understanding of SEP event probabilities over time. We employ a range of models encompassing nonparametric, semiparametric, and parametric approaches, such as the Kaplan–Meier estimator and Cox Proportional Hazards models. These are complemented by various distribution models—including exponential, Weibull, lognormal, and log-logistic distributions—to effectively tackle the challenges associated with “censored data,” a common issue in survival analysis. Employing Amazon Web Services and Python’s “lifelines” and “scikit-survival” libraries, we efficiently preprocess and analyze large data sets. This methodical approach not only enhances our current analysis, but also sets a robust statistical foundation for the development of predictive models, which will be the focus of the subsequent paper. In identifying the key determinants that affect the timing of SEP detection, we establish the vital features that will inform the machine-learning (ML) techniques explored in the second paper. There, we will utilize advanced ML models—such as survival trees and random survival forests—to evolve SEP event prediction capabilities. This research is committed to advancing space weather, strengthening the safety of space-borne technology, and safeguarding astronaut health.

Characterizations of continuous log-symmetric distributions based on properties of order statistics

The class of log-symmetric distributions is a generalization of log-normal distribution and includes some well-known distributions such as log-normal, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-student-t, log-slash, and Birnbaum-Saunders distributions. In this paper, several characterization results are obtained for log-symmetric distributions based on moments of some functions of the parent distribution and also on the basis of some properties of order statistics. Specifically, when X is identical in distribution with a decreasing continuous function h ( X ) , then a relationship is established between upper and lower order statistics which is then utilized to construct characterization results for log-symmetric distributions in terms of functions of order statistics. The established results can be used for constructing a goodness-of-fit test for log-symmetric distributions.