Search Results (6)

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

Generalized linear mixed models are commonly used in repeated measurement studies and account for the dependence between observations obtained from the same experimental unit. The designs of repeated measurements in which each experimental unit (e.g., subject) is proven in more than one experimental condition are widespread in psychology, neuroscience, medicine, social sciences and agricultural research. Estimation in generalized linear mixed models is often based on the maximum likelihood theory, which assumes that the assumptions about the underlying probability model are correct. These assumptions include the specification of the distribution of random effects. This research study aimed to identify the impact of the incorrect specification of this distribution on the probability of a type I error and the statistical power of the Wald test. This was achieved through a simulation study where different distributions were considered for random effects in generalized linear mixed models with Poisson and negative binomial responses. Evidence of the impact of the incorrect specification was presented in distributions for random effects different from the normal ones. Lognormal was used for random intercepts and bivariate exponential and Tukey for random intercepts and slopes. Lognormal has positive asymmetry and high kurtosis. Exponential has moderate asymmetry and kurtosis, and Tukey has moderate asymmetry and high kurtosis. Full article

This article presents an experimental–analytical statistical study of low-cycle fatigue to crack initiation and complete failure. The application of statistical and probability methods provides for the possibility of improving the characteristics related to the structural life and the justification for the respective values of cyclic loads in the design stage. Most studies investigating statistical descriptions of crack initiation or complete failure do not analyse the distribution of the characteristics, correlation relationships, and statistical parameters of low-cycle fatigue. Low-cycle failure may be quasistatic or (due to the fatigue) transient. Materials with contrasting cyclic properties were selected for the investigation: cyclically softening alloyed steel 15Cr2MoVA; cyclically stable structural steel C45; cyclically hardening aluminium alloy D16T1. All samples were produced in a single batch of each respective material to reduce the distribution of data. The lowest values of the variation coefficient of one of the key statistical indicators were obtained using the log-normal distribution, which is superior to the normal or Weibull distribution. Statistical analysis of the durability parameters showed that the distribution was smaller than the parameters of the distribution of the deformation diagram. The results obtained in the study enable the verification of durability and life of the structural elements of in-service facilities subjected to elastoplastic loading by assessing the distribution of characteristics of crack initiation and failure and low-cycle strain parameters as well as the permissible distribution limits. Full article

Research Highlights: Simulations of treefall patterns during tornado events have been conducted, enabling the coupled effects of tornado characteristics, tree properties and soil conditions to be assessed for the first time. Background and Objectives: Treefall patterns and forest damage assessed in post-storm surveys are dependent on the interaction between topography, biology and meteorology, which makes identification of characteristic behavior challenging. Much of our knowledge of tree damage during extreme winds is based on synoptic storms. Better characterization of tree damage will provide more knowledge of tornado impacts on forests, as well as their ecological significance. Materials and Methods: a numerical method based on a Rankine vortex model coupled with two mechanistic tree models for critical wind velocity for stem break and windthrow was used to simulate tornadic tree damage. To calibrate the models, a treefall analysis of the Alonsa tornado was used. Parametric study was conducted to assess induced tornadic tree failure patterns for uprooting on saturated and unsaturated soils and stem break with different knot factors. Results: A power law relationship between failure bending moments and diameter at breast height (DBH) for the hardwood species provided the best correlation. Observed failure distributions of stem break and windthrow along the tornado track were fitted to lognormal distributions and the mean of the critical wind speeds for windthrow were found to be higher than that for stem break. Relationships between critical wind speed and tree size were negatively correlated for windthrow and positively correlated for stem break. Higher soil moisture contents and lower knot factors reduced the critical wind speeds. The simulations show varying tree fall patterns displaying forward and backward convergence, different tornado damage widths and asymmetry of the tracks. These variations were controlled by the relative magnitudes of radial and tangential tornado velocities, the ratio between translational speed and maximum rotational wind speed and the mode of failure of the trees. Conclusions: The results show the complexity of predicting tornadic damage in forests, and it is anticipated that this type of simulation will aid risk assessments for insurance companies, emergency managers and forest authorities. Full article

The whole life cycle of an insulated gate bipolar transistor (IGBT) is a kind of asymmetry process, while the whole life cycles of a set of IGBTs can be regarded as a symmetry process. Modelling these symmetry characteristics of the IGBT life cycles enables the improvement of the remaining useful life (RUL) prediction performance. For this purpose, based on the key failure mechanism of IGBT in electric vehicles, a new method for estimating the RUL of an IGBT module is proposed based on the two-stress acceleration synthesis environment of junction temperature and vibration. The maximum likelihood estimation (MLE) was employed to estimate the logarithmic standard deviation and covariance matrix. The Shapiro–Wilk (S–W) test was performed to investigate the satisfaction degree of the RUL of the IGBT module to the lognormal distribution. The accelerated life test datasets of the IGBT module were analyzed using the Weibull++ software. The analysis results demonstrate that the IGBT lifetime is confirmed to lognormal distribution, and the accelerated model accords with the generalized Eyring acceleration model. The proposed method can estimate IGBT RUL in a short time, which provides a certain technical reference for the reliability analysis of the IGBT module. Full article

As a result of asymmetry in practical problems, the Lognormal distribution is more suitable for data modeling in biological and economic fields than the normal distribution, while biases of maximum likelihood estimators are regular of the order $O\left(n^{-1}\right)$ , especially in small samples. It is of necessity to derive logical expressions for the biases of the first-order and nearly consistent estimators by bias correction techniques. Two methods are adopted in this article. One is the Cox-Snell method. The other is the resampling method known as parametric Bootstrap. They can improve maximum likelihood estimators performance and correct biases of the Lognormal distribution parameters. Through Monte Carlo simulations, we obtain average root mean squared error and bias, which are two important indexes to compare the effect of different methods. The numerical results reveal that for small and medium-sized samples, the performance of analytical bias correction estimation is superior than bootstrap estimation and classical maximum likelihood estimation. Finally, an example is given based on the actual data. Full article