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Generation algorithms of record times and values obtained from sequences of independent and non-identically distributed random variables, the distribution functions of which are defined on a common support, are proposed in the present paper. Known algorithms of generation of record times and values are given in Introduction for the case when the initial random variables are independent and identically distributed. A brief review of the scientific literature associated with this topic is also given in Introduction. It is also pointed out there that all efficient algorithms of record generation are based on the Markov property of records. In Section 2, the distribution functions of record times and values are derived for the case when the initial random variables are independent and non-identically distributed. The corresponding record-generation algorithms are proposed for the first time. These algorithms are based on the derived distributions and the Markov property of records, which also holds in the case when the initial observations are independent but non-identically distributed. At the end of this work, in Section 3, the proposed algorithms are tested by simulation experiments. In these experiments the records are generated for the case when the initial random variables have Gumbel distribution functions.

The effect of Mo concentration on pit initiation rate and pit growth in austenitic stainless steels (SS) were investigated by electrochemical noise (EN) and confocal laser scanning microscopy (CLSM) techniques for the first time. We used 304LN, 316LN and 317LN containing different concentrations of Mo (0.02, 2.53 and 3.58 wt.%) for our studies. Using EN technique, initiation of pits and growth of pits were analyzed using Weibull and Gumbel distribution function, respectively. Pit depth was obtained using CLSM and the correlation between pit aspect ratio and Mo concentration were studied. Weibull probability plots showed that the Mo present in the alloy reduces the pit generation rates and improves the passivity. The plot for the pit size distribution using Gumbel distribution function showed the lowest metastable pit in 317LN and the highest in 304LN, suggesting the improvement in the pitting corrosion resistance due to Mo addition. The CLSM imaging showed maximum pit depth for specimen 304LN and minimum for specimen 317LN SS. Specimen 316LN SS showed intermediate pit depth. The depth of the pits observed in 304LN, 316LN and 317LN ranged from 80-100 µm, 30-40 µm and 20-30 µm, respectively. Alloy 317LN containing highest Mo concentration (3.58 wt.%) showed the lowest pit aspect ratio values followed by alloy 316LN with 2.53 wt.% Mo. These results indicate that Mo present in the stainless steel helps in arresting the pit growth and improve the resistance to pitting corrosion.

A statistic method, statistics of extreme values （SEV）, was described in detail, which can esti mate the size of maximum inclusion in steel. The characteristic size of the maximum inclusion in a high clean bearing steel （GCrl5） was evaluated by this method, and the morphology and corn position of large inclusions found were analyzed by scanning electron microscopy （SEM）. When standard inspection area （S0） is 280 mm2, the characteristic size of the biggest inclusion found in 30 standard inspection area is 23.93 μm, and it has a 99.9% probability of the characteristic size of maximum inclusion predicted being no larger than 36.85μm in the experimental steel. SEM result shows that large inclusions found are mainly composed of CaS, calcium-aluminate and MgO. Compositing widely exists in large inclusions in high clean bearing steel. Compared with traditional evaluation method, SEV method mainly focuses on inclusion size, and the esti- mation result is not affected by inclusion types. SEV method is suitable for the inclusion eval uation of high clean bearing steel.

PurposeThe purpose of this paper is to investigate the stochastic comparisons of the parallel system with independent heterogeneous Gumbel components and series and parallel systems with independent heterogeneous truncated Gumbel components in terms of various stochastic orderings.Design/methodology/approachThe obtained results in this paper are obtained by using the vector majorization methods and results. First, the components of series and parallel systems are heterogeneous and having Gumbel or truncated Gumbel distributions. Second, multiple-outlier truncated Gumbel models are discussed for these systems. Then, the relationship between the systems having Gumbel components and Weibull components are considered. Finally, Monte Carlo simulations are performed to illustrate some obtained results.FindingsThe reversed hazard rate and likelihood ratio orderings are obtained for the parallel system of Gumbel components. Using these results, similar new results are derived for the series system of Weibull components. Stochastic comparisons for the series and parallel systems having truncated Gumbel components are established in terms of hazard rate, likelihood ratio and reversed hazard rate orderings. Some new results are also derived for the series and parallel systems of upper-truncated Weibull components.Originality/valueTo the best of our knowledge thus far, stochastic comparisons of series and parallel systems with Gumbel or truncated Gumble components have not been considered in the literature. Moreover, new results for Weibull and upper-truncated Weibull components are presented based on Gumbel case results.

Statistical theory for the estimators of the location, scale and shape parameters of the approximating Generalized Pareto distribution in the Peaks-Over-Threshold approach to extreme value statistics typically requires a second-order extended regular variation condition on the tail quantile function of the distribution underlying the random variable of interest. Somewhat surprisingly, the existing sufficient criteria ensuring that this condition holds appear not to easily apply to many common light-tailed distributions, from the well-known Gaussian, log-normal and Gamma distributions to more specific examples arising in finance and reliability theory. We provide several sufficient criteria based on the computation of appropriate asymptotic expansions of a distribution function or of its quantile function when the underlying distribution has a Weibull-type tail. A list of examples to which our theory applies is given.

Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly nonstationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be nonrelevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model Xi = μ(i/n) + ε i with centered errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function μ from a functional g(μ) (such as the value μ(0) or the integral ∫ 0 1 μ ( t ) d t ) is smaller than a given threshold on a given interval [x 0, x 1] ⊆ [0, 1]. A test for this type of hypotheses is developed using an appropriate estimator, say d̂ ∞, n , for the maximum deviation d ∞ = sup t ∈ [ x 0 , x 1 ] | μ ( t ) − g ( μ ) | . We derive the limiting distribution of an appropriately standardized version of d̂ ∞, n , where the standardization depends on the Lebesgue measure of the set of extremal points of the function μ(·) − g(μ). A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example.

The manuscript deals with a new unit distribution that depends on two positive parameters. The distribution itself was obtained from the Gumbel distribution, i.e., by its transformation, using generalized logistic mapping, into a unit interval. In this way, the so-called Gumbel-logistic unit (abbr. GLU) distribution is obtained, and its key properties, such as cumulative distribution function, modality, hazard and quantile function, moment-based characteristics, Bayesian inferences and entropy, have been investigated in detail. Among others, it is shown that the GLU distribution, unlike the Gumbel one which is always positively asymmetric, can take both asymmetric forms. An estimation of the parameters of the GLU distribution, based on its quantiles, is also performed, together with asymptotic properties of the estimates thus obtained and their numerical simulation. Finally, the GLU distribution has been applied in modeling the empirical distributions of some real-world data related to telecommunications.

This paper introduces a novel three-parameter invertible bimodal Gumbel distribution, addressing the need for a versatile statistical tool capable of simultaneously modeling maximum and minimum extremes in various fields such as hydrology, meteorology, finance, and insurance. Unlike previous bimodal Gumbel distributions available in the literature, our proposed model features a simple closed-form cumulative distribution function, enhancing its computational attractiveness and applicability. This paper elucidates the behavior and advantages of the invertible bimodal Gumbel distribution through detailed mathematical formulations, graphical illustrations, and exploration of distributional characteristics. We illustrate using financial data to estimate Value at Risk (VaR) from our suggested model, considering maximum and minimum blocks simultaneously.

This paper presents a novel two-parameter G family of distributions. Relevant statistical properties such as the ordinary moments, incomplete moments and moment generating function are derived.  Using common copulas, some new bivariate type G families are derived. Special attention is devoted to the standard exponential base line model. The density of the new exponential extension can be “asymmetric and right skewed shape” with no peak, “asymmetric right skewed shape” with one peak, “symmetric shape” and “asymmetric left skewed shape” with one peak. The hazard rate of the new exponential distribution can be “increasing”, “U-shape”, “decreasing” and “J-shape”. The usefulness and flexibility of the new family is illustrated by means of two applications to real data sets. The new family is compared with many common G families in modeling relief times and survival times data sets.