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We establish a novel family of Kumaraswamy-X probability distributions in the present investigation. We discussed the KumaraswamyExponential univariate probability distribution. The new distribution with three parameters possesses density function with unimodal and reverse J-shape and hazard rate function of bathtub shaped. We study various statistical properties for it and derive the expressions for its density function, distribution function, survival and hazard rate function, Probability weighted Moments, lth moment, moment generating function, quantile function and Shannon entropy. For the derived distribution order statistics is also discussed. The parameters are estimated using the maximum likelihood estimation approach, and the performance of the estimators was evaluated using a Monte Carlo simulation. Through extensive Monte Carlo simulations and comparative analyses, we assess the performance of the Kumaraswamy-X distribution against other common probability distributions used in engineering contexts. When we apply it to real datasets, it offers a more suitable fit than other existing distributions. We explore the characteristics and potential applications of the Kumaraswamy-X distribution in the context of engineering problems through a comprehensive simulation-based investigation. [ABSTRACT FROM AUTHOR]

We adopt extreme value theory to estimate the upper limit of the next record‐breaking magnitudes of induced seismic events. The methodology is based on order statistics and does not rely on knowledge of the state of the subsurface reservoir or injection strategy. The estimation depends on the history of record‐breaking events produced by the anthropogenic activities. We apply the methodology to three different types of industrial operations: natural gas production, saltwater disposal and hydraulic fracturing. We show that the upper limit estimate provides a reliable and realistic upper bound for magnitudes of the record‐breaking events in investigated datasets including 15 publicly available datasets. The predicted magnitudes do not overestimate the observed magnitudes by more than 1.0 magnitude unit and underestimation is rare, probably resulting from insufficient sampling of the statistical distribution of the induced seismicity. The richest dataset, sourced from downhole and surface monitoring of the Preston New Road hydraulic fracturing, provides reliable estimates of the magnitudes over three orders of magnitudes with only slight underprediction of the largest observed event. While the detection of weaker events improves the performance of the method, we show that it can be applied even with a few observed record‐breaking events to provide reliable estimates of magnitudes. However, care must be taken to ensure that event catalogues are estimated consistently across a range of magnitudes. [ABSTRACT FROM AUTHOR]

Quantile regression predictions are considered for generalized order statistics which extend results previously established for record values and order statistics. In order to derive the results, some (univariate and bivariate) distortion representations of generalized order statistics are established. In particular, prediction of the sth generalized order statistic $ X_{(s)}^* $ X(s)∗ based on F given a single (generalized) order statistic $ X_{(r)}^* $ X(r)∗ with r<s will be addressed. The presentation includes results for both known and unknown baseline distributions. In the latter case, we consider exponential distributions with unknown mean as well as the proportional hazards model. Using unimodality properties of the marginal density functions (pdf) of generalized order statistics, we find a simple representation of the maximum likelihood prediction of $ X_{(s)}^* $ X(s)∗ given $ X_{(r)}^* $ X(r)∗. [ABSTRACT FROM AUTHOR]

In this study, we focused on investigating the properties of residual Tsallis entropy for order statistics. The reliability of engineering systems is highly influenced by order statistics, for example, when modeling the lifetime of a series system and the lifetime of a parallel system. The residual Tsallis entropy of the ith order statistic from a continuous distribution function and its deviation from the residual Tsallis entropy of the ith order statistics from a uniform distribution were investigated. In the mathematical framework, a method was provided to represent the residual Tsallis entropy of the ith order statistic in the continuous case with respect to the case where the distribution was uniform. This approach can provide insight into the behavior and properties of the residual Tsallis entropy for order statistics. We also investigated the monotonicity of the new uncertainty measure under different conditions. An investigation of these properties leads to a deeper understanding of the relationship between the position of the order statistics and the resulting Tsallis entropy. Finally, we presented the computational results and proposed estimators for estimating the residual Tsallis entropy of an exponential distribution. For this purpose, we derived a maximum likelihood estimator. [ABSTRACT FROM AUTHOR]

Many researchers have proposed numerous Bayesian predictive densities for Type-II censored data that is generated by ordered observations. However, their evaluations of predictive densities were insufficient because the Bayesian predictive density includes prior parameters, thus we suffer from the selection of the prior parameters. In this study, we consider two types of predictive densities, posterior predictive and plug-in, for observations from an exponential distribution of Type-II censored data. We discuss a suitable predictive density using the risk with the Kullback–Leibler loss function. In our setting, we consider a Gamma prior, which is a conjugate prior for mathematical tractability. We prove that the posterior predictive density with an improper Gamma prior provides the dominance of the posterior predictive density over the plug-in densities without depending on the selection of an unknown parameter in our setting. Finally, we show that the posterior predictive density outperforms the plug-in densities in terms of coverage probabilities for unobserved data by censoring in a simulation study. [ABSTRACT FROM AUTHOR]

More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k -out-of- n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{\mathrm{e}}}^x)$. Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions. [ABSTRACT FROM AUTHOR]

This paper proposes a new Topp-Leone Exponentiated Pareto (TLEtP) distribution. The new distribution family is derived by expanding the Topp Leone-G family of distributions with additional positive shape parameters. The corresponding density and distribution functions are derived and shown. Some of the derived mathematical properties of the distribution include quantile function, ordinary and incomplete moments generating function (mgf), hazard function, survival function, odd function, probability weighted moment, and distribution of order statistic. The parameters of the distribution are estimated using Maximum Likelihood method. The proposed distribution's validity is demonstrated by fitting two sets of real data and comparing the results with two existing same-family distributions, the Topp-Leone Pareto type I(TLPI) and Pareto (P), with the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC), respectively. The comparison of the proposed Topp-Leone Exponentiated Pareto (TLEtP) to the Topp-Leone Pareto type I(TLPI) and Pareto (P) distribution demonstrate that the TLEtP distribution offers a better fit for the data sets than the other two distributions. [ABSTRACT FROM AUTHOR]

We introduced a flexible class of continuous distributions called the generalized Kavya-Manoharan-G (GKM-G) family. The GKM-G family extended the Kavya-Manoharan class and provided greater flexibility to the baseline models. The special sub-models of the GKM-G family are capable of modeling monotone and non-monotone failure rates including increasing, reversed J shape, decreasing, bathtub, modified bathtub, and upside-down bathtub. Some properties of the family were studied. The GKM-exponential (GKME) distribution was studied in detail. Eight methods of estimation were used for estimating the GKME parameters. The performance of the estimators was assessed by simulation studies under small and large samples. Furthermore, the flexibility of the twoparameter GKME distribution was explored by analyzing five real-life data applications from applied fields such as medicine, environment, and reliability. The data analysis showed that the GKME distribution outperforms other competing exponential models. [ABSTRACT FROM AUTHOR]

Let $(X_{1},\ldots,X_{n})$ be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for $1\leq i\leq n$ , $X_{i:n}$ denote the corresponding i th-order statistic. We consider the problem of comparing the strength of dependence between any pair of X i 's with that of the corresponding order statistics. It is in particular proved that for $m=2,\ldots,n$ , the dependence of $X_{2:m}$ on $X_{1:m}$ is more than that of X 2 on X 1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that $(X_{1:m},X_{2:m})$ is more concordant than $(X_{1},X_{2})$. It will be interesting to examine whether these results can be extended to other exchangeable models. [ABSTRACT FROM AUTHOR]

Order statistics occupy an important place in statistical theory. They have an important place in many fields of applied statistics such as goodness of fit tests and parameter estimation. In addition, it is necessary to find the expected values of these order statistics in these application areas. However for some probability distributions, these expected values are very difficult to find such as the standard normal distribution. So the problem of finding the expected values of the order statistics in statistical theory is of importance. In this study, two novel approximation methods are proposed for the expected values of the order statistics of the standard normal distribution. Also, the true values with previously given approximations, simulation results and our proposed approximations are compared by using mean square error (MSE), mean absolute error (MAE) and maximum error (ME) criteria. Furthermore, to evaluate the performances of all approximation methods, we compute the differences between exact values and approximation values. Then, the plot of these differences against the exact values is given. Based on both the plots and the comparison results, novel approximations fit the true values better than the other approximations presented in this paper. [ABSTRACT FROM AUTHOR]

The expansion of the application domains is greatly aided by the generalization of distributions. To achieve this, an extension of the two–parameter Kumaraswamy distribution to a three–parameter second order transmuted Kumaraswamy distribution has been presented in this study utilizing the cubic transmutation map. Along with the order statistics and parameter estimates, a number of statistical characteristics are shown. The performance of the estimated parameters has been demonstrated through simulation study. A few applications additionally serve to demonstrate the suggested distribution’s dominating applicability. [ABSTRACT FROM AUTHOR]

The representability of the Burr distribution as a mixture of Weibull distribution is studied in order to justify its utility for modelling the statistical regularities in extreme values registered in non-stationary flows of informative events. A result of [24] is improved by extending the domain of admissible values of the parameters which provide the representability of the (generalized) Burr distribution as a scale mixture of the Weibull distribution. This result gives an argument in favour of application of the Burr distribution as a model of statistical regularities of extreme values registered within moderate regular time intervals, say, daily (short-term) extremes. In turn, if we are interested in the statistical regularities of the behaviour of the absolute extreme observation over a long period, say, a decade (the long-term extreme), then it can be noted that the daily extreme values form a sample of the Burr-distributed random variables. As is known, the Burr distribution belongs to the domain of max-attraction of the Fréchet distribution. The problem of improving the accuracy of the approximation of the distribution of the absolute extreme by the Fréchet distribution by the construction of an asymptotic expansion for the distribution of the extreme order statistics in the sample of independent identically Burr-distributed random variables is also considered. These results are illustrated by an example of fitting the Burr distribution to the data representing the extreme values of characteristics of the interplanetary magnetic field. [ABSTRACT FROM AUTHOR]

In statistical modeling, lifetime datasets play an important and significant role. Weighted distributions can be useful to attain the essential purposes relating to the developed distributions' flexibility and to improve further compliances in data forming; they provide the extension of distributions. The current study suggests a new best-fit probability distribution model based on Pareto distribution. The new model is called Survival Weighted Pareto (SWP) distribution model. Furthermore, the statistical expressions for different properties, which include moments, inverse moments, quantile function, and order statistics, are reported. Moreover, to check the performance and the behavior of different estimators using different sample sizes, a simulation study, which is based on SWP distribution, is performed. According to the simulation results, it is established that the proposed model is a most useful fitted probability model. In addition, an application of the SWP model for three datasets is provided to check the goodness of fit measures of the model. [ABSTRACT FROM AUTHOR]

Recently, extropy has emerged as an alternative measure of uncertainty instead of entropy. When it comes to quantifying uncertainty regarding the remaining lifetime of a component, entropy has proven to be less effective. Therefore, the concept of residual entropy was introduced to address this limitation. Similar to the residual entropy, the residual extropy was formulated and used to investigate the uncertainty in the residual lifetime of a unit. Systems in the real world exhibit a pervasive property of uncertainty that affects future events and past events. For this reason, the concept of past extropy was introduced to specifically capture and analyze the uncertainty associated with past events. This paper focuses on stochastic aspects, including stochastic orderings, which provide useful inequalities related to past extropy when applied to order statistics and lower record values. It is worth noting that the past extropy of the i th-order statistics and record values in the continuous case is related to the past extropy of the i th-order statistics and record values evaluated from the uniform distribution. The monotonicity of the past extropy of order statistics is examined and some insights into the past extropy of lower data set values are also given. Finally, some computational results are presented. In fact, an estimator for the extropy of the exponential distribution is proposed. For this purpose, the maximum likelihood estimator is derived. The proposed method is easy to implement and apply from a computational point of view. [ABSTRACT FROM AUTHOR]

We evaluate, in the large- N limit, the complete probability distribution P (A , m) of the values A of the sum ∑ i = 1 N | λ i | m , where λ i ( i = 1 , 2 , ... , N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N → ∞ , the rate function of P (A , m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue 'evaporates' from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p = 1 + 1 / | m − 1 | , where 0 < m < 2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m ⩽ 1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner's semicircular distribution. [ABSTRACT FROM AUTHOR]

In parametric statistical modeling, it is essential to create generalizations of current statistical distributions that are more flexible when modeling actual data sets. Therefore, this study introduces a new generalized lifetime model named the odd Weibull Inverse Gompertz distribution (OWIG). The OWIG is derived by combining the odd Weibull family of distributions with the inverse Gompertz distribution. Essential statistical properties are discussed, including reliability functions, moments, Rényi entropy, and order statistics. The proposed OWIG is particularly significant as its hazard rate functions exhibit various monotonic and nonmonotonic shapes. This enables OWIG to model different hazard behaviors more commonly observed in natural phenomena. OWIG's parameters are estimated and its flexibility in predicting unique symmetric and asymmetric patterns is shown by analyzing real-world applications from psychology, environmental, and medical sciences. The results demonstrate that the proposed OWIG is an excellent candidate as it provides the most accurate fits to the data compared with some competing models. [ABSTRACT FROM AUTHOR]

In this article we present a stochastic ordering verification algorithm between multivariate discrete distributions implemented in the C++ programming language. This algorithm is essential in problems of finding the optimal portfolio when dealing with discrete distributions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the concomitants of the extensions of Morgenstern family (Lai and Xie extensions) for generalized order statistics and dual generalized order statistics. In addition, recurrence relation between moments for the required models is obtained. Furthermore, the joint distribution between two concomitants is derived for generalized order statistics and its dual. We finally present the Shannon entropy and Fisher information with exponential distribution as an example from Lai and Xie extensions. [ABSTRACT FROM AUTHOR]

Fire related to household items poses a threat to life safety and residential property. In this study, the fire characteristics of single household items are established as a fundamental of risk assessment for performance-based design. Information on peak heat release rate, time to peak heat release rate and total heat released are collected from full-scale fire experiments in the literature. The household items are individually classified by their dimension: Television, Chair, Table, Armchair, Cabinet and Bed for statistical and probability distribution analyses. The analysis generally exhibits an increasing trend in the fire characteristics against the item dimension increment. A linear fit with a strong correlation value is achieved for the peak heat release rate and total heat released, while a weak correlation is observed to linearly fit the time to peak heat release rate. The ranked order of fitting statistics suggests Gamma, Log-Normal and Logistic distribution as the best-fit distribution respectively for the peak heat release rate, time to peak heat release rate and total heat released. The distribution parameters are provided and can be used for probabilistic fire assessment related to household items. [ABSTRACT FROM AUTHOR]

This article is set to push new boundaries with leading-edge innovations in statistical distribution for generating up-to-the-minute contemporary distributions by a mixture of the second record value of the Gompertz distribution and the classical Gompertz model (weighted Gompertz model) using T-X characterization, especially used for two-sided schemes that provide an accurate model. The quantile, ordinary, and complete moments, order statistics, probability, and moments generating functions, entropies, probability weighted moments, Lin's condition random variable, reliability in multicomponent stress strength system, reversed, and moments of residuals life and other reliability characteristics in engineering, actuarial, economics, and environmental technology were derived in their closed form. To investigate and test the flexibility, viability, tractability, and performance of the proposed Weighted Gompertz-G (WGG) generated model, the shapes of some sub-models of the WGG model were examined. The shapes of the sub-models indicated J-shapes, increasing, decreasing, and bathtub hazard rate functions. The maximum likelihood estimation of the WGG-generated model parameters was examined. An illustration with simulation and real-life data analysis indicated that the WGG-generated model provides consistently better goodness-of-fit statistics than some competitive models in the literature. [ABSTRACT FROM AUTHOR]

This article introduces a new three-parameter model with an increasing and bathtub failure rate functions as an extension of the Mustapha type-II distribution (Mu-II). The model can be very useful in statistical studies, reliability, computer sciences and engineering. Various mathematical and statistical properties of the distribution are discussed, such as moments, mean deviations, Bonferroni and Lorenz curves, entropy, order statistic, and extreme value distributions. Moreover, we consider the bivariate extension of the new model. Statistical inferences by the maximum likelihood method are discussed and assess by simulation studies. Applications of the proposed model to two rightskewed data are presented for illustration. The new model provides a better fit than some other existing distribution as measured by some model selection criteria and goodness of fits statistics. [ABSTRACT FROM AUTHOR]

In this paper, we provide a measure based on inaccuracy between distributions of the ith order statistic and the parent random variable. This measure characterizes the distribution function of parent random variable uniquely. We demonstrate that the extropy of the parent random variable is the average of the accuracy measure. It is also shown that the measure of inaccuracy defined is invariant under scale but not under location transformation. Nonparametric estimators for the proposed measures are also obtained. A Monte Carlo simulation study is performed to verify the performance of the suggested estimators. Simulation results show that the estimator based on the reflection boundary technique for probability density function estimation and the empirical method for cumulative distribution function estimation has the best performance among estimators. Also, a real dataset is considered to show an application of the proposed estimators on model selection. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel extension of the Tornumonkpe distribution, known as the length biased Tornumonkpe distribution. This distribution holds particular significance as it belongs to the family of weighted distributions, specifically the length biased variant. Through an in-depth analysis, we explore the mathematical and statistical properties of this distribution, shedding light on its unique characteristics. To estimate the model parameters of this new distribution, we employ the wellestablished technique of maximum likelihood estimation. This allows us to accurately determine the parameters and enhance our understanding of the distribution's behavior. To demonstrate the practical applicability and advantages of the length biased Tornumonkpe distribution, we showcase its performance using a real-life time data set. Through this empirical examination, we investigate the distribution's superiority and flexibility, providing valuable insights into its potential use in various domains. [ABSTRACT FROM AUTHOR]

Characterization of a probability distribution plays an important role in probability and statistics. Before a particular probability distribution model is applied to fit the real world data, it is necessary to confirm whether the given probability distribution satisfies the underlying requirements by its characterization. The aim of this paper is to find characterization results New Quasi Lindley distribution. These results are established using the relation between truncated moments and failure rate functions and conditional expectation of adjacent order statistics. The first characterization result is based on relation between left truncation moment and failure rate function while the second result is based on relation between right truncated moment and reverse failure rate function. In the third characterization result we used conditional expectation of order statistics when the conditioned one is adjacent order statistics. Further, some of its important deductions are also discussed. [ABSTRACT FROM AUTHOR]

Heavy-tailed distributions have garnered interest due to their advantageous statistical and reliability characteristics, rendering them valuable in applied fields such as economics, finance, and risk management. Such distributions offer robust properties, making them pertinent to studies in various areas like econometrics, statistics, and insurance. Thus, the primary objective of this paper is to propose a Two parameter right skewed- upside down bathtub type, heavy tailed distribution, which is a generalisation of Inverse Exponential distribution and is referred to as Modi Inverse Exponential distribution. We derive several mathematical and statistical features, including quantile function, mode, median, skewness, kurtosis, and mean deviation. Additionally, the reliability and hazard rate functions are also derived. Stochastic ordering and order statistics of the proposed distribution were derived. We also investigate the tail behaviour of the proposed model. Furthermore, estimation methods such as maximum likelihood estimation and its asymptotic confidence bound, percentile method, and Cramer-von-Mises method were examined. To demonstrate the appropriateness of the suggested model, we have considered two distinct real datasets along with three distinct models and concluded that the proposed model is more adaptable. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel class of distributions called the length biased Uma distribution. This distribution is a specific instance of the broader weighted distribution family, known for its versatility in various applications. We explore the structural properties of the length biased Uma distribution and propose a robust parameter estimation technique based on maximum likelihood estimation. To assess its efficacy, we apply the newly introduced distribution to two real-world datasets, evaluating its flexibility and performance in comparison to existing models. The results obtained demonstrate the potential of the length biased Uma distribution as a valuable addition to the repertoire of statistical distributions, offering valuable insights for a wide range of practical applications. [ABSTRACT FROM AUTHOR]

This manuscript focuses on the statistical properties and estimation methods of the exponentiated Suja distribution, which is characterized by two parameters: scale and shape. From a frequentist perspective, our primary emphasis is on estimation techniques. Additionally, we derive statistical and reliability characteristics for the model. We explore various estimation procedures, including order statistics, entropies, reliability analysis, and the maximum likelihood method. To assess the model's superiority and practical utility, we analyze real lifetime data sets. Overall, this study provides a comprehensive analysis of the exponentiated Suja distribution, offering insights into its statistical properties, estimation techniques, and real-life applications. [ABSTRACT FROM AUTHOR]

In this paper, a new continuous distribution with one parameter named weighted Arz distribution (WAD) is suggested using the idea of weighted distributions. Several properties of this distribution such as moments, moment generating function, skewness, coefficient of variation, kurtosis, and index of dispersion are investigated. Also, the distribution of order statistics, Lorenz and Bonferroni curves, Gini index, stochastic ordering, R'enyi entropy, mean deviations about mean and median are provided. The reliability functions including survival, hazard, odds, reversed hazard rate, and mean residual life are provided with supported graphical representation. The maximum likelihood estimate (MLE) of the distribution parameter is provided. A simulation study is conducted to show the consistency of the MLE. Application to COVID-19 data set is presented and showed that the proposed distribution is more flexible than some other competing distributions in fitting such data. [ABSTRACT FROM AUTHOR]

In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory. [ABSTRACT FROM AUTHOR]

For a given positive number 'δ′, we consider a sequence of δ − neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a U (0 , 1) distribution, and "stop as soon as their union contains the interval (0 , 1). " We call such a union "a cover." To find the distributions of N (δ) , the stopping time random variable, we need the joint distribution of order statistics from a U (0 , 1) distribution. For each δ > 0 and n = 1 , 2 , ... , we obtain a general expression for P (N (δ) ≤ n) , and for a fixed value of δ , it is the distribution function of N (δ). For a given n, let Δ (n) be the minimum value of δ , so that the union of the n δ − neighborhoods of the first n observations contains the interval (0 , 1). Because N (δ) ≤ n if and only if Δ (n) ≤ δ , the distributions of Δ (n) can be obtained by fixing n in the general expression for P (N (δ) ≤ n). To describe the impact of δ on the distribution of N (δ) and that of n on Δ (n) , we sketch the graphs of distribution functions and the empirical distribution functions. [ABSTRACT FROM AUTHOR]

Let $V_{(r,n,\tilde {m}_n,k)}^{(p)}$ and $W_{(r,n,\tilde {m}_n,k)}^{(p)}$ be the $p$ -spacings of generalized order statistics based on absolutely continuous distribution functions $F$ and $G$ , respectively. Imposing some conditions on $F$ and $G$ and assuming that $m_1=\cdots =m_{n-1}$ , Hu and Zhuang (2006. Stochastic orderings between p -spacings of generalized order statistics from two samples. Probability in the Engineering and Informational Sciences 20: 475) established $V_{(r,n,\tilde {m}_n,k)}^{(p)} \leq _{{\rm hr}} W_{(r,n,\tilde {m}_n,k)}^{(p)}$ for $p=1$ and left the case $p\geq 2$ as an open problem. In this article, we not only resolve it but also give the result for unequal $m_i$ 's. It is worth mentioning that this problem has not been proved even for ordinary order statistics so far. [ABSTRACT FROM AUTHOR]

Dispersive order is a type of variability order for comparing the variability in probability distributions. Star order compares the skewness of probability distributions. This work considers dispersive and star orders of extreme order statistics from dependent random variables following the proportional odds (PO) model. The joint distribution of the random variables is modeled with Archimedean copula. Numerical examples are provided to illustrate the findings. [ABSTRACT FROM AUTHOR]

In this paper, the results of El-Adll et al. [Asymptotic prediction for future observations of a random sample of unknown continuous distribution, Complexity 2022 (2022), Art. ID 4073799], are extended to the lower negative tail index distributions. Three distinct estimators of the lower negative tail index are proposed, as well as an asymptotic confidence interval. Moreover, different asymptotic predictive intervals for future observations are constructed for distributions attracted to the lower extreme value distribution with a negative tail index. Furthermore, the asymptotic maximum likelihood estimator (AMLE) of the shape parameter, as well as an asymptotic maximum likelihood predictor (AMLP), are obtained. Finally, extensive simulation studies are conducted to demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this work, we investigate spacings based on order statistics obtained from continuous distribution functions. At the beginning of the paper, we present distributional results for spacings and a method of classification of distributions according to their tails. Then we use this method to derive asymptotic results for spacings. By applying some special versions of Borel–Cantelli lemma, we obtain their strong limit results. At the end of the paper, we present some illustrative examples. [ABSTRACT FROM AUTHOR]

Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables. [ABSTRACT FROM AUTHOR]

This paper is devoted to the properties and applications of a new discrete model on Z . This model represents the difference of two independent Poisson-Lindley random variables with the same common parameter. The properties, such as, distribution function, moments, quantile function and order statistics are studied. Among its features, it is revealed to be useful for the analysis of overdispersed data sets and also has a flexible probability mass function which is unimodal in nature. Estimation of its parameter by the method of maximum likelihood is obtained. Efficiency of this model with several pertinent competitive models is proved by applying the proposed model to a data set on the change of the stock price. Thanks to its discrete nature, the changes in stock price represent a set of discrete values which suits the nature of the proposed model. [ABSTRACT FROM AUTHOR]

In this article, we consider the estimation problem of Unit Gompertz distribution with parameters α and β under the framework of dual generalized order statistics. This article is purely devoted to present the Bayesian view of estimation of Unit Gompertz distribution. For this purpose, we consider two widely popular approximation methods called Markov chain Monte Carlo and Lindley approximation methods. The results are derived under the symmetric (squared error) and asymmetric (Linear exponential and General entropy) loss functions. Since the order statistics and lower record values are the particular cases of the dual generalized order statistics, a simulation study is provided for order statistics and lower record values to observe the behavior of estimators. The average lengths of highest posterior density intervals of α, β, and R(t) are calculated for 95% confidence coefficient. Finally, real data applications are reported for lower record values and order statistics, separately, to show the practical aspects of the derived results. [ABSTRACT FROM AUTHOR]

Among diseases, cancer exhibits the fastest global spread, presenting a substantial challenge for patients, their families, and the communities they belong to. This paper is devoted to modeling such a disease as a special case. A newly proposed distribution called the binomial-discrete Erlang-truncated exponential (BDETE) is introduced. The BDETE is a mixture of binomial distribution with the number of trials (parameter n ) taken after a discrete Erlang-truncated exponential distribution. A comprehensive mathematical treatment of the proposed distribution and expressions of its density, cumulative distribution function, survival function, failure rate function, Quantile function, moment generating function, Shannon entropy, order statistics, and stress-strength reliability, are provided. The distribution's parameters are estimated using the maximum likelihood method. Two real-world lifetime count data sets from the cancer disease, both of which are right-skewed and over-dispersed, are fitted using the proposed BDETE distribution to evaluate its efficacy and viability. We expect the findings to become standard works in probability theory and its related fields. [ABSTRACT FROM AUTHOR]

Cables or suspenders are the critical force-transmitting components of cable-supported bridges, and their timely tension monitoring is consequently the most important issue in the corresponding structural health monitoring. However, very few works regarding the full automation of vibration-based tension estimation have been reported in the literature. To develop a monitoring system of cable tension based on real-time vibration signals, this research first employs an efficient stochastic subspace identification (SSI) method with tailored parameter selection to continuously identify the three frequencies of adjacent modes for the cables of Mao-Luo-Hsi Bridge. More importantly, an automated sieving algorithm is delicately established to obtain the stable modal frequencies by making the best of the specific modal frequency distribution for cables. The ratios between the frequency values identified from SSI analysis are exhaustively checked to systematically extract the qualified cable frequencies and decide their corresponding mode orders. The tension is finally computed with one available cable frequency according to the priority order predetermined by the statistics of identification rate. Demonstrated by analyzing the vibration signals measured from the stay cable of Mao-Luo-Hsi Bridge in real time for two full years, the effectiveness and robustness of this real-time monitoring system have been extensively testified. The long-term success rates for the immediate determination of dependable tension are found to be perfect for 15 of the 18 investigated cables. As for the other three cables, their corresponding success rates are still higher than 99.99% with very few cases of absent or false tension values. [ABSTRACT FROM AUTHOR]

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$ , the generalized spacing $X_{j:n} - X_{i:n}$ is more dispersed than $Y_{j:n} - Y_{i:n}$ according to dispersive ordering and for $2\le i \le n$ , the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1 :n}$ , in the sense of the more stochastically increasing ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$. [ABSTRACT FROM AUTHOR]

In this article, we propose two new scale invariant test statistics when the available data are subject to progressively Type-II censoring. The proposed tests are based on Gini index of spacings. It is observed thorough extensive Monte Carlo simulations that the proposed tests are quite powerful in compare to similar existing goodness-of-fit tests studied by Balakrishnan et al. and Wang. We also illustrate the method proposed here using a real data from reliability literature. [ABSTRACT FROM AUTHOR]

E‐Bayesian inference is of incremental significance in the area related to the reliability of industrial products dealing with censored data sets. In this article, based on the progressive Type‐II right censored order statistics (PTIIRCOS), the E‐Bayesian estimates (E‐BE) for the parameters of Gompertz distribution (GD) and its hazard function are obtained and compared with the maximum likelihood (MLE) and Bayesian (BE) estimates. The squared error loss function (SE), linear‐exponential loss function (LINEX), and Al‐Bayyati loss function (ALB) are considered for the methods of BE and E‐BE. Some properties for the E‐BE estimates are discussed based on PTIIRCOS. Real data is established to clear the theoretical procedures. Akaike information criterion (AIC) is used to prove that GD is the best to model the data compared to other competitors and could potentially be very adequate in describing and modeling industrial products data. Finally, simulation study has been operated for comparing the E‐BEs with the MLEs and BEs. [ABSTRACT FROM AUTHOR]

Knowable moments, abbreviated as K-moments, are redefined as expectations of maxima or minima of a number of stochastic variables that are a sample of the variable of interest. The new definition enables applicability of the concept to any type of variable, continuous or discrete, and generalization for transformations thereof. While K-moments share some characteristics with classical and other moments, as well as with order statistics, they also have some unique features, which make them useful in relevant applications. These include the fact that they are knowable, i.e., reliably estimated from a sample for high orders. Moreover, unlike other moment types, K-moment values can be assigned values of distribution function by making optimal use of the entire dataset. In addition, K-moments offer the unique advantage of considering the estimation bias when the data are not an independent sample but a time series from a process with dependence. Both for samples and time series, the K-moment concept offers a strategy of model fitting, including its visualization, that is not shared with other methods. This enables utilization of the highest possible moment orders, which are particularly useful in modelling extremes that are closely associated with high-order moments. [ABSTRACT FROM AUTHOR]

The Erlang Truncated (ETE) distribution is modified and the new lifetime distribution is called the Extended Erlang Truncated Exponential (EETE) distribution. Some statistical and reliability properties of the new distribution functions have been characterized based on two non-adjacent generalized and dual generalized order statistics. Moreover, we show that these characterization properties provide a beneficial strategy to predict future events, which are based on past or current events and on an arbitrary distribution function. A characterization in statistics is a specific distributional property of a statistic that uniquely identify related parametric family of distributions. In statistical applications, the researchers usually want to verify whether the data that they are dealing with belong to a certain family of DFs. Therefore, the researchers have to rely on a characterization of the assumed distribution and check if the corresponding conditions are satisfied. [ABSTRACT FROM AUTHOR]