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Being inspired by the biological eye, event camera is a novel asynchronous technology that poses a paradigm shift in acquisition of visual information. This paradigm enables event cameras to capture pixel-size fast motions much more naturally compared to classical cameras.In this paper, we present a new asynchronous event-driven algorithm for detection of high-frequency pixel-size periodic signals using an event camera. Development of such new algorithms to efficiently process the asynchronous information of event cameras is essential to utilize its special properties and potential, and being a main challenge in the research community.It turns out that this algorithm, which was developed in order to satisfy the new paradigm, is related to an untreated theoretical problem in probability: Let 0 ≤ τ1 ≤ τ2 ≤⋯ ≤ τm ≤ 1 originated from an unknown distribution. Let also 휖,δ ∈ ℝ, and d ∈ ℕ. What can be said about the probability Φ(m,d) of having more than d adjacent τi-s pairs that the distance between them is δ, up to an error 휖? This problem, which reminds the area of order statistic, shows how the new visualization paradigm is also an opportunity to develop new areas and problems in mathematics. [ABSTRACT FROM AUTHOR]

In this article, we improve the behavior of nonparametric Shewhart-type control charts, which employ order statistics and multiple runs-type rules. The proposed class of monitoring schemes includes some existing control charts. In addition, new distribution-free monitoring schemes that pertain to the class, are set up and examined extensively. Explicit expressions for determining the variability and the mean of the run length distribution for the enhanced control charts are additionally delivered. As an example, a real-life managerial application is considered, where the proposed framework is implemented in order to enhance the provided services of a company under a supply chain management environment. Based on numerical comparisons, we draw the conclusion that the new charts outperform their competitors in identifying potential changes in the fundamental distribution in almost all cases considered.

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.

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 two-parameter 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.

In this paper, we introduced a family of distributions with a very flexible shape named generalized scale mixtures of generalized asymmetric normal distributions (GSMAGN). We investigated the main properties of the new family including moments, skewness, kurtosis coefficients and order statistics. A variant of the expectation maximization (EM)-type algorithm was established by combining the proflie likihood approach (PLA) with the classical expectation conditional maximization (ECM) algorithm for parameter estimation of this model. This approach with analytical expressions in the E-step and tractable M-step can greatly improve the computational speed and efficiency of the algorithm. The performance of the proposed algorithm was assessed by some simulation studies. The feasibility of the proposed methodology was illustrated through two real datasets.

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.

We consider a random permutation Τn uniformly distributed on the set of all per- mutations of degree n whose cycle lengths lie in a fixed set A (the so-called A-permutations). Let ζn be the total number of cycles, and let ηn(1) ≥ ηn(2) ≥ • • • ≥ ηn(ζn) be the ordered sample of cycle lengths of the permutation Τn. We consider a class of sets A with positive density in the set of natural numbers. We study the asymptotic behavior of ηn(m) with numbers m in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence ηn(m) dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340--357] who considered the case A = N. [ABSTRACT FROM AUTHOR]

This paper presents several nonparametric estimators for the Sharma–Taneja–Mittal entropy measure of a continuous random variable with known support, utilizing spacing, a local linear model, and a kernel function. The properties of these estimators are discussed. Their performance was also examined through real data analysis and Monte Carlo simulations. In the Monte Carlo experiments, the proposed Sharma–Taneja–Mittal entropy estimators were employed to create a test of goodness-of-fit under the standard uniform distribution. The suggested test statistics demonstrate strong performance, as evidenced by a comparison of their power with that of other tests for uniformity. Finally, we examine a classification issue in the recognition of patterns to underscore the significance of these measures.

Moments of order statistics (OSs) characterize the Weibull–geometric and half-logistic families of distributions, of which the extended exponential–geometric (EEG) distribution is a particular case. The EEG distribution is used to create the log-extended exponential–geometric (LEEG) distribution, which is bounded in the unit interval (0, 1). In addition to the generalized Stirling numbers of the first kind, a few years ago, the polylogarithm function and the Lerch transcendent function were used to determine the moments of order statistics of the LEEG distributions. As an application based on the L-moments, we expand the features of the LEEG distribution in this work. In terms of the Gauss hypergeometric function, this work presents the precise equations and recurrence relations for the single moments of OSs from the LEEG distribution. Along with recurrence relations between the expectations of function of two OSs from the LEEG distribution, it also displays the truncated and conditional distribution of the OSs. Additionally, we use the L-moments to estimate the parameters of the LEEG distribution. We further fit the LEEG distribution on three practical data sets from medical and environmental sciences areas. It is seen that the estimated parameters through L-moments of the OSs give a superior fit. We finally determine the correspondence between the entropies and the OSs.

In this paper, a new estimator for kernel quantile estimation is given to reduce the bias. The asymptotic properties of the proposed estimator was established and it turned out that the bias has been reduced to the fourth power of the bandwidth, while the bias of the estimators considered has the second power of the bandwidth, while the variance remains at the same order. Futhermore, we calculate the optimal bandwidth which minimizes the asymptotic mean squared error. A simulation study and a real data example are carried out to illustrate the performance of the proposed estimator and compared with other existing approaches mentioned.

In this paper the local limit theorem for density of k-th maxima of independent identically distributed random variables is proved.

When the random sample size is assumed to converge weakly and to be independent of the basic variables, the asymptotic distributions of extreme, intermediate, and central order statistics, as well as record values, for a mixture of two stationary Gaussian sequences under an equi-correlated setup are derived. Furthermore, sufficient conditions for convergence are derived in each case. An interesting fact is revealed that in several cases, the limit distributions of the aforementioned statistics are the same when the sample size is random and non-random. e.g., when one mixture component has a correlation that converges to a non-zero value.

This paper explores the concept of residual extropy as an uncertainty measure for order statistics. We specifically derive the residual extropy for the ith-order statistic and establish its relationship with the residual extropy of the ith-order statistic from a random sample generated from a uniform distribution. By employing this approach, we obtain a formula for the residual extropy of order statistics applicable to general continuous distributions. In addition, we offer two lower bounds that can be applied in situations where obtaining closed-form expressions for the residual extropy of order statistics in diverse distributions proves to be challenging. Additionally, we investigate the monotonicity properties of the residual extropy of order statistics. Furthermore, we present other aspects of the residual extropy of order statistics, including its dependence on the position of order statistics and various features of the underlying distribution.

From the perspective of wireless communication, as communication systems become more complex, order statistics have gained increasing importance, particularly in evaluating the performance of advanced technologies in fading channels. However, existing analytical methods are often too complex for practical use. In this research paper, we introduce innovative statistical findings concerning the sum of ordered gamma-distributed random variables. We examine various channel scenarios where these variables are independent but not-identically distributed. To demonstrate the practical applicability of our results, we provide a comprehensive closed-form expression for the statistics of the signal-to-interference-plus-noise ratio in a multiuser scheduling system. We also present numerical examples to illustrate the effectiveness of our approach. To ensure the accuracy of our analysis, we validate our analytical results through Monte Carlo simulations.

The control chart is a fundamental tool in statistical process control (SPC), widely employed in manufacturing and construction industries for process monitoring with the primary objective of maintaining quality standards and improving operational efficiency. Control charts play a crucial role in identifying special cause variations and guiding the process back to statistical control. While Shewhart control charts excel at detecting significant shifts, EWMA and CUSUM charts are better suited for detecting smaller to moderate shifts. However, the effectiveness of all these control charts is compromised when the underlying distribution deviates from normality. In response to this challenge, this study introduces a robust mixed EWMA-CUSUM control chart tailored for monitoring processes characterized via symmetric but non-normal distributions. The key innovation of the proposed approach lies in the integration of a robust estimator, based on order statistics, that leverages the generalized least square (GLS) technique developed by Lloyd. This integration enhances the chart’s robustness and minimizes estimator variance, even in the presence of non-normality. To demonstrate the effectiveness of the proposed control chart, a comprehensive comparison is conducted with several well-known control charts. Results of the study clearly show that the proposed chart exhibits superior sensitivity to small and moderate shifts in process parameters when compared to its predecessors. Through a compelling illustrative example, a real-life application of the enhanced performance of the proposed control chart is provided in comparison to existing alternatives.

The modern trend in distribution theory is to propose hybrid generators and generalized families using existing algebraic generators along with some trigonometric functions to offer unique, more flexible, more efficient, and highly productive G-distributions to deal with new data sets emerging in different fields of applied research. This article aims to originate an odd sine generator of distributions and construct a new G-family called "The Odd Lomax Trigonometric Generalized Family of Distributions". The new densities, useful functions, and significant characteristics are thoroughly determined. Several specific models are also presented, along with graphical analysis and detailed description. A new distribution, "The Lomax cosecant Weibull" (LocscW), is studied in detail. The versatility, robustness, and competency of the LocscW model are confirmed by applications on hydrological and survival data sets. The skewness and kurtosis present in this model are explained using modern graphical methods, while the estimation and statistical inference are explored using many estimation approaches.

The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the α-quantile residual lifetime function was studied. Then, the change points of the α-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.

In this paper, we derive the L-moments for some distributions, such as logistic, generalized logistic, doubly truncated logistic, and doubly truncated generalized logistic distributions. We also establish some new axioms and identities, including recurrence relations satisfied by the L-moment from the underlying derivations. In addition, we establish some new general recurrence relations satisfied by the L-moment from any distribution.

We come up with a new class called log-logistic tan generalized family which provides sub-models with left skewed, symmetrical, right skewed, unimodal, bimodal and reversed-J densities, and increasing, decreasing, modified bathtub, bathtub, unimodal, reversed-J shaped, and J-shaped hazard rates. Some of its sub-models are provided along with some general structural properties. The parameter estimation has been conducted via maximum likelihood. Moreover, the estimators behavior are assessed using various simulation results. The capability of the log-logistic tan-Weibull model is proved using two real-life data sets. It provides higher quality fit than competing Weibull extensions, among others.

This paper introduces a new extension of the traditional Lomax distribution. The current distribution, which has one scale and two shape parameters, is known as logarithmic transformed Lomax distribution. The hazard rate function of the new distribution based on parameter values, making monotonically decreasing, upside-down and bathtub upside-down shaped. Some properties of the proposed distribution including mixture expansion, moments, entropies, probability weighted moments, order statistics, reversed failure rate and mean past life time are discussed. The distribution parameters are estimated using four frequentist methods: namely, maximum likelihood, least squares, weighted least squares, and maximum product of spacing. The efficiency of the different estimators is compared using a simulation analysis using mean square error. Finally, two real data sets for service times of Aircraft Windshield and survival times of a group of patients given chemotherapy medication are analyzed and compared to some other competing distributions. Based on the results of data analysis, the estimates of minimum and maximum values for the service times of Aircraft Windshield and the survival times of patients given chemotherapy medication are obtained.

In this work, we reveal some distributional traits of concomitants of order statistics (COSs) arising from the extended Farlie–Gumbel–Morgenstern (FGM) bivariate distribution, which was developed and studied in recent work. The joint distribution and product moments of COSs for this family are discussed. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. In addition, the asymptotic behavior of the concomitant’s rank for order statistics (OSs) is studied. The information measures, differential entropy, Kullback–Leibler (KL) distance, Fisher information number (FIN), and cumulative past inaccuracy (CPI) are theoretically and numerically studied.

In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and 2-out-of-n/2-out-of-(n+1) systems. Finally, some examples are given to illustrate the obtained results.

We suggest a new method for constructing an efficient point predictor for the future order statistics when the sample size is a random variable. The suggested point predictor is based on some characterization properties of the distributions of order statistics. For several distributions, including the mixture distribution, the performance of the suggested predictor is evaluated by means of a comprehensive simulation study. Three examples of real lifetime data-sets are analyzed by using this method and compared with an efficient recent method given by Barakat et al. [1], that deals with non-random sample sizes. One of these examples predicts the accumulative new cases per million for infection of the new Coronavirus (COVID-19).

In this paper, we carry out stochastic comparisons of lifetimes of series and parallel systems with dependent heterogeneous Topp-Leone generated components. The usual stochastic order and the reversed hazard rate order are developed for the parallel systems with the help of vector majorization under Archimedean copula dependence, and the results for the usual stochastic order are also obtained for the series systems. Finally, some numerical examples are provided to illustrate the effectiveness of our theoretical findings.

This article introduces a new three-parameter Marshall-Olkin Burr-R (MOB-R) family which extends the generalize Burr-G class. Some of its general properties are discussed. One of its special models called the MOB-Lomax distribution is studied in detail for illustrative purpose. A modified chi-square test statistic is provided for right censored data from the MOB-L distribution. The model parameters are estimated via the maximum likelihood and simulation results are obtained to assess the behavior of the maximum likelihood approach. Applications to real data sets are provided to show the usefulness of the proposed MOB-Lomax distribution. The modified chi-square test statistic shows that the MOB-Lomax model can be used as a good candidate for analyzing real censored data.

Multiple comparisons methodology has experienced a resurgence of interest due to the increase in high-dimensional datasets generated from various biological, medical and scientific fields. An outstanding problem in this area is how to perform testing in the presence of dependence between the p-values. We propose a novel approach to this problem based on a spacings-based representation of the Benjamini-Hochberg procedure. The representation leads to a new application of the wavelet transform to effectively decorrelate p-values. Theoretical justification for the procedure is shown. The power gains of the proposed methodology relative to existing procedures is demonstrated using both simulated and real datasets.

The objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

Abstract Let {Xni,1≤i≤mn,n≥1} $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}$ be an array of independent random variables with uniform distribution on [0,θn] $[0, \theta _{n}]$, and {Xn(k),k=1,2,…,mn} $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}$ be the kth order statistics of the random variables {Xni,1≤i≤mn} $\{X_{ni}, 1 \leq i \leq m_{n}\}$. We study the limit properties of ratios {Rnij=Xn(j)/Xn(i),1≤i

We consider the reliability function of a k-out-of-n system under conditions that failures of its components lead to an increase in the load on the remaining ones and, consequently, to a change in their residual lifetimes. Development of models able to take into account that failures of a system’s components lead to a decrease in the residual lifetime of the surviving ones is of crucial significance in the system reliability enhancement tasks. This paper proposes a novel approach based on the application of order statistics of the system’s components lifetime to model this situation. An algorithm for calculation of the system reliability function and two moments of its uptime has been developed. Numerical study includes sensitivity analysis for special cases of the considered model based on two real-world systems. The results obtained show the sensitivity of system’s reliability characteristics to the shape of lifetime distribution, as well as to the value of its coefficient of variation at a fixed mean.

In the present paper, we deliver a reliability study of the weighted-r-consecutive-k-out-of-n: F reliability systems consisting of independent and identically distributed components. The signature vector of the structures is arithmetically determined by the aid of an appropriate Monte-Carlo simulation scheme. Several conclusions for the corresponding reliability polynomials are also drawn. An extensive numerical investigation is carried out to evaluate the performance of the weighted-r-consecutive-k-out-of-n: F reliability systems. Some stochastic signature-based comparisons among several members of the underlying family of reliability structures are also discussed.

In this paper, we introduced a generalization of the raised cosine distribution. We also provided its several distributional properties and characterizations, including percentiles and some applications.

In this paper, we have considered that ranked set sampling is able to estimate the parameters of exponentiated Pareto distribution. The method with which the maximum likelihood estimators for the parameters of exponentiated Pareto distribution is studied is numerical since there is no presence or possibility of a closed-form at the hands of estimators or any other intellectual. The numerical approach is a well-suited one for this study as there has been struggles in achieving it with any other technique. In order to compare the different sampling methods, simulation studies are performed as the main technique. As for the illustrative purposes, analysis of a simulated dataset is desired for the objective of the presentation. The conclusion that we can reach based on these is that the estimators based on the ranked set sample have far better efficiency than the simple random sample at the same sample size.

The generalized Bilal (GB) distribution can be defined as the distribution of the median of three independent random variables drawn from the Weibull distribution. Its failure rate function can be monotonic (decreasing or increasing) or upside-down bathtub-shaped. In this study, we aim to reveal some important properties of the GB distribution that have not been considered before. The findings are both theoretical and practical. From the theoretical viewpoint, we present explicit expressions for both single and product moments of order statistics from the GB distribution. The L-moments are derived as well. From the practical viewpoint, the parameter estimations are accomplished using the maximum likelihood (ML) method, which is based on two different sampling schemes: simple random sampling (SRS) and ranked set sampling (RSS) schemes. Furthermore, the asymptotic confidence intervals for the SRS and RSS estimators are discussed. For the sake of comparison and illustration, a simulation study and a real data example are presented. Concluding remarks are given at the end.

In this article, Burr III distribution is proposed with a significantly improved functional form. This new modification has enhanced the flexibility of the classical distribution with the ability to model all shapes of hazard rate function including increasing, decreasing, bathtub, upside-down bathtub, and nearly constant. Some of its elementary properties, such as rth moments, sth incomplete moments, moment generating function, skewness, kurtosis, mode, ith order statistics, and stochastic ordering, are presented in a clear and concise manner. The well-established technique of maximum likelihood is employed to estimate model parameters. Middle-censoring is considered as a modern general scheme of censoring. The efficacy of the proposed model is asserted through three applications consisting of complete and censored samples.

Diagnostics of goodness-of-fit in the theory of point processes are often considered through the transformation of data into residuals as a result of a thinning or a rescaling procedure. We alternatively consider here second-order statistics coming from weighted measures. Motivated by Adelfio and Schoenberg (Ann Inst Stat Math 61(4):929–948, 2009) for the temporal and spatial cases, we consider an extension to the spatio-temporal context in addition to focussing on local characteristics. In particular, our proposed method assesses goodness-of-fit of spatio-temporal models by using local weighted second-order statistics, computed after weighting the contribution of each observed point by the inverse of the conditional intensity function that identifies the process. Weighted second-order statistics directly apply to data without assuming homogeneity nor transforming the data into residuals, eliminating thus the sampling variability due to the use of a transforming procedure. We provide some characterisations and show a number of simulation studies. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the ratios of order statistics based on samples drawn from uniform distribution and establish some limit properties such as the almost sure central limit theorem, the large deviation principle, the Marcinkiewicz-Zygmund law of large numbers and complete convergence.

We apply the result of conditional independence of blocked ordered data established by Iliopoulos and Balakrishnan, together with the algorithm of Huffer and Lin, to obtain exact expressions for the distributions of linear combinations of Laplace order statistics. These expressions can then be evaluated using symbolic math package Maple. Finally, we illustrate the computational method with some examples. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized t (new Gt) distribution based on a distribution construction approach is proposed and proved to be suitable for fitting both the data with high kurtosis and heavy tail. The main innovation of this article consists of four parts. First of all, the main characteristics and properties of this new distribution are outined. Secondly, we derive the explicit expression for the moments of order statistics as well as its corresponding variance–covariance matrix. Thirdly, we focus on the parameter estimation of this new Gt distribution and introduce several estimation methods, such as a modified method of moments (MMOM), a maximum likelihood estimation (MLE) using the EM algorithm, a novel iterative algorithm to acquire MLE, and improved probability weighted moments (IPWM). Through simulation studies, it can be concluded that the IPWM estimation performs better than the MLE using the EM algorithm and the MMOM in general. The newly-proposed iterative algorithm has better performance than the EM algorithm when the sample kurtosis is greater than 2.7. For four parameters of the new Gt distribution, a profile maximum likelihood approach using the EM algorithm is developed to deal with the estimation problem and obtain acceptable.

This article presents a new method for generating distributions. This method combines two techniques—the transformed—transformer and alpha power transformation approaches—allowing for tremendous flexibility in the resulting distributions. The new approach is applied to introduce the alpha power Weibull—exponential distribution. The density of this distribution can take asymmetric and near-symmetric shapes. Various asymmetric shapes, such as decreasing, increasing, L-shaped, near-symmetrical, and right-skewed shapes, are observed for the related failure rate function, making it more tractable for many modeling applications. Some significant mathematical features of the suggested distribution are determined. Estimates of the unknown parameters of the proposed distribution are obtained using the maximum likelihood method. Furthermore, some numerical studies were carried out, in order to evaluate the estimation performance. Three practical datasets are considered to analyze the usefulness and flexibility of the introduced distribution. The proposed alpha power Weibull–exponential distribution can outperform other well-known distributions, showing its great adaptability in the context of real data analysis.

Abstract In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for R n i j $R_{nij}$ with 1 ≤ i < j < m n = m $1\leq i< j< m_{n}=m$ . We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of R n i j $R_{nij}$ with 2 ≤ i < j < m n = m $2\leq i< j< m_{n}=m$ .

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions.

We obtain explicit expressions for single and product moments of the order statistics of an omega distribution. We also discuss seven methods to estimate the omega parameters. Various simulation results are performed to compare the performance of the proposed estimators. Furthermore, the maximum likelihood method is adopted to estimate the omega parameters under the type II censoring scheme. The usefulness of the omega distribution is proven using a real data set.

In this article, we introduce the exponentiated Fréchet-G family of distributions. Several models of the introduced exponentiated Fréchet-G family are presented. The proposed family is precisely more flexible and effective in modeling complex data and is instrumental in reliability analysis. It covers a wide variety of shapes, such as unimodal, reverse J, right-skewed, symmetrical, and asymmetrical shapes. Various structural mathematical properties, such as the quantile, moment, incomplete moment, entropy, and order statistics, are derived. The parameters are evaluated using a parametric estimation method. The performance and flexibility of the exponentiated Fréchet-G family are analyzed via a simulation and two applications; one deals with reliability data, and the other deals with medical data.

In Statistical theory, inclusion of an additional parameter to standard distributions is a usual practice. In this study, a new distribution referred to as Alpha-Power Pareto distribution is introduced by including an extra parameter. Several properties of the proposed distribution, including moment generating function, mode, quantiles, entropies, mean residual life function, stochastic orders and order statistics are obtained. Parameters of the proposed distribution have been estimated using maximum likelihood estimation technique. Two real datasets have been considered to examine the usefulness of the proposed distribution. It has been observed that the proposed distribution outperforms different variants of Pareto distribution on the basis of model selection criteria. [ABSTRACT FROM AUTHOR]

The unit-Rayleigh distribution is a one-parameter distribution with support on the unit interval. It is defined as the so-called unit-Weibull distribution with a shape parameter equal to two. As a particular case among others, it seems that it has not been given special attention. This paper shows that the unit-Rayleigh distribution is much more interesting than it might at first glance, revealing closed-form expressions of important functions, and new desirable properties for application purposes. More precisely, on the theoretical level, we contribute to the following aspects: (i) we bring new characteristics on the form analysis of its main probabilistic and reliability functions, and show that the possible mode has a simple analytical expression, (ii) we prove new stochastic ordering results, (iii) we expose closed-form expressions of the incomplete and probability weighted moments at the basis of various probability functions and measures, (iv) we investigate distributional properties of the order statistics, (v) we show that the reliability coefficient can have a simple ratio expression, (vi) we provide a tractable expansion for the Tsallis entropy and (vii) we propose some bivariate unit-Rayleigh distributions. On a practical level, we show that the maximum likelihood estimate has a quite simple closed-form. Three data sets are analyzed and adjusted, revealing that the unit-Rayleigh distribution can be a better alternative to standard one-parameter unit distributions, such as the one-parameter Kumaraswamy, Topp–Leone, one-parameter beta, power and transmuted distributions.

Statistical analysis of attacks on symmetric ciphers often requires assuming the normal behaviour of a test statistic. Typically such an assumption is made in an asymptotic sense. In this work, we consider concrete versions of some important normal approximations that have been made in the literature. To do this, we use the Berry–Esséen theorem to derive explicit bounds on the approximation errors. A basic mathematical requirement is that such approximation errors should be within reasonable bounds, a point which appears to have been overlooked in many of the earlier works on statistical aspects of cryptanalysis. Interpreting the error bounds in the cryptanalytic context yields several surprising results. One important implication is that this puts in doubt the applicability of the order statistics based approach for analysing key recovery attacks on block ciphers. This approach has been earlier used to obtain several results on the data complexities of (multiple) linear and differential cryptanalysis. The non-applicability of the order statistics based approach puts a question mark on the data complexities obtained using this approach. Fortunately, we are able to recover all of these results by utilising the hypothesis testing framework. This, however, necessitates using normal approximations for the χ2${\chi ^2}$ and the LLR test statistics considered in earlier works. These approximations themselves have issues which seem to be difficult to resolve satisfactorily. More generally, the message of our work is that all cryptanalytic attacks should properly derive and interpret the error bounds for any (normal) approximation that is made.

In this paper, we consider the ratios of order statistics in samples from uniform distribution and establish strong and weak laws for these ratios.

Some distributional properties of the symmetric arcsine distribution on (−1, 1) is presented. Based on the distributional properties, several characterizations of the arcsine distribution are given.

In this note we give some recurrence relations satisfied by single and product moments of k-th upper record values from the exponential-Weibull lifetime distribution. Using a recurrence relation for single moments we obtain a characterization of exponential-Weibull, Weibull, exponential, Rayleigh and two parameter linear failure rate distributions.