Moments Of Order Statistics Research Articles

Modal identification of wind turbine tower based on optimal fractional order statistical moments

AbstractIn vibration testing of civil engineering structures, the first two vibration modes are crucial in representing the global dynamic behavior of the structure measured. In the present study, a comprehensive method is proposed to identify the first two vibration modes of wind turbine towers, which is based on the analysis of fractional order statistical moments (FSM). This study offers novel contributions in two key aspects: (1) theoretical derivations of the relationship between FSM and vibration mode; and (2) successful use of 32/7‐order displacement statistical moment as the optimal FSM to identify wind turbine tower modes, by combining with noise resistance analysis, sensitivity analysis, and stability analysis, respectively. Using the proposed method, the FSM was first used to identify the modal vibration of wind turbine towers. By obtaining the response of the structure on the same vertical line, FSM was then calculated to estimate the corresponding structural modal vibration. Considering other influencing factors in the field test, the modal identification results of this index under different excitation forms and noise conditions were analyzed based on numerical simulation and verified with field wind tower test data. The results of the evaluation show that the proposed statistical moments of can accurately identify the first two vibration modes of wind turbine towers. This presents a new robust method for modal vibration identification, that is, simple and effective in its implementation.

Cumulative entropies and sums of moments of order statistics

Generalized weighted cumulative residual entropies and generalized weighted cumulative entropies are represented by means of weighted sums of moments of upper and lower order statistics, respectively. A variety of examples is shown by specifying the weight function.

Order Statistics and Actuarial Measures from Powered Inverse Rayleigh Distribution

Nashaat [25] introduced the powered inverse Rayleigh (PIR) distribution. It provides a better fit other than (inverse Rayleigh, Rayleigh, and Weibull) distributions. The moments of order statistics and recurrence relations for the single and double moments have been established. The computation of the means and variances are enumerated. These computations can be trulyinteresting and applied in numerous domains of study. Moreover, cumulative entropy (C.E.) and actuarial measures (A.M.) are also calculated to address the uncertainty in portfolio optimization. The usages of C. E. and A.M. are widespread in many real-word applications specifically in physical sciences and insurance science.

A point mapping strategy-based sparse grid integration method for statistical moments estimation and structural reliability analysis

The high-order moment-based method is an attractive non-intrusive approach for structural reliability analysis, while it remains a challenge to strike a balance between accuracy, efficiency, and versatility in estimating statistical moments of performance functions. In this paper, a point mapping strategy-based sparse grid integration (SGI) method is proposed for statistical moment estimation and structural reliability analysis, with improved efficiency while ensuring the accuracy. Firstly, a point mapping strategy (PMS) is developed based on the adaptive high dimensional model representation (HDMR), by which the traditional SGI is reconstructed as a sum of multiple integrations over the sparse grid point sets with different effective dimensions. With this new form of SGI, the estimation of statistical moments only requires the evaluation of the performance function at a small number of necessary reference points, once the effects of some interaction terms are negligible. The validity and feasibility of the proposed PMS-based SGI method for statistical moments estimation are verified by several numerical examples. Then, the proposed method is applied to structural reliability analysis with the aid of an improved maximum entropy method. Two engineering examples are investigated, including the strength failure problem of tubular tower K-shape connection joints and the fatigue failure problem of a marine riser resulting from the vortex-induced vibration (VIV). The results indicate that the proposed method provides a better trade-off between accuracy and efficiency in evaluating the first four order statistical moments and structural failure probability compared to conventional methods.

A novel skewed generalized normal distribution: properties, statistical inference, and its applications

In this paper, a novel skewed generalized normal distribution (NSGN), which encompasses several well-known distributions, is a two-component mixture model based on the generalized normal distribution. The NSGN distribution offers flexibility in modeling skewed data characterized by high kurtosis and strong skewness, making it highly applicable in finance, medicine, and engineering. The primary characteristics and properties of this new distribution are introduced, and the explicit expressions for the moments of order statistics are provided using recursive relations. The maximum likelihood estimation based on a profile likelihood approach, L-moments estimation, and a two-step estimation method combining Bayesian posterior maximum estimation with moment estimates are presented to estimate the four parameters of NSGN distribution. A simulation study compares the performance of these methods across different sample sizes and different parameters. The appropriateness of the proposed distribution has been tested by comparing it with several skewed distributions. Additionally, applications to real datasets showcase the beneficial properties of the NSGN for applied statistical research.

Approximations to the Moments of Order Statistics for Normal Distribution

Approximations to the Moments of Order Statistics for Normal Distribution

Linear Combination of Order Statistics Moments from Log-Extended Exponential Geometric Distribution with Applications to Entropy

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.

A statistical framework for a new Kavya-Manoharan Bilal distribution using ranked set sampling and simple random sampling

In survival and stochastic lifespan modeling, numerous families of distributions are sometimes considered unnatural, unjustifiable theoretically, and occasionally superfluous. Here, a novel parsimonious survival model is developed using the Bilal distribution (BD) and the Kavya-Manoharan (KM) parsimonious transformation family. In addition to other analytical properties, the forms of probability density function (PDF) and behavior of the distributions' hazard rates are analyzed. The insights are theoretical as well as practical. Theoretically, we offer explicit equations for the single and product moments of order statistics from Kavya-Manoharan Bilal Distribution. Practically, maximum likelihood (ML) technique, which is based on simple random sampling (SRS) and ranked set sampling (RSS) sample schemes, is employed to estimate the parameters. Numerical simulations are used as the primary methodology to compare the various sampling techniques.

An active multiphase probabilistic power flow based on a clustering approach

Probabilistic methods for analyzing power systems have become more widespread, mainly due to the increase in load and generation uncertainties and the current need for planning problems that are sufficiently accurate to take appropriate actions. However, some characteristics need to be better investigated in the Distribution Systems (DS) analysis, such as the input random variables’ modeling using real databases and, mainly, the probabilistic treatment of unbalance and scalability studies. This paper proposes an active Multiphase Probabilistic Power Flow (MPPF) evaluation, considering real measurement databases to represent customer demand uncertainties. It is called Multiphase Probabilistic Power Flow Based on K-Means (MPPFKM), and it encompasses a mixture of a clustering technique and an analytical method. The K-Means algorithm divides the input database into a set of clusters that models the scenarios to be evaluated in the multiphase power flow problem, which considers the distribution network with its inherent unbalance. Subsequently, the output variables’ higher order statistical moments are calculated using the obtained responses for each cluster. Finally, their PDFs and CDFs are recovered using the Gram-Charlier Expansion analytical method. While the clustering algorithm allows working with real data, giving the proposed work greater proximity to real-world results, the analytical tool provides an accurate recovery of the probability functions, because it detects their distortions caused by the nature of the multiphase problem. Estimating the correct number of clusters is a key factor in obtaining the precision required by power systems engineers and achieving significant computational time reduction, especially for large-scale unbalanced systems. Simulations are carried out using several DS feeders to test the proposed method, which specifically addresses DS issues not yet well explored in the literature, such as probabilistic simulations under different unbalance situations, applications with single-phase DGs, and scalability studies are conducted using the IEEE 8500 test feeder. Comparisons are made with usual MPPF algorithms, including Monte Carlo Simulation (MCS) and Point Estimation Methods (PEM). Simulations using large-scale unbalanced systems proved that the output variables' probability functions might present distortions, becoming more challenging for other methods. Under this perspective, MPPFKM presents better exactness in the calculation of higher order statistical moments and, consequently, can estimate the PDFs of the MPPF output variables with great precision, which is crucial for statistical models that use actual input data.

On the theory of order statistics of the flexible Lomax distribution

This paper studies the flexible Lomax distribution’s order statistics with graphical and numerical findings. Along with the quantitative measurements, some plots are furnished, including those for the skewness and kurtosis measures. We will dwell on the numerous results that relate to statistics of moments of order. We consider the single and product moment of order statistics from the new distribution. Further, we establish some recurrence relation for single moments of order statistics. We have sought to apply the derived relations to empirically evaluate the moments of smallest (largest) order statistics to establish well-known moments and related measures. For order statistics of a flexible Lomax distribution, exact analytical expressions of entropy, residual entropy, and past latent entropy are determined.

Integral Turbulent Length and Time Scales of Higher Order Moments

Turbulent length and time scales represent a fundamental quantity for analysing and modelling turbulent flows. Although higher order statistical moments have been conveniently used for decades to describe the mean behaviour of turbulent fluid flow, the definition of the integral turbulent scales seems to be limited to the velocity or its fluctuation itself (i.e. the first moment). Higher order moments are characterized by smaller integral scales and a framework is proposed for estimating autocorrelation functions and integral turbulent length or time scales of higher order moments under the assumption that the probability distribution of the velocity field is Gaussian. The new relations are tested for synthetic turbulence as well as for DNS data of a turbulent plane jet at Reynolds number 10000. The present results in particular suggest that the length or time scales of higher order moments can be markedly smaller than those of the turbulent variable itself, which has implications for statistical uncertainty estimates of higher order moments.

Non-Gaussian buffeting analysis of large structures by means of a Proper Orthogonal Decomposition

Proper Orthogonal Decomposition is implemented in a Bispectral Stochastic Analysis of large MDOF structures. Although Higher Order Stochastic Analysis has been introduced few decades ago, its establishment has found place only in a theoretical way, or applied to very small systems. Its heavy computational cost as well as resource consumption burden has let it aside for practical civil engineering applications. Nevertheless, in wind engineering, the community is increasingly considering the importance of non-Gaussian nature of wind induced vibrations. This is of interest for the extreme value analysis, for which many theoretical and empirical models exist for non-Gaussian random processes. In this context, a (bi-)spectral approach is realized in a 2-D frequency space where the bispectrum is computed then the 3rd statistical moment is obtained by means of a twofold integration. These operations are quite heavy as soon as the dimension of the problem increases. A novel algorithm implementation is proposed. It hinges on (i) the use of Proper Orthogonal Decomposition and the (ii) development of an optimized numerical method. The proposed algorithmic arrangement minimizes the number of operations, and consequently saves time and memory, while conserving precision in estimating 3rd order statistical moments of structural responses.

Higher-order statistical moments of predicted sonic boom waveforms through turbulence

Sonic booms generated by supersonic aircraft are affected by turbulence in the atmospheric boundary layer through which they propagate. Turbulence effects lead to random variability of the sonic boom waveforms measured on the ground, complicating the prediction of such waveforms. As an initial effort to predict the waveform variability, the solution of the coherent or mean sonic boom waveform has previously been formulated by the author. The current paper extends the formulation to derive an expression of the second-order statistical moment necessary to calculate the variance of the spectral amplitudes. Since the derivation uses a full wave equation, results using the derived expression will be compared with those obtained using a parabolic approximation. In order to fully quantify the uncertainty of our predictions, the higher order statistical moments are also formulated and are shown to be approximately zero. Consequently, the probability density function (pdf) of the spectral amplitudes is predicted to be Gaussian. The formulation is further extended to determine the pdf of the loudness of sonic booms and quantify the uncertainties associated with the loudness prediction. Results from the formulation are compared with available flight test data.

Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime

In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range 1,200, common in industrial and environmental processes. The proposed expressions are derived by fitting the results obtained by means of Particle-Resolved Direct Numerical Simulations (PR-DNS) for a uniform flow around different sets of irregular particles and considering a large number of random orientations. The resulting flow resistance coefficients can be approximated by normal distributions whose first and second order statistical moments (i.e., mean and standard deviation) are described in this work by fitting functions depending of particle Reynolds number and sphericity. The derived correlations are simple, can be easily implemented and allow the tracking of irregular particles in a Lagrangian stochastic framework.

An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia

ABSTRACT In this research paper, we introduce an extension of the Gamma-Lindley distribution using a particular exponentiation of its cumulative distribution function, which offers a more flexible model for lifetime data. Another attractive feature of this extension is that it has several particular cases: Weibull, generalized Pareto, Lindley, exponential and Gamma-Lindley distributions. Different statistical properties of this distribution are explored, such as the density, failure rate and the r th moments. We attempt to prove that the extended Gamma-Lindley distribution is characterized by its truncated moment of order statistics. One of the main merits of this work lies in the fact that it provides the ability of this characterization to simulate the new distribution compared with the inverse transform sampling method. Estimation of the parameters using the tailed regression method is investigated. The COVID-19 real data evolution in Tunisia illustrates the performance of the EGL distribution compared with its special cases through some criteria such as the Kolmogorov-Smirnov test, Mean Squared error and Kullback-Leibler divergence. The predictive ability of the extended Gamma-Lindley distribution proved to provide a better fit to expect the number of deaths by Corona-virus in Tunisia in the future period.

Easily Changeable Kurtosis Distribution

The goal of this paper is to introduce the easily changeable kurtosis (ECK) distribution. The uniform distribution appears as a special cases of the ECK distribution. The new distribution tends to the normal distribution. Properties of the ECK distribution such as PDF, CDF, modes, inflection points, quantiles, moments, moment generating function, Moors’ measure, moments of order statistics, random number generator and the Fisher Information Matrix are derived. The unknown parameters of the ECK distribution are estimated by the maximum likelihood method. The Shannon, Renyi and Tsallis entropies are calculated. Illustrative examples of applicability and flexibility of the ECK distribution are given. The most important R codes are presented in the Appendix.

Parameter estimation of a model using maximum likelihood function and Bayesian analysis through moment of order statistics

Due to advancements in Bayesian modeling, the likelihood-free posterior estimate is now feasible. These estimation methods are crucial for a deeper understanding of simulation-based systems since it might be difficult, if not impossible, to estimate probability values. This compares the effectiveness of such updated estimate approaches to earlier maximum likelihood forecasting models and offers some adjustments to maximum likelihood estimation for estimating the parameters of the Bayesian analysis in this work. Recent improvements in Bayesian modeling have made it feasible to obtain the likelihood-free posterior estimate. These estimate methods are essential for evaluating simulation-based theories since it might be difficult, if not impossible, to determine probability values. Simulation-based concepts such as the Leaky Competing Accumulator (LCA) theory and Feed-Forward Inhibition (FFI) theory have not yet benefited from Bayesian techniques. As assessment criteria, total relative deviation (TRD), total mean square error (TMSE), and Stein Loss Function (SLF) are used. Maximum likelihood (ML) estimates are modified because the original statistic's Cumulative Distribution Function (CDF) is better than traditional ML estimations and other modified estimators emphasize average and coefficient variability. Prior estimator performance was unaffected by response rate or real parameter settings. These systems have not been formally evaluated by the Bayesian factor.

Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data

Abstract The compounded Bell generalized class of distributions is proposed in this article as an alternative to the compounded Poisson generalized family of distributions. Some properties and actuarial measures are presented. The properties of a special model named Bell Weibull (BellW) are obtained such as the linear representation of density, rth moment, incomplete moment, moment generating function using Wright generalized hypergeometric function and Meijer’s G function, the pth moment of order statistics, reliability, stochastic ordering, and residual and reversed residual life. Moreover, some commonly used entropy measures, namely, Rényi, Havrda and Charvat, and Arimoto and Tsallis entropy are obtained for the special model. From the inferential side, parameters are estimated using maximum likelihood estimation. The simulation study is performed to highlight the behavior of estimates. Some actuarial measures including expected shortfall, value at risk, tail value at risk, tail variance, and tail variance premium for the BellW model are presented with the numerical illustration. The usefulness of the proposed family is evaluated using insurance claims and COVID-19 datasets. Convincing results are obtained.

Extension of Exponential Pareto Distribution with the Order Statistics: Some Properties and Application to Lifetime Dataset

The Exponential Pareto (EP) model has been extended by applied and theoretical statisticians for wider applications and new knowledge using different techniques but the Weibull-X technique has not been considered. This article proposed a new extension of the EP model called the Weibull-Exponential Pareto (WEP) distribution to provide better modeling that fits real-life datasets and to explore the statistical theory of order statistics from the proposed distribution. Statistical properties investigated include the Shannon and Renyi entropies; the moments and moment generating function. Distribution of order statistics and the moment of order statistics were derived including the mean and variance of order statistics. WEP distribution has unimodal, decreasing, and increasing failure rates; and it can be negatively or positively skewed and approximately symmetric with the potential for fitting platykurtic, mesokurtic, and leptokurtic lifetime data. The parameters of the distribution were estimated using the method of maximum likelihood estimation (MLE), which was examined for consistency through a simulation study. The performance of the proposed distribution was investigated by application to flood peaks exceedances and some lifetime datasets from engineering. The results from data analysis using the R-software revealed that the WEP distribution has the potential to provide a superior model that fits the three data sets better than some notable existing distributions and previous extensions of the EP model in the literature. The statistical property of order statistics extended in the study established some important results that characterized some notable lifetime distributions in the literature.

Entropic Central Limit Theorem for Order Statistics

It is well known that central order statistics exhibit a central limit behavior and converge to a Gaussian distribution as the sample size grows. This paper strengthens this known result by establishing an entropic version of the central limit theorem that ensures a stronger mode of convergence using the relative entropy. This upgrade in convergence is shown at the expense of extra regularity conditions, which can be considered as mild. To prove this result, ancillary results on order statistics are derived, which might be of independent interest. For instance, a rather general bound on the moments of order statistics, and an upper bound on the mean squared error of estimating the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p \in (0,1)$ </tex-math></inline-formula> -th quantile of an unknown cumulative distribution function, are derived. Finally, a discussion on the necessity of the derived conditions for convergence and on the rate of convergence and monotonicity of the relative entropy is provided.