Rth Order Statistic Research Articles

Extropy Based on Concomitants of Order Statistics in Farlie-Gumbel-Morgenstern Family for Random Variables Representing Past Life

In this paper, we refined the concept of past extropy measure for concomitants of order statistics from Farlie-Gumbel-Morgenstern family. In addition, cumulative past extropy measure and dynamic cumulative past extropy measure for concomitant of rth order statistic are also conferred and their properties are studied. The problem of estimating the cumulative past extropy is investigated using empirical technique. The validity of the proposed estimator has been emphasized using simulation study.

Order Statistics of Doubly Truncated Additive Uniform Exponential Distribution

Abstract: In this paper we truncated the Additive Uniform Exponential Distribution (AUED) proposed by Venkata Subbarao Uppu (2010).The probability density functions of rth order Statistics, l th moment of the rth order Statistic, minimum, maximum order statistics, mean of the maximum and minimum order statistics, the joint density function of two order statistics of the truncated distribution were calculated and discussed in detailed .

On Beta Exponentiated Moment Exponential Distribution with Mathematical Characteristics and Application to Engineering Sectors

A new BEME distribution known as beta Exponentiated moment exponential (BEME) distribution is proposed. We provide here some shape properties, moments in the form of special functions, mean deviations of BEME distribution. We derive mathematical properties of the BEME distribution including the reliability measures, the Bonferroni and the Lorenz curves, rth order statistics, measures of uncertainty: the Shannon entropy measure and the s-entropy measure. The parameters of the BEME distribution are estimated by the method of maximum likelihood estimation and estimated non-linear equations for these estimates are presented. The application of BEME distribution is explored in three different fields of engineering.

Order Statistics of Additive Uniform Exponential Distribution

Abstract: In this paper we investigated the order statistics by using Additive Uniform Exponential Distribution (AUED) proposed by Venkata Subbarao Uppu (2010).The probability density functions of rth order Statistics, lth moment of the rth order Statistic, minimum, maximum order statistics, mean of the maximum and minimum order statistics, the joint density function of two order statistics were calculated and discussed in detailed . Applications and several aspects were discussed Keywords: Additive Uniform Exponential Distribution, Moments, Minimum order statistic, Maximum order statistic, Joint density of the order Statistics, complete length of service.

A New Double Truncated Generalized Gamma Model with Some Applications

The generalized Gamma model has been applied in a variety of research fields, including reliability engineering and lifetime analysis. Indeed, we know that, from the above, it is unbounded. Data have a bounded service area in a variety of applications. A new five-parameter bounded generalized Gamma model, the bounded Weibull model with four parameters, the bounded Gamma model with four parameters, the bounded generalized Gaussian model with three parameters, the bounded exponential model with three parameters, and the bounded Rayleigh model with two parameters, is presented in this paper as a special case. This approach to the problem, which utilizes a bounded support area, allows for a great deal of versatility in fitting various shapes of observed data. Numerous properties of the proposed distribution have been deduced, including explicit expressions for the moments, quantiles, mode, moment generating function, mean variance, mean residual lifespan, and entropies, skewness, kurtosis, hazard function, survival function, r th order statistic, and median distributions. The delivery has hazard frequencies that are monotonically increasing or declining, bathtub-shaped, or upside-down bathtub-shaped. We use the Newton Raphson approach to approximate model parameters that increase the log-likelihood function and some of the parameters have a closed iterative structure. Six actual data sets and six simulated data sets were tested to demonstrate how the proposed model works in reality. We illustrate why the Model is more stable and less affected by sample size. Additionally, the suggested model for wavelet histogram fitting of images and sounds is very accurate.

A Log-Dagum Weibull Distribution: Properties and Characterization

This article proposes a new family of continuous distributions generated from a log dagum random variable (named Log-Dagum Weibull Distribution) on the basis of T-X family technique. mathematical and statistical properties including survival function, hazard and reverse hazard function, Rth moments, L-moments, incomplete rth moments, quantile points, Order Statistics, Bonferroni and Lorenz curves as well as entropy measures for this class of distributions are presented also LDW distribution characterized by truncated moments order statistics and upper record values. Simulation study of the proposed family of distribution has been derived. The model parameters are obtained by the method of maximum likelihood estimation. We illustrate the performance of the proposed new family of distributions by means of four real data sets and the data sets show the new family of distributions is more appropriate as compared to Exponentiated exponential distribution (EED), Weibull distribution (WD), Gamma distribution (GD), NEED Nadarajah Exponentiated exponential distribution and Lomax distribution (LD). Moreover, perfection of competing models is also tested via the Kolmogrov-Simnorov (K S), the Anderson Darling (A*) and the Cramer-von Misses (W*). The measures of goodness of fit including the Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) are computed to compare the fitted models.

Gumbel-Geometric Distribution: Properties and Applications

A three-parameter generalization of the Gumbel distribution, which we call Gumbel-geometric distribution, is defined and investigated. The shape of the density and hazard function is examined and discussed. Explicit expressions for the moment generating function, the characteristics function and the rth order statistic are obtained. Other properties of the distribution are also discussed. The method of maximum likelihood is proposed for the estimation of the parameter of the model and discussed. A simulation experiment is carried out to examine the asymptotic properties of the distribution. The result shows that the MSE decreases to zero as while the bias either increases or decreases (depending on the sign) for each of the parameters. The new distribution is applied to two datasets and compared to some existing generalization to illustrate its flexibility.

Concomitant of Order Statistics from New Bivariate Gompertz Distribution

For the new bivariate Gompertz distribution, the expression for probability density function (pdf) of rth order statistics and pdf of concomitant arising from rth order statistics are derived. The properties of concomitant arising from the corresponding order statistics are used to derive these results. The exact expression for moment generating function (mgf) of concomitant of order rth statistics is derived. Also, the mean of concomitant arising from rth order statistics is computed using the mgf of concomitant of rth order statistics, and the exact expression for joint density of concomitant of two non-adjacent order statistics are derived.

Inferences for the Type-II Exponentiated Log-Logistic Distribution Based on Order Statistics with Application

In this paper, we first derive the exact explicit expressions for the single and product moments of order statistics from the type-II exponentiated log-logistic distribution, and then use these results to compute the means, variances, skewness and kurtosis of rth order statistics. Besides, best linear unbiased estimators (BLUEs) for the location and scale parameters for the type-II exponentiated log-logistic distribution with known shape parameters are studied. Finally, the results are illustrated with a real data set.

On Generalized Moment Exponential Distribution and Power Series Distribution

In this research paper, a new life time family is introduced. Sadaf [1] proposed a moment exponential power series (MEPS) distribution. Generalized moment exponential power series (GMEPS) distribution is a general form of MEPS distribution. It is characterized by compounding GME distribution and power series (PS) distribution. This new family has some new sub models such as GME geometric distribution, GME Poisson (GMEP) distribution, GME logarithmic (GMEL) distribution and GME binomial (GMEB) distribution. We provide statistical properties of GMEPS family of distributions. We find here expression of quantile function based on Lambert W function, the density function of rth order statistic and moments of GMEPS distribution. Descriptive expressions of Shannon entropy and Rényi entropy of new general model are found. We provide special sub-models of the GMEPS family of distributions. The maximum likelihood (ML) estimation method is used to find estimates of the parameters of GMEPS distribution. Simulation study is carried out to check the convergence of new estimators. We apply GMEPS family of distributions on two sets of real data.

Joint distribution of a random sample and an order statistic: A new approach with an application in reliability analysis

This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an (n−r+1)-out-of-n system, which is intact until at least n−r+1 of the components are alive, and it fails if the number of failed components exceeds r. The (n−r+1)-out-of-n system is a preferred type of redundancy of fault-tolerant systems used in reliability engineering and finds wide applications in industrial and military systems. Well-known examples of this system are multiengine systems in airplane, multipump systems in hydraulic control, multicylinder engines in automobile, the radar systems, communication systems with multiple transmitters, a web host with several servers. The lifetime of the system is the rth order statistic. Assuming that some of the components failed but the system is still functioning, using the results presented in this paper it is possible to find an expected value of the number of inspections we need to do for detecting certain number of failed components.

On the joint distribution of order statistics from independent non-identical bivariate distributions

In this note, the exact joint probability density function (jpdf) of bivariate order statistics from independent non-identical bivariate distributions is obtained. Furthermore, this result is applied to derive the joint distribution of a new sample rank obtained from the rth order statistics of the first component and the sth order statistics of the second component.

Moments of order statistics from nonidentically Distributed Lomax, exponential Lomax and exponential Pareto Variables

In this paper, the probability density function and the cumulative distribution function of the rth order statistic arising from independent nonidentically distributed (INID) Lomax, exponential Lomax and exponential Pareto variables are presented. The moments of order statistics from INID Lomax, exponential lomax and exponential Pareto were derived using the technique established by Barakat and Abdelkader. Also, numerical examples are given.

Transmuted Kumaraswamy-G Family of Distributions for Modelling Reliability Data

Abstract This research introduced the new class of distributions, called the transmuted Kumaraswamy (TKw) G family for modelling life testing problems. The new extended family is obtained by using the quadratic rank transmutation map method, which possesses a bathtub shape for its hazard rate. Some special models in the new family are discussed. We studied some mathematical properties of the new family, including the ordinary moments, moment generating functions, probability weighted moment, quantile function, and formulated the pdf of rth order statistics. The TKw G-distribution parameters are estimated using the method of maximum likelihood. We illustrate the potentiality of the new family with two applications to the failure and service times of Aircraft windshield data.

New family of distributions using the pdf of the rth order statistic from independent non-identically distributed random variables

New family of distributions using the pdf of the rth order statistic from independent non-identically distributed random variables

On an application of concomitants of order statistics

ABSTRACTLet (Xi, Yi), i = 1, …, n be a pair where the first coordinate Xi represents the lifetime of a component, and the second coordinate Yi denotes the utility of the component during its lifetime. Then the random variable Y[r: n] which is known to be the concomitant of the rth order statistic defines the utility of the component which has the rth smallest lifetime. In this paper, we present a dynamic analysis for an n component system under the above-mentioned concomitant setup.

Exact Moments of order Statistics from The exponentiated Lomax distribution

In this paper, order statistics from the exponentiated Lomax distribution (ELD) are obtained. Exact form for the single, product and Triple moment of order statistics from ELD are derived. Measures of skewness and kurtosis of the probability density function of the rth order statistic are presented. Some recurrence relations for the single and product moments of order statistics from ELD are established. Also, the percentage points of single order statistics from ELD are computed.

On a Left Truncated Generalized Gaussian Distribution

The generalized Gaussian distribution is useful in analyzing the data sets arising at places like Image processing, Signal processing, Speech recognition, Statistical Quality Control, Industrial experimentation, and Biological experiments. In this paper, a Left Truncated Generalized Gaussian distribution is introduced. The various distributional properties such as the distribution function, the four moments, skewness, kurtosis, hazard function, survival function are derived. The distribution of the rth order statistics, the median distribution also obtained. Some inferential aspects of the distribution are studied. An application of the suggested model is given with respect to the data on the length of fish in a catch.

An identity on order statistics of a set of random variables.

When an n × 1 random vector X = (X1, …, Xn ) T has a sign-invariant distribution, Strait [J. Multivariate Anal. 4 (1974) 494-496] proved that the expectations of max(0, X1, X1 + X2, …, X1 + Xn ) and max(0, X1, …, Xn ) are equal. In this note we assume a weaker condition that (X1, X2, …, Xn ) and (-X1, X2, …, Xn ) are equal in distribution and prove a more general result that the expectations of Lr (0, X1, X1 + X2, …, X1 + Xn ) and Lr (0, X1, …, Xn ) are equal, where Lr (0, X1, …, Xn ) is the rth order statistic of 0, X1, …, Xn for r = 1, …, n + 1.

Statistical analysis of the Lomax–Logarithmic distribution

In this paper we introduce a three-parameter lifetime distribution following the Marshall and Olkin [New method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997;84(3):641–652] approach. The proposed distribution is a compound of the Lomax and Logarithmic distributions (LLD). We provide a comprehensive study of the mathematical properties of the LLD. In particular, the density function, the shape of the hazard rate function, a general expansion for moments, the density of the rth order statistics, and the mean and median deviations of the LLD are derived and studied in detail. The maximum likelihood estimators of the three unknown parameters of LLD are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance–covariance matrix. Finally, a real data set is analysed to show the potential of the new proposed distribution.