Functions Of Order Statistics Research Articles

Characterizations of continuous log-symmetric distributions based on properties of order statistics

The class of log-symmetric distributions is a generalization of log-normal distribution and includes some well-known distributions such as log-normal, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-student-t, log-slash, and Birnbaum-Saunders distributions. In this paper, several characterization results are obtained for log-symmetric distributions based on moments of some functions of the parent distribution and also on the basis of some properties of order statistics. Specifically, when X is identical in distribution with a decreasing continuous function h ( X ) , then a relationship is established between upper and lower order statistics which is then utilized to construct characterization results for log-symmetric distributions in terms of functions of order statistics. The established results can be used for constructing a goodness-of-fit test for log-symmetric distributions.

A Conversation with Stephen M. Stigler

Stephen M. Stigler received his Ph.D. in Statistics from the University of California, Berkeley, with a dissertation on the asymptotic distribution of linear functions of order statistics. Starting in 1967, he taught at the University of Wisconsin, Madison, then in 1979 moved to the University of Chicago where he taught from 1979 to 2021. Stigler has worked on a variety of topics in mathematical statistics, ranging from asymptotic theory to the theory of experimental design, and on applications of statistics including in anthropology, forensic science, paleontology, psychology, information transfer and sports. In recent years, he has concentrated on the history of statistics, with inquiries ranging from the development of statistical methods in astronomy and geodesy and their spread to biological and social sciences, to lotteries, to the modern development of statistical theory. He has published four books, The History of Statistics (1986), Statistics on the Table (1999), The Seven Pillars of Statistical Wisdom (2016) and Casanova’s Lottery (2022). A recent research focus has been upon the way the work of Francis Galton on the statistics of inheritance led to the creation of modern multivariate analysis and made a true Bayesian inference possible, and on how R. A. Fisher’s transformation of Karl Pearson’s path breaking research led to a modern period of statistical enlightenment. Stigler is an elected member of the American Academy of Arts and Sciences and of the American Philosophical Society; he has served as President of the Institute of Mathematical Statistics and of the International Statistical Institute. In 2005, he received the Humboldt Foundation Research Award; in 2010, he was elected Membre Associé of the Académie royale de Belgique, Classe des Sciences. Stigler served as Theory and Methods Editor for the Journal of the American Statistical Association 1979–1982. He was a Guggenheim Fellow in 1977, and received awards for undergraduate teaching at the University of Wisconsin (1971) and University of Chicago (1998). This interview with Stigler was conducted remotely in July 2021.

Least Trimmed Squares for Regression Models with Stable Errors

Least Trimmed Squares (LTS) is a robust regression method with respect to outliers. It is based on performing Ordinary Least Squares (OLS) estimates on sub-datasets and determining the optimal solution corresponding to the minimum sum of squared residuals. Since the method of LTS is based on OLS, errors in regression models have finite variance. This work aims to generalize LTS for heavy tail data with infinite variance. When errors have infinite variance, it is impossible to benefit from OLS estimates. We use the property of variance existence of most ordered errors to find an initial robust OLS estimate. We polish the LTS method with a maximum likelihood estimator based on the density function of order statistics and determine the optimal solution for stable regression models. The proposed algorithm is implemented for linear regression models.

Adequate Mathematical Models of the Cumulative Distribution Function of Order Statistics to Construct Accurate Tolerance Limits and Confidence Intervals of the Shortest Length or Equal Tails

The technique used here emphasizes pivotal quantities and ancillary statistics relevant for obtaining tolerance limits (or confidence intervals) for anticipated outcomes of applied stochastic models under parametric uncertainty and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the experimental data are complete or Type II censored. The exact tolerance limits on order statistics associated with sampling from underlying distributions can be found easily and quickly making tables, simulation, Monte-Carlo estimated percentiles, special computer programs, and approximation unnecessary. The proposed technique is based on a probability transformation and pivotal quantity averaging. It is conceptually simple and easy to use. The discussion is restricted to one-sided tolerance limits. Finally, we give practical numerical examples, where the proposed analytical methodology is illustrated in terms of the exponential distribution. Applications to other log-location-scale distributions could follow directly.

Quantile Based Geometric Vitality Function of Order Statistics

Quantile Based Geometric Vitality Function of Order Statistics

ON THE PROPERTIES OF TOPP-LEONE KUMARASWAMYWEIBUL DISTRIBUTION WITH APPLICATIONS TO BIOMEDICAL DATA

In this study, a new four-parameter lifetime distribution called the Topp Leone KumaraswamyWeibull distribution was derived using the Topp-Leone Kumaraswamy-G family of distributions. The model includes several important sub-models as special cases such as Topp-Leone exponentiatedWeibull,Topp Leone Weibull, exponentiatedWeibull and Weibull distributions. An expansion for the probability distribution function was carried out which was used to derive some of the mathematical properties. Some mathematical properties of the distribution were presented such as moments, moment generating function, quantile function, survival function, hazard function as well as mean, 1st quartile, median and 3rd quartile. The probability distribution function of order statistics of the Topp-Leone KumaraswamyWeibull distribution was obtained. Estimation of the parameters by maximum likelihood estimation method was discussed. Two real-life application of the distribution was presented and the analysis showed the fit and flexibility of the new distribution over some lifetime models considered. The analysis showed that the model is effective in fitting biomedical data.

ESTIMATES OF THE CONCENTRATION FUNCTION FOR STATISTICS

The results of this work were published in the journal Reports of the Academy of Sciences of the Republic of Uzbekistan without proof. In this paper, all the results obtained with complete proofs. Several theorems for a linear combination of functions of order statistics are proved in this work.

Recurrence relations for conditional moment generating functions of order statistics and record values

Recurrence relations for conditional moment generating functions of order statistics and record values

Power Darna Distribution with Right Censoring: Estimation, Testing, and Applications

The current paper proposes a new three-parameter probability distribution, called power Darna distribution (PDD), as an extension to the Darna distribution. An extra parameter is added to the base model and aims to increase the flexibility of the model in fitting various real data sets. Different statistical properties as reliability functions, hazard functions, and reversed hazard rate functions are obtained. Additionally, the shapes of the model, moments, inverse moments, and quantile function are derived and discussed. The distribution functions of order statistics from the PDD and moments of the smallest and largest order statistics are presented and a simulation study is conducted for explanation. An estimation of the parameters of PDD based on maximum likelihood estimation is presented. Finally, to demonstrate the PDD applicability in real life situations, a right-censored data set of carcinogenic DMBA in the vaginas of rats is considered and analyzed.

A Two Parameters Rani Distribution: Estimation and Tests for Right Censoring Data with an Application

In this paper, we developed a new distribution, namely the two parameters Rani distribution (TPRD). Some statistical properties of the proposed distribution are derived including the moments, moment-generating function, reliability function, hazard function, reversed hazard function, odds function, the density function of order statistics, stochastically ordering, and the entropies. The maximum likelihood method is used for model parameters estimation. Following the same approach suggested by Bagdonavicius and Nikulin (2011), modified chi squared goodness-of-fit tests are constructed for right censored data and some tests for right data is considered. An application study is presented to illustrate the ability of the suggested model in fitting aluminum reduction cells sets and the strength data of glass of the aircraft window.

Muth Distribution and Estimation of a Parameter Using Order Statistics

In this work, we have considered a lifetime distribution namely Muth distribution and pointed out instances where it appears as a good model to study the stochastic nature of the variable under consideration. We have derived the best linear unbiased estimator (BLUE) of the scale parameter of the Muth distribution based on order statistics for some known values of the shape parameter.We have further estimated the scale parameter of Muth distribution by U-statistics based on best linear functions of order statistics as kernels. The efficiency of the BLUE relative to the usual unbiased estimator has been also evaluated. An illustration describing the performance of U-statistics estimation method when compared with the classical maximum likelihood method is also given.

Forecasting structural equation modelling of lean manufacturing using high order statistical functions

Forecasting structural equation modelling of lean manufacturing using high order statistical functions

Forecasting Structural Equation Modelling of Lean Manufacturing using High Order Statistical Functions

Typically, the primary goal of lean technology (LT) is to improve profits and create value by minimising waste. The consideration of demand variability in a multi-dimensional lean manufacturing environment is an innovation in production system engineering. However, manufacturing systems that fail to recognise demand variability generate high work-in-process (WIP) and low throughput in the lean manufacturing process. Moreover, many companies have found it difficult to successfully implement and sustain lean manufacturing. It is, therefore, very important for companies to identify and understand the critical success factors for successfully implementing lean manufacturing. To overwhelm the challenges in the implementation of lean manufacturing, this paper plan to introduce an advanced model and this model is analysed using structural equation modelling.

On the Mean Square Error of Smooth $L$-Estimator

An $L$-estimator is a form of linear function of order statistics. Xiang [8] investigated the bias of this estimator. In this investigation, we derive a formula for the mean square error of the smooth $L$-estimator.

Inference on Exponentiated Power Lindley Distribution Based on Order Statistics with Application

Exponentiated power Lindley distribution is proposed as a generalization of some widely well-known distributions such as Lindley, power Lindley, and generalized Lindley distributions. In this paper, the exact explicit expressions for moments of order statistics from the exponentiated power Lindley distribution are derived. By using these relations, the best linear unbiased estimates of the location and scale parameters, based on type-II right-censored sample, are obtained. Next, the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics are calculated and then used to derive the approximate confidence interval for the location and scale parameters using the Edgeworth approximation. Finally, some numerical illustrations and two real data applications are presented.

On past geometric vitality function of order statistics

In this article, we propose geometric vitality function introduced by Nair and Rajesh (IAPQR Trans 25(1):1–8, 2000) for the past lifetime of a random variable. This measure plays a vital role in analysing different characteristics of a system/component when it fails in the interval (0, t). The monotonic behaviour and some ordering properties in terms of the proposed measure were studied under certain conditions. Similar properties of the proposed measure were analysed for order statistics as well. Further, bounds were obtained for the past geometric vitality function of order statistics. Apart from this, characterizations of some lifetime probability distributions with respect to order statistics were also discussed.

On the Strong Law of Large Numbers for Linear Combinations of Concomitants

A theorem on the strong law of large numbers for linear functions of concomitants (induced order statistics) for sequences of independent identically distributed two-dimensional random vectors is proved in this paper. The result complements previous work by S.S. Yang (1981) and N. Gribkova and R. Zitikis (2017, 2019). The proof is based on the conditional independence property of the concomitants established by P. K. Bhattacharya (1974); the van Zwet strong law of large numbers for linear functions of order statistics (1980) is used and classical inequalities apply, including the Rosenthal inequality (1970).

Forecasting the Impact of Social Media Advertising among College Students using Higher Order Statistical Functions

Nowadays, social media has emerged as one of the activity among the users on their day-to-day life activity. This research work plans to develop a statistical review that concerns on social media advertising among college students from diverse universities. The review analysis on social media advertising is given under six sections such as (i) Personal Profile; (ii) Usage; (iii) Assessment; (iv) Higher Order statistics like Community, Connectedness, Openness, Dependence and Participation; (v) Trustworthiness such as Trust, Perceived value and Perceived risk; and (vi) Towards advertisement which involves attitude towards advertisement, response towards advertisement and purchase intension. The initial stage is on questionnaire preparation based on social media networking. The records stated in the questionnaire are intimately taken about the usage of social media sites and the advertisement on social media networking. In the second stage, the planned questionnaire is distributed over college students from diverse universities. The entire questions are made mandatory in this questionnaire and subsequent to this, the students from various universities are demanded to fill up their response to this questionnaire. These responses from the students are then taken for analysis purpose. In this research work, the analysis is performed on the basis of higher-order statistical analysis that favorably concerned on correlation coefficients and entropy. This in turns helps to determine the correlation and entropy among the response towards the social media network.

Introduction to the Special Issue on Intelligence on Scalable computing for Recent Applications

Introduction to the Special Issue on Intelligence on Scalable computing for Recent Applications

The Type II Topp Leone-Power Lomax Distribution with Analysis in Lifetime Data

In this article, a new distribution, namely the type II Topp Leone-power Lomax (TIITL-PL) distribution, is proposed. Some statistical properties of the proposed distribution are derived including the survival function, hazard rate function, quantile function, moments, moment-generating function, the density function of order statistics, Lorenz curve and Bonferroni curve. The proposed distribution has a special submodel, that is, the type II Topp Leone-Lomax distribution. The maximum likelihood method is employed for model parameter estimation. For application study, the goodness-of-fit study is presented to illustrate the ability of the proposed model to fit various types of real data sets. The result shows that the TIITL-PL model is a flexible alternative to fit and analysis of lifetime data.