Distribution Functions Of Order Statistics Research Articles

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.

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.

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.

Unit Burr-III distribution with application

In this paper we introduce a new Unit Burr-III probability distribution with support interval (0,1). The proposed distribution possesses a flexible density function and bathtub shape hazard rate function. We derive the expressions for its mathematical and statistical properties. Probability distribution function of order statistics and expression for pth moment for derived distribution is also obtained. We calculate parameter estimates by using the technique of maximum likelihood estimation. We apply it on two real datasets and found that it provides better fit than other well known distributions.

Distribution bounds for order statistics when each k-tuple has the same piecewise uniform copula

ABSTRACTLet be an n-tuple of random variables each having the same known distribution function F and such that each k-tuple has the same known copula. We extend the Kemperman [Bounding moments of an order statistic when each k-tuple is independent. In: Benes̆ V, S̆tĕpán J, editors. Distributions with given marginals and moment problems. Dordrecht: Kluwer Academic Publishers; 1997. p. 291–304] characterization and resulting method for determining sharp lower and upper bounds for the distribution functions of the associated order statistics to the case of (i) dependence structures other than independence and (ii) arbitrary linear combinations of distribution functions of order statistics.

Bounds on expectations of L-estimates for maximally and minimally stable samples

The random variables are said to be maximally (resp. minimally) stable of order j () if the distribution of (resp. of ) is the same for any j element subset of Under the assumption of maximal (resp. minimal) stability of order j, we give lower and upper bounds on linear combinations of distribution functions of order statistics in terms of (resp. ) and present the corresponding sharp two-sided bounds on expected L-estimates. Moreover, we evaluate the upper and lower deviations of the expectation of L-estimate from observations with a given dependence structure (copula) under maximally stable violations of the dependence assumption.

Signatures in Markov-Modulated Processes

□ In recent years, signatures are widely used for analysis of coherent systems consisting of unreliable components. If component lifetimes are independent and identically distributed, then system lifetime distribution function is a convex combination of distribution functions of order statistics for component lifetimes. Coefficients of this convex combination are called signatures. This article considers the case when a system operates in a so-called random environment, i.e., component failure rates are jointly modulated by a finite-state continuous-time Markov chain. In this model, component lifetimes remain exchangeable. An expression for distribution function of time to system failure is derived. Here, a crucial role is played by an elaborated procedure of deriving a distribution function of order statistics for system component lifetimes. A numerical example illustrates the suggested approach and analyzes the influence of random environment on the distribution function of system lifetime.

Bounds for mixtures of order statistics from exponentials and applications

This paper deals with the stochastic comparison of order statistics and their mixtures. For a random sample of size n from an exponential distribution with hazard rate λ , and for 1 ≤ k ≤ n , let us denote by F k : n ( λ ) the distribution function of the corresponding k t h order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates λ 1 , … , λ m . Assume that p 1 , … , p m > 0 , such that ∑ i = 1 m p i = 1 , and let U and V be two random variables with the distribution functions F k : n ( λ ) and ∑ i = 1 m p i F k : n ( λ i ) , respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if λ ≥ ∑ i = 1 m p i λ i k k , and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if λ ≤ min 1 ≤ i ≤ m λ i , for all k = 1 , … , n . These properties are used to find the best bounds for the survival functions of order statistics from independent heterogeneous exponential random variables. For the proof, we will use a mixture type representation for the distribution functions of order statistics.

Permanent Expansions and Distributions of Order Statistics in the INID Case

Expansions of permanents are applied to derive expressions for the distribution functions of order statistics arising from n INID random variables. For this purpose, we apply two types of expansions. The first one is based on Ryser's method and it leads to a representation of the distribution function as a generalized mixture of distributions of order statistics from IID random variables. The second approach, which is based on the calculation of the permanent of a sum of two matrices, yields an expression in terms of products of both distributions of minima and maxima. It turns out that the most efficient way of calculating the distribution function is the one based on Ryser's expansion.

On copulas and their diagonals

We formulate the necessary and sufficient conditions for a function δ : [ 0 , 1 ] → [ 0 , 1 ] to be the diagonal section of a multivariate absolutely continuous copula. Moreover we provide some simple analytic formulas for copulas having given diagonal section or given distribution functions of order statistics.

Fast Computation by Block Permanents of Cumulative Distribution Functions of Order Statistics from Several Populations

The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution functions of the populations. The computational cost of our formula in the case of two populations is still exponential in the worst case, but it is a dramatic improvement compared to the general formula by Bapat and Beg. In the case when only the joint distribution function of a subset of the order statistics of fixed size is needed, the complexity is polynomial, for the case of two populations.

A General Recurrence Relation for the Moments of Bivariate and Trivariate Order Statistics

In this article, by using the dropping argument, a general recurrence relation satisfied by the joint cumulative distribution functions of order statistics from any arbitrary bivariate distribution function is established. This recurrence relation is the first bivariate version of the basic triangle rule for order statistics arisen from univariate distribution function. Finally, this relation is extended to the trivariate case. These lead to similar identities for product moments (of any order) of order statistics.

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of- n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.

Adaptive sampling based on the cumulative distribution function of order statistics to delineate heavy-metal contaminated soils using kriging

Correctly classifying “contaminated” areas in soils, based on the threshold for a contaminated site, is important for determining effective clean-up actions. Pollutant mapping by means of kriging is increasingly being used for the delineation of contaminated soils. However, those areas where the kriged pollutant concentrations are close to the threshold have a high possibility for being misclassified. In order to reduce the misclassification due to the over- or under-estimation from kriging, an adaptive sampling using the cumulative distribution function of order statistics (CDFOS) was developed to draw additional samples for delineating contaminated soils, while kriging. A heavy-metal contaminated site in Hsinchu, Taiwan was used to illustrate this approach. The results showed that compared with random sampling, adaptive sampling using CDFOS reduced the kriging estimation errors and misclassification rates, and thus would appear to be a better choice than random sampling, as additional sampling is required for delineating the “contaminated” areas.

On special linear identities for order statistics

Linear identities for the distribution functions of order statistics from an iid sample are defined. It is shown that such identities are true for all distributions or to some discrete distributions taking a finite number of values.

Kriging method evaluation for assessing the spatial distribution of urban soil lead contamination.

Describing contaminant spatial distribution is an integral component of risk assessment. Application of geostatistical techniques for this purpose has been demonstrated previously. These techniques may provide both an estimate of the concentration at a given unsampled location, as well as the probability that the concentration at that location will exceed a critical threshold concentration. This research is a comparative study between multiple indicator kriging and kriging with the cumulative distribution function of order statistics, with both local and global variograms. The aim was to determine which of the four methods is best able to delineate between "contaminated" and "clean" soil. The four methods were validated with a subset of data values that were not used in the prediction. Method performance was assessed by calculating the root mean square error (RMSE), analysis of variance, the proportion of sites misclassified by each method as either "clean" when they were actually "contaminated" or vice versa, and the expected loss for each misclassification type. The data used for the comparison were 807 topsoil Pb concentrations from the inner-Sydney suburbs of Glebe and Camperdown, Australia. While there was very little difference between the four methods, multiple indicator kriging was found to produce the most accurate predictions for delineating "clean" from "contaminated" soil.

Comparison of order statistics between dependent and independent random variables

The closeness of distribution functions of order statistics between positively (negatively) associated and independent random variables is compared. One application is presented.

Saddlepoint approximations for multinomial distributions,with applications to the joint

The saddlepoint approximation for the multinomial distribution is derived and expressed in terms of the corresponding generating function, so that the asymptotic formula of Good (1957) for the equiprobable multinomial distribution is generalized. Based on the multinomial argument, the representations and the saddlepoint approximation for the joint distribution function of order statistics from a continuous or discrete population are obtained. Numerical examples illustrate the great accuracy of the approximations.

Computing distributions of order statistics

Recurrence relationships among the distribution functions of order statistics of independent, but not identically distributed, random quantities are derived. These results extend known theory and provide computationally practicable algorithms for a variety of problems.