Kth Order Statistic Research Articles

Distributed convex optimization as a tool for solving [formula omitted]-consensus problems

For a group of networked agents, f-consensus means reaching consensus upon the value of a desired function, f, of the initial state of the individual agents. This paper shows how one can often convert a given f-consensus problem into a suitable distributed convex optimization (DCO) problem, which can be readily solved with existing DCO algorithms in the literature. A computational advantage may then accrue. Particular classes of f-consensus problems shown to be solvable with this approach include weighted power mean consensus, and kth smallest value or kth order statistic consensus (which includes max/min consensus and median consensus as special cases).

Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised kth order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised kth order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyse various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.

A New Fractional Poisson Process Governed by a Recursive Fractional Differential Equation

This paper proposes a new fractional Poisson process through a recursive fractional differential governing equation. Unlike the homogeneous Poison process, the Caputo derivative on the probability distribution of k jumps with respect to time is linked to all probability distribution functions of j jumps, where j is a non-negative integer less than or equal to k. The distribution functions of arrival times are derived, while the inter-arrival times are no longer independent and identically distributed. Further, this new fractional Poisson process can be interpreted as a homogeneous Poisson process whose natural time flow has been randomized, and the underlying time randomizing process has been studied. Finally, the conditional distribution of the kth order statistic from random number samples, counted by this fractional Poisson process, is also discussed.

An Inferential Aptness of a Weibull Generated Distribution and Application

In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kth order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

Characterization and Goodness-of-Fit Test of Pareto and Some Related Distributions Based on Near-Order Statistics

In this paper, a new definition of the number of observations near the kth order statistics is developed. Then some characterization results for Pareto and some related distributions are established in terms of mass probability function, first moment of these new counting random variables, and using completeness properties of the sequence of functions xn,0<x<1,n≥1. Finally, new goodness-of-fit tests based on these new characterizations for Pareto distribution are presented. And the power values of the proposed tests are compared with the power values of well-known tests such as Kolmogorov–Smirnov and Cramer-von Mises tests by Monte Carlo simulations.

Limit theorems for ratios of order statistics from uniform distributions

Let {X_{ni}, 1 leq i leq m_{n}, ngeq 1} be an array of independent random variables with uniform distribution on [0, theta _{n}], and {X_{n(k)}, k=1, 2, ldots , m_{n}} be the kth order statistics of the random variables {X_{ni}, 1 leq i leq m_{n}}. We study the limit properties of ratios {R_{nij}=X_{n(j)}/X_{n(i)}, 1leq i < j leq m_{n}} for fixed sample size m_{n}=m based on their moment conditions. For 1=i < j leq m, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for 2=i < j leq m, we obtain some classical limit theorems and self-normalized limit theorems.

ON STOCHASTIC COMPARISONS OF ORDER STATISTICS FROM HETEROGENEOUS EXPONENTIAL SAMPLES

AbstractWe show that the kth order statistic from a heterogeneous sample of n ≥ k exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan [14]. As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea [11]. Extensions to general spacings are also presented.

Extended Exponentiated Inverse Lindley Distribution: Model, Properties and Applications

In this article, a four-parameter generalization of inverse Lindley distribution is obtained, with the purpose of obtaining a more flexible model relative to the behavior of hazard rate functions. Various statistical properties such as density, hazard rate functions, moments, moment generating functions, stochastic ordering, Renyi entropy, distribution of kth order statistics has been derived. The method of maximum likelihood estimation has been used to estimate parameters. Further confidence intervals are also obtained. Finally, applicability of the proposed model to the real data is analyzed. A comparison has also been made with some existing distributions.

Robust visual data segmentation: Sampling from distribution of model parameters

This paper approaches the problem of geometric multi-model fitting as a data segmentation problem. The proposed solution is based on a sequence of sampling hyperedges from a hypergraph, model selection and hypergraph clustering steps. We developed a sampling method that significantly facilitates solving the segmentation problem using a new form of the Markov-Chain-Monte-Carlo (MCMC) method to effectively sample from hyperedge distribution. To sample from this distribution effectively, our proposed Markov Chain includes new ways of long and short jumps to perform exploration and exploitation of all structures. To enhance the quality of samples, a greedy algorithm is used to exploit nearby structure based on the minimization of the Least kth Order Statistics cost function. Unlike common sampling methods, ours does not require any specific prior knowledge about the distribution of models. The output set of samples leads to a clustering solution by which the final model parameters for each segment are obtained. The method competes favorably with the state-of-the-art both in terms of computation power and segmentation accuracy.

Exact distributions of order statistics from ln,p-symmetric sample distributions

Abstract We derive the exact distributions of order statistics from a finite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end,we first review the special cases of order statistics fromspherical aswell as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure.We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ {1, . . . , n − 1}.

Top-k Algorithm for User Preferences based on Selection Strategy

In order to deal with multiple user preferences and improve query efficiency, selection strategy is adopted for top-k query to depress the compare operations. Firstly, the kth order statistics are selected randomly along with partitioning the data set basing on it, and the top-k result set can be received after several recursive partitions. Secondly, to select the kth order statistics accurately, the approximate kth order statistics is choose as threshold according to the similarity of user preference and system preference, and the top-k query result set can be accessed through simple comparison. Finally, the time complexities of presented algorithms are analyzed and their correctness and completeness are proved respectively. The experimental results show that our algorithms improve the efficiency of top-k query greatly.

Towards a queueing-based framework for in-network function computation

We seek to develop network algorithms for function computation in sensor networks. Specifically, we want dynamic joint aggregation, routing, and scheduling algorithms that have analytically provable performance benefits due to in-network computation as compared to simple data forwarding. To this end, we define a class of functions, the Fully-Multiplexible functions, which includes several functions such as parity, MAX, and kth-order statistics. For such functions we characterize the maximum achievable refresh rate of the network in terms of an underlying graph primitive, the min-mincut. In acyclic wireline networks we show that the maximum refresh rate is achievable by a simple algorithm that is dynamic, distributed, and only dependent on local information. In the case of wireless networks we provide a MaxWeight-like algorithm with dynamic flow-splitting, which is shown to be throughput-optimal.

Stochastic Comparisons for k-out-of-N(F) System of Random Numbers of Units

In this paper, stochastic comparisons for -out-of- ( ) system with random numbers of identical components are studied. The lifetime of -out-of- ( ) system corresponds to the kth order statistic of random numbers of independent identically distributed random variables. Character- izations based on some stochastic orders, for example, star order, hazard rate order and reversed hazard order system are studied. Stochastic order has been widely applied in computer queuing networks, communications systems, automated production line system, inventory systems.

Results on residual Rényi entropy of order statistics and record values

This paper explores properties of the residual Rényi entropy of some ordered random variables. The residual Rényi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual Rényi entropy of the kth order statistic from uniform distribution. The monotone behavior of the residual Rényi entropy of order statistic under various conditions is discussed. Analogues results for the residual Rényi entropy of record values are also given.

NUMBERS OF OBSERVATIONS NEAR ORDER STATISTICS

SummaryLimit theorems are obtained for the numbers of observations in a random sample that fall within a left‐hand or right‐hand neighbourhood of the kth order statistic. The index k can be fixed, or tend to infinity as the sample size increases unboundedly. In essence, the proofs are applications of the classical Poisson and De Moivre–Laplace theorems.

Weak Hölder Convergence of Smoothed Uniform Quantile Process

It is well known that the paths of the Brownian bridge B are (with probability one) of Hölder regularity α for any α < 1/2 and the uniform quantile process converges weakly to the Brownian bridge in the Skorokhod space D[0,1]. We prove that this weak convergence holds too in Hölder spaces H α [0,1]. The polygonal smoothing of the quantile process (instead of the quantile function, we use its interpolate at points (k/n,U k:n ) where U k:n is the kth order statistic of a uniform sample (U 1,…, U n )) gives us this convergence for any order α < 1/2. The convolution smoothing brings rather to an order α depending on the choice of window of the convolution kernel in this convergence.

TIGHT BOUNDS ON EXPECTED ORDER STATISTICS

In this article, we study the problem of finding tight bounds on the expected value of the kth-order statistic E [Xk:n] under first and second moment information on n real-valued random variables. Given means E [Xi] = μi and variances Var[Xi] = σi2, we show that the tight upper bound on the expected value of the highest-order statistic E [Xn:n] can be computed with a bisection search algorithm. An extremal discrete distribution is identified that attains the bound, and two closed-form bounds are proposed. Under additional covariance information Cov[Xi,Xj] = Qij, we show that the tight upper bound on the expected value of the highest-order statistic can be computed with semidefinite optimization. We generalize these results to find bounds on the expected value of the kth-order statistic under mean and variance information. For k &lt; n, this bound is shown to be tight under identical means and variances. All of our results are distribution-free with no explicit assumption of independence made. Particularly, using optimization methods, we develop tractable approaches to compute bounds on the expected value of order statistics.

Small-Sample Quantile Estimators in a Large Nonparametric Model

The large nonparametric model in this note is a statistical model with the family ℱ of all continuous and strictly increasing distribution functions. In the abundant literature of the subject, there are many proposals for nonparametric estimators that are applicable in the model. Typically the kth order statistic X k:n is taken as a simplest estimator, with k = [nq], or k = [(n + 1)q], or k = [nq] + 1, etc. Often a linear combination of two consecutive order statistics is considered. In more sophisticated constructions, different L-statistics (e.g., Harrel–Davis, Kaigh–Lachenbruch, Bernstein, kernel estimators) are proposed. Asymptotically the estimators do not differ substantially, but if the sample size n is fixed, which is the case of our concern, differences may be serious. A unified treatment of quantile estimators in the large, nonparametric statistical model is developed.

Comparison of order statistics in a random sequence to the same statistics with i.i.d. variables

The paper is motivated by the stochastic comparison of the reliability of non-repairable k-out-of-n systems. The lifetime of such a system with nonidentical components is compared with the lifetime of a system with identical components. Formally the problem is as follows. Let Ui ,i =1 , ..., n, be positive independent random variables with common distribution F.F orλi > 0a nd µ> 0, let consider Xi = Ui/λi and Yi = Ui/µ, i =1 , ..., n. Remark that this is no more than a change of scale for each term. For k ∈{ 1, 2, ..., n}, let us define Xk:n to be the kth order statistics of the random variables X1, ..., Xn, and similarly Yk:n to be the kth order statistics of Y1, ..., Yn. If Xi ,i =1 , ..., n, are the lifetimes of the components of a n+1-k-out-of-n non-repairable system, then Xk:n is the lifetime of the system. In this paper, we give for a fixed k a sufficient condition for Xk:n ≥st Yk:n where st is the usual ordering for distributions. In the Markovian case (all components have an exponential lifetime), we give a necessary and sufficient condition. We prove that Xk:n is greater that Yk:n according to the usual stochastic ordering if and only if � n k � µ k ≥ �

Exact Laws for Sums of Order Statistics from a Generalized Pareto Distribution

We select the kth order statistic from each row from a sequence of independent and identically distributed random variables from a distribution that generalizes the Pareto distribution. We then examine weighted sums of these order statistics to see whether or not Laws of Large Numbers with nonzero limits exist.