Properties Of Order Statistics Research Articles

Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.

Random translations, contractions and dilations of order statistics and records

We introduce a new tool to investigate some distributional properties of order statistics and records related by a random translation (contraction or dilation) scheme. This technique is based on the property of uniqueness of solutions of certain non-linear integral equations of Volterra type. We show how this tool is used to obtain new characterizations of distributions.

From Student's t 2 Distribution to Student's t 3 Distribution Through Characterizations

Some simple regressional properties of order statistics, similar to those obtained for t 2 distribution in the recent statistical literature, are shown to characterize Student's t 3 distribution.

Review on Order Statistics and Record Values from F^{α} Distributions

Abstract: Both order statistics and records arise naturally in many fields of studies such as climatology, sports, science, engineering, medicine, traffic, and industry, among others. Studies of their properties and applications play important roles in many areas of statistical research, for example, statistical inference, nonparametric statistics, among others. In recent years, the F^{α} distributions have been widely studied in statistics because of their wide applicability in the modeling and analysis of life time data. An absolutely continuous positive random variable X is said to have an F^{α} or exponentiated distribution if its cumulative distribution function (cdf) is given by G(x)=F^{α}(x)=[F(x)]^{α}, α>0, x>0, which is the αth power of the base line distribution function F(x). Many researchers and authors have developed various classes of F^{α} distributions. It appears from the literature that not much attention has been paid to the analysis of record values and order statistics from these members of F^{α} family of distributions, and, therefore, need further and special investigation. In this paper, some distributional properties of order statistics and record values from F^{α} distributions are reviewed. We hope that the findings of this paper will be a useful for the practitioners in various fields of studies and further enhancement of research in order statistics and record value theory, and their applications.

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by De Vylder and Goovaerts (1999, 2000). First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.

AGING PROPERTIES OF SEQUENTIAL ORDER STATISTICS

Sequential order statistics describe the ordered failure times in a k-out-of-n system, where the failures of components might affect the performance of remaining working components. In this article aging properties of sequential order statistics are examined and conditions are given such that the distribution of a sequential order statistic is ILR, IFR, IFRA, or NBU. Moreover, conditions for aging properties of spacings of sequential order statistics are obtained.

Some characterizations of families of distributions, including logistic and exponential ones, by properties of order statistics

New characterizations of distributions based on properties of the maximal order statistics are obtained. The families of distributions that are characterized by some properties of maxima include exponential and logistic distributions as partial cases. Bibliography: 4 titles.

On some regression relationships relating sample means and successive maxima

In recent years, a number of papers have appeared in which characterizations of a series of distributions in terms of regression properties of order statistics were obtained. In particular, characterizations of the Student’s t2-distribution in terms of linear regression relation-ships, including the sample median X2, 3 and the sample mean T3, were obtained in 2002. Later, similar results were also established for samples of larger sizes, and characterizations of families of distributions containing the t2-distribution were put forward. These characterizations are based, in particular, on linear regression relationships relating sample means T2k + 1 and sample medians Xk + 1, 2k + 1, k = 1, 2, ...

Characterizations of Student's t-distribution via regressions of order statistics

Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov et al. [New characterizations by properties of midrange and related statistics, Commun. Stat. Theory Methods 33(12) (2004), pp. 3133–3143], which includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.

Inference Under Progressively Type II Right-Censored Sampling for Certain Lifetime Distributions

In this article, estimation of the parameters of a certain family of two-parameter lifetime distributions based on progressively Type II right-censored samples (including ordinary Type II right censoring) is studied. This family, of proportional hazard distributions, includes the Weibull, Gompertz, and Lomax distributions. A type of parameter estimation named inverse estimation is investigated for both parameters. Exact confidence intervals for one of the parameters and generalized confidence intervals for the other are explored; inference for the first parameter can be accomplished by our methodology independently of the unknown value of the other parameter in this family of distributions. Derivation of the estimation method uses properties of order statistics. A simulation study concentrating mainly on the Weibull distribution illustrates the accuracy of these confidence intervals [and the shorter length of our exact confidence interval compared with an alternative of Wu (2002)] and compares inverse estimators favorably with maximum likelihood estimators. A numerical example is used to illustrate the proposed procedures.

Renyi Entropy Properties of Order Statistics

In this article, we discuss some properties of Renyi entropy and Renyi information of order statistics. Some bounds for Renyi entropy of order statistics are obtained. Also, we relate Renyi entropy ordering of order statistics to Renyi entropy ordering and other well known orderings of parent random variables. Then it is proved that the Renyi information between order statistics and parent random variable is distribution free, and it is shown, as expected, the distance is minimum for the median.

Correlation-based approach to analysis of spiking networks

The correlation structure of neural activity is believed to play a major role in the encoding and possibly decoding of information in neural populations. In addition, some of the most fundamental and widely applied tools for the identification of systems rely on the use of second order statistical properties (correlation or spectral). An arsenal of tools for identifying spike train models from their correlations, rather than from their full observed realizations could thus form a welcome bridge between ‘classical’ signal processing ideas and tools and the field of neural spike train analysis. Recently, several methods were developed for controlling the correlation structure of multi-channel synthetic spike trains [1,3-5], and in related work, correlation based analysis of spike trains was used for blind identification of single-neuron models [5], for identifying compact autoregressive models for multi-channel spike trains, and for facilitating their causal network analysis [6]. However, the diversity of correlation structures that can be explained by the non-recurrent generative models used in these studies is limited, and hence, methods based on such models occasionally fail while analyzing correlation structures that are observed in neural activity. Here, we extend this framework by deriving closed-form expressions for the correlation structure of a more powerful multivariate self- and mutually-exciting Hawkes model class that is driven by exogenous non-negative inputs. We demonstrate that the resulting Linear-Nonlinear-Hawkes (LNH) framework is capable of capturing the dynamics of spike trains with a generally richer multi-correlation structure. We explore several new applications of this framework including highly compact representation of multi-channel spike train data and causal analysis of network information flow.

Maximum entropy models for antibody diversity

Recognition of pathogens relies on families of proteins showing great diversity. Here we construct maximum entropy models of the sequence repertoire, building on recent experiments that provide a nearly exhaustive sampling of the IgM sequences in zebrafish. These models are based solely on pairwise correlations between residue positions but correctly capture the higher order statistical properties of the repertoire. By exploiting the interpretation of these models as statistical physics problems, we make several predictions for the collective properties of the sequence ensemble: The distribution of sequences obeys Zipf's law, the repertoire decomposes into several clusters, and there is a massive restriction of diversity because of the correlations. These predictions are completely inconsistent with models in which amino acid substitutions are made independently at each site and are in good agreement with the data. Our results suggest that antibody diversity is not limited by the sequences encoded in the genome and may reflect rapid adaptation to antigenic challenges. This approach should be applicable to the study of the global properties of other protein families.

A Covariance Function Based Approach to Networked System Identification

System identification for networked control is considered. Due to the random time-delays in the network, it can be difficult to work with a discrete-time model and a continuous-time model is therefore chosen here. A covariance function based method is proposed for estimating the parameters, which relies on the second order statistical properties of the output signal and that the input signal samples can be taken from a discrete-time white noise sequence. Numerical studies indicate that the method is robust to additive measurement noise and packet dropouts.

Trace data characterization and fitting for Markov modeling

We propose a trace fitting algorithm for Markovian Arrival Processes (MAPs) that can capture statistics of any order of interarrival times between measured events. By studying real traffic and workload traces often used in performance evaluation studies, we show that matching higher order statistical properties, in addition to first and second order descriptors, results in increased queueing prediction accuracy with respect to algorithms that only match the mean, the coefficient of variation, and the autocorrelations of the trace. This result supports the approach of modeling traces by the interarrival time process instead of the counting process that is more frequently used in the literature. We proceed by first characterizing the general properties of MAPs using a spectral approach. Based on this result, we show how different MAPs can be combined together using Kronecker products to define a larger MAP with predefined properties of interarrival times. We then devise an algorithm that is based on this Kronecker composition and can accurately fit data traces. This MAP fitting algorithm uses nonlinear optimization that can be customized to fit an arbitrary number of moments and to meet the desired cost-accuracy tradeoff. Numerical results of the fitting algorithm on real data, such as the Bellcore Aug89 trace and a Seagate disk drive trace, indicate that the proposed fitting technique achieves increased prediction accuracy with respect to other state-of-the-art fitting methods.

Noise-induced transitions in slow wave neuronal dynamics

Many neuronal systems exhibit slow random alternations and sudden switches in activity states. Models with noisy relaxation dynamics (oscillatory, excitable or bistable) account for these temporal, slow wave, patterns and the fluctuations within states. The noise-induced transitions in a relaxation dynamics are analogous to escape by a particle in a slowly changing double-well potential. In this formalism, we obtain semi-analytically the first and second order statistical properties: the distributions of the slow process at the transitions and the temporal correlations of successive switching events. We find that the temporal correlations can be used to help distinguish among biophysical mechanisms for the slow negative feedback, such as divisive or subtractive. We develop our results in the context of models for cellular pacemaker neurons; they also apply to mean-field models for spontaneously active networks with slow wave dynamics.

A class of distributions that includes student’s t 2-distribution

Some new characterizations of distributions by regressional properties of extremal order statistics are obtained. Bibliography: 5 titles.

Nonlinearity Reduction by Tone Reservation with Null Subcarriers for WiMAX System

This paper proposes an effective tone reservation method with null subcarriers (TRNS) to reduce the peak to average power ratio (PAPR) for so-called WiMAX system. The TRNS method utilizes a part of null subcarriers in guard band to reduce the PAPR, which satisfies the spectrum mask regulation and avoids bandwidth sacrifice. Furthermore, this paper analyzes the properties of order statistics of orthogonal frequency division multiplexing (OFDM) signals. A type of low complexity TRNS method (LC-TRNS) is proposed to solve problems of the TRNS method, including high complexity and the performance degradation under low predefined threshold. Finally, a joint nonlinearity reduction scheme by predistortion and TRNS method is proposed. Not only the nonlinear distortion in the WiMAX system is reduced, but also the good tradeoff between power efficiency and system performance is achieved. Both analytical and simulation results demonstrate that those proposed methods are effective in terms of spectrum mask regulation, total degradation, complementary cumulative probability function versus PAPR threshold, and bit error rate in presence of nonlinear distortion.

A New Method for Goodness-of-Fit Testing Based on Type-II Right Censored Samples

We present a simple method for testing goodness-of-fit based on type-II right censored samples. Applying the property of order statistics due to Malmquist, we can transform any conventional type-II right censored sample of size r out of n from a uniform distribution to a complete sample of size r from a uniform distribution. This result is used to develop the proposed goodness-of-fit test procedure. The simulation studies reveal that the proposed approach provides as good or better overall power than the method of Michael & Schucany.

Invariance, symmetry and irreversibility in coherence theory

The intrinsic degrees of coherence characterize several properties of partially polarized and partially coherent light. In particular, their properties of invariance by the multiplication of the electric fields by non singular deterministic Jones matrices is reviewed and their ability to exhibit different possible symmetry configurations of the second order statistical properties is discussed. Their evolution when the electric fields of the light are multiplied by random Jones matrices is also analyzed and it is shown that two kinds of irreversible behavior can be exhibited.