Order Statistics In Samples Research Articles

On the effect of rounding on hypothesis testing when sample size is large

It is well known that sample moments are more sensitive and less robust than order statistics for robustness with respect to outliers. In this article, we show that the situation is exactly the opposite for robustness with respect to rounding. For large and very large sample sizes, statistical procedures based on order statistics become non‐applicable even for very mild data rounding while procedures based on sample moments work perfectly for this rounding level. The comparison of sample moments and order statistics is made for tests for normality and tests for exponentiality.

Estimation Based on Optimally Selected Single Order Statistic in Pareto: Rayleigh Distribution

Abstract: A special case of the well known T – X family of distributions proposed by Ayman Alzaatreh et al. (2012) called Pareto – Rayleigh (P – R) distribution is considered. Its scale parameter is estimated using a single order statistic in small samples. Optimal criterion for the choice of single order statistic in a given small sample is worked out. Comparison is made with the corresponding optimal choice of single order statistic with respect to the criterion of asymptotic variance. The results are extended to estimate parametric functions like reliability and hazard rate. Keywords: T-X family, Pareto-Rayleigh distribution, order statistics, optimal estimation, asymptotic variance.

Accurate approximation of the expected value, standard deviation, and probability density function of extreme order statistics from Gaussian samples

We show that the expected value of the largest order statistic in Gaussian samples can be accurately approximated as ( 0.2069 ln ( ln ( n ) ) + 0.942 ) 4 , where n ∈ [ 2 , 10 8 ] is the sample size, while the standard deviation of the largest order statistic can be approximated as − 0.4205 arctan ( 0.5556 [ ln ( ln ( n ) ) − 0.9148 ] ) + 0.5675 . We also provide an approximation of the probability density function of the largest order statistic which in turn can be used to approximate its higher order moments. The proposed approximations are computationally efficient, and improve previous approximations of the mean and standard deviation given by Chen and Tyler (1999).

On the order statistics of exponentiated moment exponential distribution and associated inference

Explicit algebraic expressions for both single and product moments of order statistics from the exponentiated moment exponential (EME) distribution are derived. By using these expressions, we have computed the means, variances and covariances of order statistics for and for arbitrarily chosen parameter values. Further, these moments are used to determine the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) of the location and scale parameters based on complete as well as Type-II right censored samples. Prediction of unobserved order statistics in Type-II right censored samples is also discussed. A simulation study and a real data example are presented for the sake of comparison and illustration. The concluding remarks are given at the end.

Sampling properties and empirical estimates of extreme events

The statistical characteristics of the largest observations in a sample are highly uncertain. In this work we consider the problem of how to define empirical estimates of exceedance probabilities and return periods associated with an ordered sample of observations. Understanding the sampling properties of these quantities is important for assessing the fit of a statistical model and also for placing confidence bounds on estimates of extreme events from Monte Carlo simulations. The empirical distribution function (EDF) is often defined as the expected non-exceedance probability (NEP) associated with sample order statistics. Yet, due to the non-linearity of the relations between return periods, quantiles and NEP, the return period (or quantile) associated with the expected NEP is not equal to the expected return period (or quantile), leading to ambiguity. However, the sampling distributions of exceedance probabilities, return periods and quantiles are, in fact, linked by a simple relation. From this relation, it follows that defining the EDF in terms of the median NEP of the order statistics gives a consistent framework for defining empirical estimates of all three quantities. We demonstrate that the median value of the return period of the largest observation is 44% larger than the return period calculated using the common definition of the EDF in terms of the expected NEP of the order statistics. We also derive some new results about the size of the confidence intervals for exceedance probabilities and return periods.

Ordering results of second order statistics from random and non-random number of random variables with Archimedean copulas

In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and $ 2 $-out-of-$ n $/$ 2 $-out-of-$ (n+1) $ systems. Finally, some examples are given to illustrate the obtained results.

Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics

In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are proved. Unlike the conventional analytical approach, here the presentation follows the lines of a kind of arithmetical approach in the space of random variables or vectors. Here the operation of scale mixing in the space of distributions is replaced with the operation of multiplication in the space of random vectors/variables under the assumption that the multipliers are independent. By doing so, the reasoning becomes much simpler, the proofs become shorter and some general features of the distributions under consideration become more vivid. The first part of the paper concerns the univariate case. Some known results are discussed and simple alternative proofs for some of them are presented as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions. The results obtained here open the way to deal with natural multivariate analogs of EP distributions. In the second part of the paper, we discuss the conventionally defined multivariate EP distributions and introduce the notion of projective EP (PEP) distributions. The properties of multivariate EP and PEP distributions are considered as well as limit theorems establishing the conditions for the convergence of multivariate statistics constructed from samples with random sizes (including random sums of random vectors) to multivariate elliptically contoured EP and projective EP laws. The results obtained here give additional theoretical grounds for the applicability of EP and PEP distributions as asymptotic approximations for the statistical regularities observed in data in many fields.

Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions

In the subject of statistics for engineering, physics, computer science, chemistry, and earth sciences, one of the sampling challenges is the accuracy, or, in other words, how representative the sample is of the population from which it was drawn. A series of statistics were developed to measure the departure between the population (theoretical) and the sample (observed) distributions. Another connected issue is the presence of extreme values—possible observations that may have been wrongly collected—which do not belong to the population selected for study. By subjecting those two issues to study, we hereby propose a new statistic for assessing the quality of sampling intended to be used for any continuous distribution. Depending on the sample size, the proposed statistic is operational for known distributions (with a known probability density function) and provides the risk of being in error while assuming that a certain sample has been drawn from a population. A strategy for sample analysis, by analyzing the information about quality of the sampling provided by the order statistics in use, is proposed. A case study was conducted assessing the quality of sampling for ten cases, the latter being used to provide a pattern analysis of the statistics.

Fast Spectrum Sensing Using Spatial Higher Order Statistics With Large Receive Antenna Array

A novel spectrum sensing (SS) technique based on the higher order statistics (HOS) of spatial samples, obtained through a large receive antenna array, is proposed. Unlike conventional SS schemes, our approach, relying mainly on large number of spatial samples, exploits the spatial properties of the primary user’s signals and differentiates them from the Gaussian noises, white or colored, across different antennas. The novelty of our approach lies in treating the spatial samples as though they are the time-domain samples of a non-stationary random process (NSRP). This, in turn, enables the application of HOS-based detection of NSRP while taking advantage of its asymptotic properties as the number of antennas grows large. Simulation results show that, over one time instant, the proposed method is able to detect primary signals under correlated Gaussian noise and is superior to existing approaches in terms of its robustness against any variation of the correlation coefficient of the noise.

A New Technique of Intelligent Constructing Unbiased Prediction Limits on Future Order Statistics Coming from an Inverse Gaussian Distribution under Parametric Uncertainty

This paper provides a new technique for constructing unbiased statistical prediction limits on order statistics of future samples using the results of a previous sample from the same underlying inverse Gaussian distribution. Statistical prediction limits for the inverse Gaussian distribution are obtained from a classical frequentist viewpoint. The results have direct application in reliability theory, where the time until the first failure in a group of several items in service provides a measure of assurance regarding the operation of the items. The statistical prediction limits are required as specifications on future life for components, as warranty limits for the future performance of a specified number of systems with standby units, and in various other applications. Prediction limit is an important statistical tool in the area of quality control. The lower prediction limits are often used as warranty criteria by manufacturers. The technique used here does not require the construction of any tables. It requires a quantile of the beta distribution and is conceptually simple and easy to use. The discussion is restricted to one-sided tolerance limits. For illustration, a numerical example is given.

A New Technique for Intelligent Constructing Exact γ-content Tolerance Limits with Expected (1 – α)-confidence on Future Outcomes in the Weibull Case Using Complete or Type II Censored Data

The logical purpose for a statistical tolerance limit is to predict future outcomes for some (say, production) process. The coverage value γ is the percentage of the future process outcomes to be captured by the prediction, and the confidence level (1 – α) is the proportion of the time we hope to capture that percentage γ. Tolerance limits of the type mentioned above are considered in this paper, which presents a new technique for constructing exact statistical (lower and upper) tolerance limits on outcomes (for example, on order statistics) in future samples. Attention is restricted to the two-parameter Weibull distribution under parametric uncertainty. The technique used here emphasizes pivotal quantities relevant for obtaining tolerance factors 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 numerical examples, where the proposed analytical methodology is illustrated in terms of the two-parameter Weibull distribution. Applications to other log-location-scale distributions could follow directly.

Scale Mixtures of Frechet Distributions as Asymptotic Approximations of Extreme Precipitation

Based on the negative binomial model for the duration of wet periods measured in days, an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Frechet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor-Fisher distribution. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions the limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. Several methods are proposed for the estimation of the parameters of the distribution of the maximum daily precipitation volume. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability ("Bayesian") models.

The probability distribution of extreme precipitation

On the basis of the negative binomial distribution of the duration of wet periods calculated per day, an asymptotic model is proposed for distributing the maximum daily rainfall volume during the wet period, having the form of a mixture of Frechet distributions and coinciding with the distribution of the positive degree of a random variable having the Fisher–Snedecor distribution. The method of proving the corresponding result is based on limit theorems for extreme order statistics in samples of a random volume with a mixed Poisson distribution. The adequacy of the models proposed and methods of their statistical analysis is demonstrated by the example of estimating the extreme distribution parameters based on real data.

MODERATE DEVIATION PRINCIPLE OF SAMPLE QUANTILES AND ORDER STATISTICS

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m-dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m-dependent sequence.

Tolerance limits on order statistics in future samples coming from the Pareto distribution

It is often desirable to have statistical tolerance limits available for the distributions used to describe time-to-failure data in reliability problems. For example, one might wish to know if at least a certain proportion, say β, of a manufactured product will operate at least T hours. This question cannot usually be answered exactly, but it may be possible to determine a lower tolerance limit L(X), based on a random sample X, such that one can say with a certain confidence γ that at least 100β% of the product will operate longer than L(X). Then reliability statements can be made based on L(X), or, decisions can be reached by comparing L(X) to T. Tolerance limits of the type mentioned above are considered in this paper, which presents a new approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors 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 past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the Pareto distribution is considered. The discussion is restricted to one-sided tolerance limits. A practical example is given.

A New Approach to Constructing Tolerance Limits on Order Statistics in Future Samples Coming from a Normal Distribution

Although the concept of statistical tolerance limits has been well recognized for long time, surprisingly, it seems that their applications remain still limited. Analytic formulas for the tolerance limits are available in only simple cases, for example, for the upper or lower tolerance limit for a univariate normal population. Thus it becomes necessary to use new or innovative approaches which will allow one to construct tolerance limits on future order statistics for many populations. In this paper, a new approach to constructing lower and upper tolerance limits on order statistics in future samples is proposed. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors 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 past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the normal distribution is considered. The discussion is restricted to one-sided tolerance limits. A practical example of finding a warranty assessment of image quality is given.

Inequalities for order statistics of samples from distributions with monotone hazard rate

A number of inequalities relating the mathematical expectations of order statistics or functions of order statistics in the case where the sample is taken from a distribution with monotone hazard rate function are presented. These inequalities can be applied in reliability theory and other areas of probability theory and mathematical statistics. In particular, in the paper, they are used to obtain new characterizations of the exponential distribution.

Various limit theorems for ratios from the uniform distribution

Abstract In this paper, we consider the ratios of order statistics in samples from uniform distribution and establish strong and weak laws for these ratios.

Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution

This paper presents an innovative approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors 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 past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the two-parameter exponential distribution is considered. A practical example is given.

On regression relations between sample means and order statistics

New characterizations of certain probability distributions by regression equalities involving sample means and various elements of the variational series are obtained. As a particular case, relations characterizing Student’s t2-distribution are presented.