Evidential inference based on record data and inter-record times

Science looks to statistics for an objective measure of the strength of evidence in a given body of observations. In this paper, we shall use a criterion defined as a combination of probabilities of weak and strong misleading evidence to do the comparison between only record values and the same number of record values and inter-record times. Also, a simulation is presented to illustrate the results.     

Comparing the Fisher information in record data and random observations

We compare the Fisher information (FI) contained in the firstn record values and record times with the FI inn i. i. d. observations. General results are established for exponential family and Weibull type setups, and a summary table is provided listing several common distributions. We show that the FI in record data improves notably once the record times are included, often changing from being less to being equal or greater than the FI in a random sample of the same size. The behavior in the Weibull case is surprising. There it depends onn, whether the record or the i.i. d. observations have more FI. We propose new estimators based on record data. The results may be of interest in some life testing situations. Supported in part by Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grant # 1010222 of Chile.     

Bayesian inference on extreme value distribution using upper record values

In this paper we address estimation and prediction problems for extreme value distributions under the assumption that the only available data are the record values. We provide some properties and pivotal quantities, and derive unbiased estimators for the location and rate parameters based on these properties and pivotal quantities. In addition, we discuss mean-squared errors of the proposed estimators and exact confidence intervals for the rate parameter. In Bayesian inference, we develop objective Bayesian analysis by deriving non informative priors such as the Jeffrey, reference, and probability matching priors for the location and rate parameters. We examine the validity of the proposed methods through Monte Carlo simulations for various record values of size and present a real data set for illustration purposes.     

Linearity of regression for non-adjacent record values

Let X₁,X₂,… be a sequence of iid random variables having a continuous distribution; by R₁,R₂,… denote the corresponding record values. All the distributions allowing linearity of regressions either E(R_m+k|R_m) or E(R_m|R_m+k) are identified.     

A note on characterization based on shannon entropy of record statistics

In this note we compare the Shannon entropy of record statistics with the Shannon entropy of the original data and give an application to characterization of the generalized Pareto distribution,     

Distributions and moments of record values in a sequence of maximally dependent random variables

A sequence of possibly dependent random variables is maximally dependent if all the sample maxima in the sequence have stochastically maximal distributions in the class of all distributions with the same marginals. For a sequence of maximally dependent standard uniform random variables, we determine the distribution functions of record times and values. We show that the distribution of the record occurrence times coincides with the respective distribution for the i.i.d. sequence, and the distributions of the record values are stochastically maximal in the class of sequences with the same record times distributions, containing all the exchangeable sequences. We also derive analytic formulae for the moments of record values from the maximally dependent sequence, and compare them with those of the i.i.d. case.     

The Effect of Sample Size on the Meaning of Significance Tests

Contradictory interpretations of how the meaning of a significance test depends on the sample size are examined.     

Comparing Fisher information in record values and their concomitants with random observations

In a sequence of bivariate random variables {(X _i , Y _i ), i≥1} from a continuous distribution with a real parameter θ, general comparison results between the amount of Fisher information about θ contained in the sequence of the first n records and their concomitants, and the desired information in an i.i.d. sample of size n from the parent distribution are established. Some relationships between reliability properties and the proposed criteria are obtained in situations in which the univariate counterpart of the underlying bivariate family belongs to location, scale or shape families. It is also shown that in some classes of bivariate families, the concerned information property is equivalent to that of its univariate counterpart. The proposed procedure is illustrated by considering several examples.     

On the characterization of generalized extreme value, power function, generalized Pareto and classical Pareto distributions by conditional expectation of record values

In the present paper, we give some theorems to characterize the generalized extreme value, power function, generalized Pareto (such as Pareto type II and exponential, etc.) and classical Pareto (Pareto type I) distributions based on conditional expectation of record values. Received: June 23, 1998; revised version: September 20, 1999     

E-Bayesian estimation for the proportional hazard rate model based on record values

In this article, a new parameter estimation method, named E-Bayesian method, is considered to obtain the estimates of the unknown parameter and reliability function based on record values. The maximum likelihood, Bayesian, E-Bayesian, and hierarchical Bayesian estimates of the unknown parameter and reliability function are obtained when the underlying distribution belongs to the proportional hazard rate model. The Bayesian estimates are obtained based on squared error and linear-exponential loss functions. The previously obtained some relations for the E-Bayesian estimates are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations are obtained under the same loss functions. The comparison of the derived estimates are carried out by using Monte Carlo simulations. Real data are analyzed for an illustration of the findings.     

A statistical multivariable optimization method using improved orthogonal algorithm based on large data

Multivariable optimization under large data environment concerns with how to reliably obtain a set of optimization results from a mass of data that influence the object function complexly. This is an important issue in statistical calculation because the complexity between variable parameters leads to repeated statistical calculation analysis and a significant amount of data waste. A statistical multivariable optimization method using improved orthogonal algorithm based on large data is proposed. Considering the optimization problem with multi-parameters under large data environment, a multi-parameter optimization model used for improved orthogonal algorithm is established based on large data. Furthermore, an extensive simulation study on temperature field distribution of anti-/de-icing component was conducted to verify the validity of the statistical calculation analysis optimization method. The optimized temperature field distribution meets the anti-/de-icing requirements through numerical simulation. Simulation results show that the optimization effect is more evident and accurate than the non-optimized temperature distribution with the optimized results of the proposed method. Results verify the effectiveness of the proposed method.     

Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values

Let X _{U (1)} < X _{U (2)} < … < X _{U ( n )} < … be the sequence of the upper record values from a population with common distribution function F. In this paper, we first give a theorem to characterize the generalized mixtures of geometric distribution by the relation between E[(X _{U ( n +1)}–X _{U ( n )})²|X _{U ( n )} = x] and the function of the failure rate of the distribution, for any positive integer n. Secondly, we also use the same relation to characterize the generalized mixtures of exponential distribution. The characterizing relations were motivated by the work of Balakrishnan and Balasubramanian (1995). Received: March 31, 1999; revised version: November 22, 1999     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

On the estimation and application of max-stable processes

The theory of max-stable processes generalizes traditional univariate and multivariate extreme value theory by allowing for processes indexed by a time or space variable. We consider a particular class of max-stable processes, known as M4 processes, that are particularly well adapted to modeling the extreme behavior of multiple time series. We develop procedures for determining the order of an M4 process and for estimating the parameters. To illustrate the methods, some examples are given for modeling jumps in returns in multivariate financial time series. We introduce a new measure to quantify and predict the extreme co-movements in price returns.     

Extreme quantiles and tail index of a distribution based on kernel estimator

In this paper, we reveal the relationship between the tail exponent introduced by Parzen (1979) and tail index for a distribution function. Furthermore, we analyze the domain of attraction of the weighted sum of the distributions and its tail index. We show that the extreme quantiles can estimate directly, through knowing only the tail index of the kernel distribution function used in estimating the distribution function. Moreover, we give a smoothing parameter of extreme quantiles, which does not depend on any distribution function. The simulations and the application to reals data show that the proposed smoothed parameter gives better results for a heavy-tailed distribution, and for small sizes sample in extremes level.     

Exponentiated generalized Pareto distribution: Properties and applications towards extreme value theory

The GPD is a central distribution in modelling heavy tails in many applications. Applying the GPD to actual datasets however is not trivial. In this paper we propose the Exponentiated GPD (exGPD), created via log-transform of the GPD variable, which has less sample variability. Various distributional quantities of the exGPD are derived analytically. As an application we also propose a new plot based on the exGPD as an alternative to the Hill plot to identify the tail index of heavy tailed datasets, and carry out simulation studies to compare the two.     

Comparison of approaches for estimating the probability of coastal flooding

Coastal flooding is typically caused by combinations of extreme water-levels and large waves. Two extreme value methods, one univariate and the other multivariate, have been used for estimating the probability of coastal flooding at an existing flood defence structure and for aiding the design of a new structure. The properties of these two methods are compared in terms of extrapolation, sophistication and use of information for a range of extremal dependence structures. We find that, when applied to the assessment of the safety offered by an existing Dutch dike, the multivariate approach provides the more useful and accurate design information and has the substantial benefits of consistency and reduced statistical analysis when applied to several sites along a Dutch coastline.     

Estimation of location and scale parameters of a distribution by U-statistics based on best linear functions of order statistics

In this work we propose a technique of estimating the location parameter μ$\mu$ and scale parameter σ$\sigma$ of a distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The method has been illustrated for estimating the location and scale parameters of type-I extreme value distribution. We have computed the asymptotic relative efficiencies of the proposed U-statistics with the appropriate maximum likelihood estimators based on samples drawn from each of type-I extreme value, logistic and normal distributions. In all cases very high asymptotic relative efficiencies are obtained.     

Tail Behavior and Extremes of Short-Tailed Symmetric Distribution

Tiku and Vaughan (1999 Tiku , M. L. , Vaughan , D. C. ( 1999 ). A Family of Short-tailed Symmetric Distributions. Technical Report, McMaster University, Canada . [Google Scholar]) introduced a short-tailed symmetric family recently. In the article, the tail properties of the short-tailed symmetric distribution are studied and the asymptotic distribution of the maximum of i.i.d. random variables obeying the short-tailed distribution is gained.