Statistical Evidence in Experiments and in Record Values

According to the law of likelihood, statistical evidence for one (simple) hypothesis against another is measured by their likelihood ratio. When the experimenter can choose between two or more experiments (of approximately the same cost) to obtain data, he would want to know which experiment provides (on average) stronger true evidence for one hypothesis against another. In this article, after defining a pre-experimental criterion for the potential strength of evidence provided by an experiment, based on entropy distance, we compare the potential statistical evidence in lower record values with that in the same number of iid observations from the same parent distribution. We also establish a relation between Fisher information and Kullback–Leibler distance.     

Shannon Information Properties of the Endpoints of Record Coverage

This paper addresses the largest and the smallest observations, at the times when a new record of either kind (upper or lower) occurs, which are it called the current upper and lower record, respectively. We examine the entropy properties of these statistics, especially the difference between entropy of upper and lower bounds of record coverage. The results are presented for some common parametric families of distributions. Several upper and lower bounds, in terms of the entropy of parent distribution, for the entropy of current records are obtained. It is shown that mutual information, as well as Kullback–Leibler distance between the endpoints of record coverage, Kullback–Leibler distance between data distribution, and current records, are all distribution-free.     

Some Results Based on a Version of the Generalized Dynamic Entropies

We propose new dynamic measures of uncertainty based on the notion of generalized dynamic entropy introduced in Di Crescenzo and Longobardi (2006). These can uniquely determine distribution functions in continuous and discrete cases, and the characterizations of some well-known distributions are provided. We also define some orderings and aging notions based on the generalized dynamic measures, and prove some of their properties, obtaining as corollaries results that have recently appeared in the literature.     

Sample Entropy Statistics and Testing for Order in Complex Physiological Signals

Sample Entropy (SampEn) statistics have provided insight into the amount of order present in several types of complex physiological time series, particularly the heart rate dynamics of premature infants. Very little, however, is known of SampEn's statistical properties and this has hindered strategies for optimization and significance testing. This article shows that SampEn statistics are asymptotically Gaussian under general conditions. A straightforward point estimate of the statistic's variance is developed and compared to empirical results obtained from complex surrogate data. Statistical tests are developed to quantify the amount and scale of order detected in a signal. These tests are used to show that significant order is, in fact, being detected in the heart rate dynamics of neonates, and to suggest strategies for optimizing the analysis parameters.     

Some Properties of Weighted Distributions in the Context of Repairable Systems

ABSTRACT

In this article we introduce some structural relationships between weighted and original variables in the context of maintainability function and reversed repair rate. Furthermore, we prove some characterization theorems for specific models such as power, exponential, Pareto II, beta, and Pearson system of distributions using the relationships between the original and weighted random variables.     

On Properties of Reversed Mean Residual Life Order for Weighted Distributions

In this article, first, in order to compare X and X ^w (the weighted version of X with weight function w(·)) according to reversed mean residual life order, we provide an equivalent condition. We then try to provide conditions under which the reversed mean residual life order is preserved by weighted distributions. For this end, we obtain several independent results. Finally, the problem of preservation of increasing reversed mean residual life class under weighting is investigated, as well. Some examples are also given to illustrate the results.     

Testing asymptotic independence in bivariate extremes

Bivariate extreme value condition (see (1.1) below) includes the marginal extreme value conditions and the existence of the (extreme) dependence function. Two cases are of interest: asymptotic independence and asymptotic dependence. In this paper, we investigate testing the existence of the dependence function under the null hypothesis of asymptotic independence and present two suitable test statistics. Small simulations are studied and the application for a real data is shown. The other case with the null hypothesis of asymptotic dependence is already investigated.     

On the Influence of Record Terms in the Addition of Independent Random Variables

We discuss some problems connected with the role of record values and maximal values generated by sequences of random variables X₁, X₂,…, X _n in the process of the growth of sums X₁ +···+ X_n, n = 1, 2,….     

Some Characterizations Based on Entropy of Order Statistics and Record Values

In the literature of information theory, Shannon entropy plays an important role and in the context of reliability theory, order statistics and record values are used for statistical modeling. The aim of this article is characterizing the parent distributions based on Shannon entropy of order statistics and record values. It is shown that the equality of the Shannon information in order statistics or record values can determine uniquely the parent distribution. The exponential distribution is characterized through maximizing Shannon entropy of record values under some constraints. The results are useful in the modeling problems.     

Failure Rate and Aging Properties of Generalized Beta- and Gamma-generated Distributions

In the present article we study several characteristics of the families of generalized beta- and gamma- generated distributions introduced by Alexander et al. (2011) and Zografos and Balakrishnan (2009), respectively. Simple formulas are established for calculating the failure rate of the members of the aforementioned families by exploiting the failure rate of the parent distribution. In addition, the aging properties of the generalized beta- and gamma-generated distributions are explored in terms of the corresponding aging behavior of the parent family.     

On an Extended Version of Skew Generalized Normal Distribution and Some of its Properties

Abstract

A generalization of Chauvenet's test (see Bol'shev, L. N. 1969 Bol'shev, L. N. 1969. On tests for rejecting outlying observations. Trudy In-ta prikladnoi Mat. Tblissi Gosudart. univ., 2: 159–177. (In Russian) [Google Scholar]. On tests for rejecting outlying observations. Trudy In-ta prikladnoi Mat. Tblissi Gosudart. univ. 2:159–177. (In Russian); Voinov, V. G., Nikulin, M. N. 1996 Voinov, V. G. and Nikulin, M. N. 1996. Unbaised Estimators and Their Applications Vol. 2, Kluwer Academic Publishers.  [Google Scholar]. Unbaised Estimators and Their Applications. Vol. 2. Kluwer Academic Publishers.) suitable to applied the problem of detecting r outliers in an univariate data set is proposed. In the exponential case, the Chauvenet's test can be used. Various modifications of this test were considered by Bol'shev, Ibrakimov and Khalfina (Ibrakimov, I. A., Khalfina 1978 Ibrakimov, I. A. and Khalfina. 1978. Some asymptotic results concerning the Chauvenet test. Ter. Veroyatnost. i Primenen., 23(3): 593–597.  [Google Scholar]. Some asymptotic results concerning the Chauvenet test. Ter. Veroyatnost. i Primenen. 23(3):593–597.), Greenwood and Nikulin (Greenwood, Nikulin, P. E. 1996 Greenwood and Nikulin, P. E. 1996. A Guide to Chi-Squared Testing New York: John Wiley and Sons, Inc..  [Google Scholar]. A Guide to Chi-Squared Testing. New York: John Wiley and Sons, Inc.) depending on the choice of the estimation method used: MLE or MVUE. As procedures for testing one outlier in exponential model have been investigated by a number of authors including Chikkagoudar and Kunchur (Chikkagoudar, M. S., Kunchur, S. H. 1983 Chikkagoudar, M. S. and Kunchur, S. H. 1983. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth., 12: 2127–2142. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth. 12:2127–2142.), Lewis and Fieller (Lewis, T., Fiellerm N. R. J. 1979 Lewis, T. and Fiellerm, N. R. J. 1979. A recursive algorithm for null distribution for outliers: I. Gamma samples. Technometrics, 21: 371–376. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. A recursive algorithm for null distribution for outliers : I. Gamma samples. Technometrics 21:371–376.), Likes (Likes, J. 1966 Likes, J. 1966. Distribution of Dixon's statistics in the case of an exponential population. Metrika, 11: 46–54. (91, 96, 136, 198–200, 204, 209, 210)[Crossref] , [Google Scholar]. Distribution of Dixon's statistics in the case of an exponential population. Metrika 11:46–54. (91, 96, 136, 198–200, 204, 209, 210).) and Kabe (Kabe, D. G. 1970 Kabe, D. G. 1970. Testing outliers from an exponential population. Metrika, 15: 15–18. [Crossref], [Web of Science ®] , [Google Scholar]. Testing outliers from an exponential population. Metrika 15:15–18.); only two types of statistics for testing multiple outliers exist. First is Dixon's while the second is based on the ratio of the sum of the observations suspected to be outliers to the sum of all observations of the sample. In fact, most of these authors have considered a general case of gamma model and the results for exponential model are given a special case. The object of the present communication is to focus on alternative models, namely slippage alternatives (see Barnett, Vic., Toby Lewis 1978 Barnett, Vic. and Toby, Lewis. 1978. Outlier in Statistical Data New York: John Wiley and Sons, Inc..  [Google Scholar]. Outlier in Statistical Data. New York: John Wiley and Sons, Inc.) in exponential samples. We propose a statistic different from the well known Dixon's statistic D_r to test for multiple outliers. Distribution of the test based on this new statistic under slippage alternatives is obtained and hence the tables of critical values are given, for various n (size of the sample) and r (the number of outliers). The power of the new test is also calculated, it is compared to the power of the Dixon's statistic (Chikkagoudar, M. S., Kunchur, S. H. 1983 Chikkagoudar, M. S. and Kunchur, S. H. 1983. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth., 12: 2127–2142. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth. 12:2127–2142.). Notice that the new statistic based test power is greater the Dixon's statistic based test one.     

An Overview of Asymptotic Properties of Estimators in Truncated Distributions

In this article, a method is proposed to get the limiting distributions and asymptotic properties of estimators based on the minimum and/or maximum of a given srs of a truncated distribution. Following a common outline, a review is carried out by considering different kinds of truncated distributions, some new results are also developed.     

Properties and Inference on the Skew-Curved-Symmetric Family of Distributions

In this article, we study some results related to a specific class of distributions, called skew-curved-symmetric family of distributions that depends on a parameter controlling the skewness and kurtosis at the same time. Special elements of this family which are studied include symmetric and well-known asymmetric distributions. General results are given for the score function and the observed information matrix. It is shown that the observed information matrix is always singular for some special cases. We illustrate the flexibility of this class of distributions with an application to a real dataset on characteristics of Australian athletes.     

On an Application of Concomitants of Order Statistics in Characterizing a Family of Bivariate Distributions

In this article, we consider a family of bivariate distributions which includes the well-known Morgenstern family of bivariate distributions as its subclass. We identify some properties of concomitants of order statistics which characterize this generalized class of distributions. An application of the characterization result in modeling a bivariate distribution to a data is also explained.     

Fisher Information in Record Values and Their Concomitants: A Comparison of Two Sampling Schemes

Two sampling designs via inverse sampling for generating record data and their concomitants are considered: single sample and multisample. The purpose here is to compare the Fisher information in these two sampling schemes. It is shown that the comparison criterion depends on the underlying distribution. Several general results are established for some parametric families and their well known subclasses such as location-scale and shape families, exponential family and proportional (reversed) hazard model. Farlie-Gumbel-Morgenstern (FGM) family, bivariate normal distribution, and some other common bivariate distributions are considered as examples for illustrations and are classified according to this criterion.     

Comparison of record data and random observations based on statistical evidence

“Science looks to Statistics for an objective measure of the strength of evidence in a given body of observations. The Law of Likelihood explains that the strength of statistical evidence for one hypothesis over another is measured by their likelihood ratio” (Blume, 2002). In this paper, we compare probabilities of weak and strong misleading evidence based on record data with those based on the same number of iid observations from the original distribution. We shall also use a criterion defined as a combination of probabilities of weak and strong misleading evidence to do the above comparison. We also give numerical results of a simulated comparison.     

Characterizations of Bivariate and Multivariate Life Distributions Based on Reciprocal Subtangent

The subtangent is the projection of the tangent upon the axis of abscissa. The usefulness of the reciprocal subtangent as a measure of the survival and density curves has earlier been reported in the literature for univariate distributions. This measure was generalized for bivariate and multivariate setups and related characterization problems were examined. The conditionally specified bivariate exponential distribution has been uniquely determined from the local constancy of the bivariate reciprocal subtangents. The case of global constancy and other related results have been studied.

Conditionally specified bivariate Lomax distribution and normal distribution were also studied. Further, the conditionally specified multivariate exponential distribution was uniquely determined from the local constancy of the multivariate reciprocal subtangents.