The Folded Logistic Distribution

ABSTRACT

Physical measurements like dimensions, including time, and angles in scientific experiments are frequently recorded without their algebraic sign. The directions of those physical quantities measured with respect to a frame of reference in most practical applications are considered to be unimportant and are ignored. As a consequence, the underlying distribution of measurements is replaced by a distribution of absolute measurements. When the underlying distribution is logistic, the resulting distribution is called the “folded logistic distribution”. Here, the properties of the folded logistic distribution will be presented and the techniques for estimating parameters will be given. The advantages of using this folded logistic distribution over the folded normal distribution will be discussed and some examples will be cited.     

New Folded Models for the Log-Transformed Norwegian Fire Claim Data

Folded distributions are useful models in statistics. However, not much is known beyond the folded normal distribution introduced in the 1960s. Here, we introduce several new folded distributions. Statistical properties of each distribution are derived. Applications are provided to the Norwegian fire claim data.     

Utilizing the Flexibility of the Epsilon-Skew-Normal Distribution for Common Regression Problems

In this paper we illustrate the properties of the epsilon-skew-normal (ESN) distribution with respect to developing more flexible regression models. The ESN model is a simple one-parameter extension of the standard normal model. The additional parameter ~ corresponds to the degree of skewness in the model. In the fitting process we take advantage of relatively new powerful routines that are now available in standard software packages such as SAS. It is illustrated that even if the true underlying error distribution is exactly normal there is no practical loss n power with respect to testing for non-zero regression coefficients. If the true underlying error distribution is slightly skewed, the ESN model is superior in terms of statistical power for tests about the regression coefficient. This model has good asymptotic properties for samples of size n>50.     

A Marshall–Olkin family of heavy-tailed distributions which includes the lognormal one

Abstract

A new class of heavy-tailed distribution functions,, containing the lognormal distribution as a particular case is introduced. The class thus obtained depends on a set of three parameters, incorporating an additional distribution to the classical lognormal one. This new class of heavy-tailed distribution is presented as an alternative to other useful heavy-tailed distributions, such as the lognormal, Weibull, and Pareto distributions. The density and distribution functions of this new class are given by a closed expression which allows us to easily compute probabilities, quantiles, moments, and related measurements. Finally, some applications are shown as examples.     

Moments Determine the Tail of a Distribution (But Not Much Else)

How much information does a finite collection of moments carry about the underlying distribution? We revive an old bound, give a new, simple formula for its calculation, and demonstrate that although very little can be said about the central part of the distribution, the tail is much more sharply defined.     

On the sample characterization criterion for normal distributions

Conventionally, it was shown that the underlying distribution is normal if and only if the sample mean and sample variance from a random sample are independent. This paper focusses on the normal population characterization theorem by showing that, if the joint distribution of a skew normal sample follows certain multivariate skew normal distribution, the sample mean and sample variance are still independent.     

Description of a subfamily of the discrete pearson system as generalized-binomial distributions

In this paper we prove that a subfamily of distributions of the discrete Pearson, system, containing the Pólya distribution without replacement and hence the hypergeometric distribution, can be described as generalized-binomial distributions, i.e., the distribution of the number of successes which occur in independent trials. It is also shown that the probability of success will necessarily be different in each trial, with the exception of deterministic ones. As a consequence, all the properties of the generalized-binomial distribution will apply to this subfamily. Thus, applications to hypothesis testing and confidence intervals in the Pólya distribution are considered.     

A bootstrap control chart for inverse Gaussian percentiles

The conventional Shewhart-type control chart is developed essentially on the central limit theorem. Thus, the Shewhart-type control chart performs particularly well when the observed process data come from a near-normal distribution. On the other hand, when the underlying distribution is unknown or non-normal, the sampling distribution of a parameter estimator may not be available theoretically. In this case, the Shewhart-type charts are not available. Thus, in this paper, we propose a parametric bootstrap control chart for monitoring percentiles when process measurements have an inverse Gaussian distribution. Through extensive Monte Carlo simulations, we investigate the behaviour and performance of the proposed bootstrap percentile charts. The average run lengths of the proposed percentage charts are investigated.     

中国居民资本要素收入有多少?

资本要素收入作为国民总收入中的重要组成部分,测算、厘清其在政府、企业和住户之间的分配状况以及居民资本要素收入的结构是必要的基础性工作。利用资金流量表和统计局住户调查数据,对我国资本要素收入及其部门分配和居民资本要素收入的结构进行测算,研究发现:(1)居民资本要素收入长期以来高速增长,在国民总收入中所占比例基本稳定在10%左右,但在总资本要素收入中所占比重经历了先上升(20世纪90年代)后下降(21世纪初)的过程;(2)居民税前、税后资本要素收入对比反映出我国对居民资本要素收入征税力度尚小,最高不超过4%;(3)伴随改革和经济社会发展,我国居民资本要素收入结构由改革开放初期主要以农村居民的经营性收入为主转变为经营性、财产性收入并驾齐驱;(4)宏、微观口径测算的结果存在一定缺口,主要由劳动者报酬存在低估以及住户数据的系统性误差导致形成,国民经济核算的完善应得到关注。     

Analysing the interevent time distribution to identify seismicity phases: a Bayesian nonparametric approach to the multiple-changepoint problem

In the study of earthquakes, several aspects of the underlying physical process, such as the time non-stationarity of the process, are not yet well understood, because we lack clear indications about its evolution in time. Taking as our point of departure the theory that the seismic process evolves in phases with different activity patterns, we have attempted to identify these phases through the variations in the interevent time probability distribution within the framework of the multiple-changepoint problem. In a nonparametric Bayesian setting, the distribution under examination has been considered a random realization from a mixture of Dirichlet processes, the parameter of which is proportional to a generalized gamma distribution. In this way we could avoid making precise assumptions about the functional form of the distribution. The number and location in time of the phases are unknown and are estimated at the same time as the interevent time distributions. We have analysed the sequence of main shocks that occurred in Irpinia, a particularly active area in southern Italy: the method consistently identifies changepoints at times when strong stress releases were recorded. The estimation problem can be solved by stochastic simulation methods based on Markov chains, the implementation of which is improved, in this case, by the good analytical properties of the Dirichlet process.     

企业异质性、市场化与生产率分布

本文运用工业企业数据库估计了企业的全要素生产率,并对我国东中西部生产率分布进行了比较,统计检验发现东中西部生产率存在显著差异,差异有缩小的趋势。用回归模型进一步验证,发现区位优势对生产率存在显著溢价。接下来检验了生产率分布情况,发现企业生产率分布是右偏的,大量的企业生产率处于低水平范围。用伽马分布估计了省级层面的生产率分布参数,发现伽马分布的形状参数和尺度参数是逐渐变大的。接着用估计出来的参数构造省级面板数据,发现市场化程度与分布参数显著正相关,但对形状参数和尺度参数的影响程度不一样,这间接解释了我国区域间生产率差异为什么持续存在。     

Weighted-bootstrap alignment of explanatory variables

Adjustment for covariates is a time-honored tool in statistical analysis and is often implemented by including the covariates that one intends to adjust as additional predictors in a model. This adjustment often does not work well when the underlying model is misspecified. We consider here the situation where we compare a response between two groups. This response may depend on a covariate for which the distribution differs between the two groups one intends to compare. This creates the potential that observed differences are due to differences in covariate levels rather than “genuine” population differences that cannot be explained by covariate differences. We propose a bootstrap-based adjustment method. Bootstrap weights are constructed with the aim of aligning bootstrap–weighted empirical distributions of the covariate between the two groups. Generally, the proposed weighted-bootstrap algorithm can be used to align or match the values of an explanatory variable as closely as desired to those of a given target distribution. We illustrate the proposed bootstrap adjustment method in simulations and in the analysis of data on the fecundity of historical cohorts of French-Canadian women.     

Modeling asset returns with alternative stable distributions

In the 1960's Benoit Mandelbrot and Eugene Fama argued strongly in favor of the stable Paretian distribution as a model for the unconditional distribution of asset returns. Although a substantial body of subsequent empirical studies supported this position, the stable Paretian model plays a minor role in current empirical work.

While in the economics and finance literature stable distributions are virtually exclusively associated with stable Paretian distributions, in this paper we adopt a more fundamental view and extend the concept of stability to a variety of probabilistic schemes. These schemes give rise to alternative stable distributions, which we compare empirically using S&P 500 stock return data. In this comparison the Weibull distribution, associated with both the nonrandom-minimum and geometric-random summation schemes dominates the other stable distributions considered-including the stable Paretian model.     

Modeling asset returns with alternative stable distributions *

In the 1960's Benoit Mandelbrot and Eugene Fama argued strongly in favor of the stable Paretian distribution as a model for the unconditional distribution of asset returns. Although a substantial body of subsequent empirical studies supported this position, the stable Paretian model plays a minor role in current empirical work.

While in the economics and finance literature stable distributions are virtually exclusively associated with stable Paretian distributions, in this paper we adopt a more fundamental view and extend the concept of stability to a variety of probabilistic schemes. These schemes give rise to alternative stable distributions, which we compare empirically using S&P 500 stock return data. In this comparison the Weibull distribution, associated with both the nonrandom-minimum and geometric-random summation schemes dominates the other stable distributions considered-including the stable Paretian model.     

Experimental designs for binary data in switching measurements on superconducting Josephson junctions

Summary.  We study the optimal design of switching measurements of small Josephsonjunction circuits which operate in the macroscopic quantum tunnelling regime. In the experiment, sequences of current pulses are applied to the Josephson junction sample, while the voltage over the structure is monitored. The appearance of a voltage pulse to a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Starting from the D -optimality criterion we derive the optimal design for the estimation of the unknown parameters of the underlying Gumbel-type distribution. As a practical method for the measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The design presented has immediate applications in the area of superconducting electronics, implying faster data acquisition. The experimental results presented confirm the usefulness of the method.     

Model selection using union-intersection principle for non nested models

In this article, we have extended the Vuong’s (1989 Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypothesis. Econometrica. 57:307–333.[Crossref], [Web of Science ®] , [Google Scholar]) model selection test to three models in accordance to union-intersection principle. Using the Kullback–Leibler criterion to measure the closeness of a model to the truth, we propose a simple likelihood ratio-based statistics for testing the null hypothesis that the competing models are equally close to the true data-generating process against the alternative hypothesis that at least one model is closer. We show that the distribution of the test statistic is asymptotically equal to the distribution of the maximum of dependent random variables with bivariate folded standard normal distribution. The density function of the maximum of dependent random variables with elliptically contoured distributions has been obtained by other researchers, but, not for distributions which do not belong to the elliptically contoured distributions family. In this article, the exact distribution of the maximum of dependent random variables with bivariate folded standard normal distribution is calculated as an asymptotic distribution of the proposed test statistic. The test is directional and is derived successively for the cases where the competing models are non nested and whether three, two, one, or none of them are misspecified.     

On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data

We study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our results via several simulation studies and a real data application and show the advantages of PROS estimates over their counterparts under the simple random and ranked set sampling designs.     

STRATEGY, NONTRANSITIVE DOMINANCE AND THE EXPONENTIAL DISTRIBUTION

An easily programmed recursive formula for the evaluation of the distribution function of ratios of linear combinations of independent exponential random variables is developed. This formula is shown to yield the probability that one team beats another in a contest we call the special gladiator game. This game generates tournaments which exhibit nontransitive dominance and have some surprising consequences. Similar results are obtained for a recursive formula based on the geometric distribution.     

A Proposal for a New Bound for Discrete Distributions

In this work we re-examine some classical bounds for non negative integer-valued random variables by means of information theoretic or maxentropic techniques using fractional moments as constraints. The proposed new bound, no more analytically expressible in terms of moments or moment generating function (mgf), is built by mixing classical bounds and the Maximum Entropy (ME) approximant of the underlying distribution; such a new bound is able to exploit optimally all the information content provided by the sequence of given moments or by the mgf. Particular care will be devoted to obtain fractional moments from the available information given in terms of integer moments and/or moment generating function. Numerical examples show clearly that the bound improvement involving the ME approximant based on fractional moments is not trivial.     

Exact permutation inference for two sample repeated measures data

Experiments in which very few units are measured many times sometimes present particular difficulties. Interest often centers on simple location shifts between two treatment groups, but appropriate modeling of the error distribution can be challenging. For example, normality may be difficult to verify, or a single transformation stabilizing variance or improving normality for all units and all measurements may not exist. We propose an analysis of two sample repeated measures data based on the permutation distribution of units. This provides a distribution free alternative to standard analyses. The analysis includes testing, estimation and confidence intervals. By assuming a certain structure in the location shift model, the dimension of the problem is reduced by analyzing linear combinations of the marginal statistics. Recently proposed algorithms for computation of two sample permutation distributions, require only a few seconds for experiments having as many as 100 units and any number of repeated measures. The test has high asymptotic efficiency and good power with respect to tests based on the normal distribution. Since the computational burden is minimal, approximation of the permutation distribution is unnecessary.