Some results on conditional expectations of lower record values

In this paper, we propose a measure for obtaining the expectation of time between two lower k-records under the condition that the greater one is given. Several properties of the proposed measure are derived. Some characterization results and stochastic comparisons based on the new measure are also provided.     

Sample quantiles and additive statistics: Information,sufficiency, estimation

The order of the increase in the Fisher information measure contained in a finite number k of additive statistics or sample quantiles, constructed from a sample of size n, as n → ∞, is investigated. It is shown that the Fisher information in additive statistics increases asymptotically in a manner linear with respect to n, if 2 + δ moments of additive statistics exist for some δ > 0. If this condition does not hold, the order of increase in this information is non-linear and the information may even decrease. The problem of asymptotic sufficiency of sample quantiles is investigated and some linear analogues of maximum likelihood equations are constructed.     

Mixed-spectra analysis for stationary random fields

We consider a, discrete time, weakly stationary bidimensional process, for which the spectral measure is the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. In this paper we are interested in estimating the spectral density of the absolutely continuous measure and of the density on the lines. For this aim, by using the double kernel method, we construct consistent estimators of these densities and we study their asymptotic behaviors in term of the mean squared error with rate.     

基于非参数核估计的股市风险理论及应用研究

根据中国股市非市场化特点与股权分置改革的影响等问题,在股票市场价格严重失真情况下,运用非参数核估计方法,拟合了股市的理论收益率指标,通过对比股市实际收益率指标显著性偏离关系,提出了中国股市风险理论的独特含义与计量方法。实证显示：非参数核估计能够很好估计股市理论收益率指标,准确地计量中国股市风险动态变化情况。     

Using a Markov Chain to Construct a Tractable Approximation of an Intractable Probability Distribution

Abstract.  Let π denote an intractable probability distribution that we would like to explore. Suppose that we have a positive recurrent, irreducible Markov chain that satisfies a minorization condition and has π as its invariant measure. We provide a method of using simulations from the Markov chain to construct a statistical estimate of π from which it is straightforward to sample. We show that this estimate is 'strongly consistent' in the sense that the total variation distance between the estimate and π converges to 0 almost surely as the number of simulations grows. Moreover, we use some recently developed asymptotic results to provide guidance as to how much simulation is necessary. Draws from the estimate can be used to approximate features of π or as intelligent starting values for the original Markov chain. We illustrate our methods with two examples.     

On Two Forms of Fisher's Measure of Information

ABSTRACT

Fisher's information number is the second moment of the “score function” where the derivative is with respect to x rather than Θ. It is Fisher's information for a location parameter, and also called shift-invariant Fisher information. In recent years, Fisher's information number has been frequently used in several places regardless of parameters of the distribution or of their nature. Is this number a nominal, standard, and typical measure of information? The Fisher information number is examined in light of the properties of classical statistical information theory. It has some properties analogous to those of Fisher's measure, but, in general, it does not have good properties if used as a measure of information when Θ is not a location parameter. Even in the case of location parameter, the regularity conditions must be satisfied. It does not possess the two fundamental properties of the mother information, namely the monotonicity and invariance under sufficient transformations. Thus the Fisher information number should not be used as a measure of information (except when Θ a location parameter). On the other hand, Fisher's information number, as a characteristic of a distribution f(x), has other interesting properties. As a byproduct of its superadditivity property a new coefficient of association is introduced.     

The stability of several measures of association in small contingency tables

Measures of association are often used to describe the relationship between row and column variables in two—dimensional contingency tables. It is not uncommon in biomedical research to categorize continuous variables to obtain a two—dimensional table. In these situations it is desirable that the measure of association not be too sensitive to changes in the number of categories or to the choice of cut points. To accomplish this objective we attempt to find a measure of association that closely approximates the corresponding measure of association for the underlying distribution.Measures that are close to the underlying measure for various table sizes andcutpoints are called stable measures.     

Estimation of the term structure of interest rates with German government bonds

In this paper we estimate the term structure of interest rates using multiple linear regression. The regression equation is derived, from a no-arbitrage condition. Considering only bonds with the same coupon payment date and excluding possible collinearities results in a substantial reduction of the mean squared error as measure of mispricing in comparison to Bußmann (1989).     

An overlook on statistical inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition

ABSTRACT

In this paper, we are interested in nonparametric inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition. This condition relates the magnitude of the dissipative term and the magnitude of the random term. The precise balance between the drift term which removes energy in average and the stochastic term provided by the fluctuation-dissipation relation insures that the canonical measure is preserved by the dynamics. In this framework, it is possible to give an explicit construction of a Lyapunov function and thus to prove exponential ergodicity. Then, we consider various estimation procedures and provide also a numerical section, where simulations are conducted.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

不确定性决策中的时间价值

一、引言在经济决策中 ,时间价值问题是决策者必须加以考虑的问题。现有有关时间价值的研究表明 ,相同的货币收益或成本在不同的时期具有不同的价值 ,这种因时间上的差异所导致的价值差别就构成了决策中的时间价值量 ,为了对不同时期的货币收益或成本加以比较与综合 ,通常的办法是对未来的货币量进行折现 ,从而使不同时期的货币量均以现值的形式表现。从理论上分析 ,未来时期货币量的现值为 :Ｖ0 =ｋｔ(1+Ｒ1 ) (1+Ｒ2 )… (1+Ｒｔ) (1)　　其中 :Ｒｉ 为第ｉ期的贴现率 ,ｋｔ 为第ｔ期的货币量 :由于在现实经济活动中Ｒｉ(ｉ =1,2 ,… ,ｔ)…     

A NOTE ON EVANESCENT PROCESSES

This note examines the connection between μ-invariant measures for the transition function of a continuous-time Markov chain and those of its q-matrix, Q. The major result establishes a necessary and aufficient condition for a convergent μ-invariant measure for Q to be μ-inhant for the minimal transition function, P, under the assumption that P is honest. This corrects Theorem 6 of Vere-Jones (1969) and the first part of Corollary 1 of Pollett (1986), both of which assert that the above conclusion holds in the absence of this condition. The error was pointed out by E.A. van Doom (1991) and the counterexample which be presented provides the basis for the present arguments. In determining where the error occurred in the original proof, we are able to identify a simple sufficient condition for μ-invariance.     

How Important Is the Effect of Rounding in Goodness-of-Fit

Abstract

In the area of goodness-of-fit there is a clear distinction between the problem of testing the fit of a continuous distribution and that of testing a discrete distribution. In all continuous problems the data is recorded with a limited number of decimals, so in theory one could say that the problem is always of a discrete nature, but it is a common practice to ignore discretization and proceed as if the data is continuous. It is therefore an interesting question whether in a given problem of test of fit, the “limited resolution” in the observed recorded values may be or may be not of concern, if the analysis done ignores this implied discretization. In this article, we address the problem of testing the fit of a continuous distribution with data recorded with a limited resolution. A measure for the degree of discretization is proposed which involves the size of the rounding interval, the dispersion in the underlying distribution and the sample size. This measure is shown to be a key characteristic which allows comparison, in different problems, of the amount of discretization involved. Some asymptotic results are given for the distribution of the EDF (empirical distribution function) statistics that explicitly depend on the above mentioned measure of degree of discretization. The results obtained are illustrated with some simulations for testing normality when the parameters are known and also when they are unknown. The asymptotic distributions are shown to be an accurate approximation for the true finite n distribution obtained by Monte Carlo. A real example from image analysis is also discussed. The conclusion drawn is that in the cases where the value of the measure for the degree of discretization is not “large”, the practice of ignoring discreteness is of no concern. However, when this value is “large”, the effect of ignoring discreteness leads to an exceded number of rejections of the distribution tested, as compared to what would be the number of rejections if no rounding is taking into account. The error made in the number of rejections might be huge.     

Information Loss in Grade Point Conversion

Some information gets lost when numerical scores evaluating performances are converted into letter grades. We propose to measure this information loss through the proportion of variance lost due to grouping. We study various properties of this measure, including its invariance in location and scale equivariant families. The information loss typically decreases with an increase in the number of levels of letter grades. However, it is not appropriate to have too many levels. The optimum number of levels may be determined, either by visual inspection when the information loss becomes marginal/stable, or by minimizing the sum of the information loss and a penalty term, the latter being taken as linear in the number of levels. We also address the problem of determining the groups, or equivalently, the boundaries so that the information loss is minimized, given a fixed number of groups. Finding these optimal boundaries is a computationally intensive exercise even for moderate size data, unless the number of groups is very small. We recommend an alternative way by fitting an appropriate probability distribution. When the probabilistic nature of the data is known, the boundary points turn out to be the solutions to a system of equations; however these solutions may not necessarily have any closed form. We derive the exact or approximate solutions of these equations when the composite scores follow a probability distribution belonging to the Uniform, Triangular, and Gaussian family.     

A measure of asymmetry

It is a general practice to make assertions about the symmetry or asymmetry of a probability density function based on the coefficients of skewness. Since most of the coefficients of skewness are designed to be zero for a symmetric density, they overall do provide an indication of symmetry. However, skewness is primarily influenced by the tail behavior of a density function, and the skewness coefficients are designed to capture this behavior. Thus they do not calibrate asymmetry in the density curves. We provide a necessary condition for a probability density function to be symmetric and use that to measure asymmetry in a continuous density curve on the scale of ?1 to 1. We show through examples that the proposed measure does an admirable job of capturing the visual impression of asymmetry of a continuous density function.     

Exact small sample properties of an operational variant of the minimum mean squared error estimator

In this paper, we show a sufficient condition for an operational variant of the minimum mean squared error estimator (simply, the minimum MSE estimator) to dominate the ordinary least squares (OLS) estimator. It is also shown numerically that the minimum MSE estimator dominates the OLS estimator if the number of regression coefficients is larger than or equal to three, even if the sufficient condition is not satisfied. When the number of regression coefficients is smaller than three, our numerical results show that the gain in MSE of using the minimum MSE estimator is larger than the loss.     

设计包含最多纯净两因子交互效应分量条件

对于任意的素数或者素数幂 ,本文通过分析 设计中不纯净两因子交互效应(Two-Factor Interaction, 2FI)分量的个数,推导出某些 设计包含最多纯净2FI分量的一个条件。该结论对构造包含最多纯净2FI分量的 设计具有重要意义。     

菲茨杰拉德创作能力的变迁——基于词汇测度的统计分析

传统上,不同评论者对同一作家作品的好恶往往会导致对作家创作能力截然不同的评价,这种主观定性评判并没有考虑作品本身的内在特质。计算语言学发现幂律分布在各类文本的词汇测度中具有普遍性,这为定量刻画文学作品的内在特质提供了统计手段。因为作家的创作能力不是一成不变的,不同创作时期的作品反映了作家创作能力的波动,为检验这一论断,以菲茨杰拉德的四部小说为例,对包括类符形符比、词汇增长率和词汇相对熵等在内的词汇测度进行统计分析,发现这三个测度与文中的形符数之间服从幂律关系。故提出一种计算作家创作能力的新测度以及比较其创作能力变化的新方法。菲茨杰拉德作品的实证分析表明：该测度不仅能够有力地印证现有的定性评论观点,而且还能够有效地反映作家创作能力的巅峰与低谷状态。     

Sensitivity analysis approaches to high-dimensional screening problems at low sample size

Sensitivity analysis is an essential tool in the development of robust models for engineering, physical sciences, economics and policy-making, but typically requires running the model a large number of times in order to estimate sensitivity measures. While statistical emulators allow sensitivity analysis even on complex models, they only perform well with a moderately low number of model inputs: in higher dimensional problems they tend to require a restrictively high number of model runs unless the model is relatively linear. Therefore, an open question is how to tackle sensitivity problems in higher dimensionalities, at very low sample sizes. This article examines the relative performance of four sampling-based measures which can be used in such high-dimensional nonlinear problems. The measures tested are the Sobol' total sensitivity indices, the absolute mean of elementary effects, a derivative-based global sensitivity measure, and a modified derivative-based measure. Performance is assessed in a ‘screening’ context, by assessing the ability of each measure to identify influential and non-influential inputs on a wide variety of test functions at different dimensionalities. The results show that the best-performing measure in the screening context is dependent on the model or function, but derivative-based measures have a significant potential at low sample sizes that is currently not widely recognised.     

University student enrollment forecasts by analysis structural ratios using ARIMA-methods

Summary Forecasts for the number of students in Germany are conducted by the Kultusministerkonferenz. They use a transition model which does not allow for prediction intervals and therefore lack a measure of uncertainty of the forecast. Since the uncertainty is high for such forecasts, this lack is of importance. In this paper, structural ratios, relating the number of university students to the population of the same age, are analyzed and forescasted using ARIMA-models with outliers. Multiplying these ratios with official population forecasts for Germany provides the future number of students, additionally giving prediction intervals. This number will increase from 1.94 million in 2002 to 2.35 million in 2015. The uncertainty of the forecast is high; the forecast interval in 2015 will range between 1.72 and 2.98 million at a 95% confidence level. Supported by the German Research Foundation (DFG). We are grateful to an anonymous referee for some helpful comments.