Laguerre and Hermite bases for inverse problems

We present inverse problems of nonparametric statistics which have a smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are $Y_i=X_i{U}_i,Z_i=X_i+{\Sigma }_i,$ where the $X_i$’s are i.i.d. with unknown density $f$, the ${\Sigma }_i$’s are i.i.d. with known density $f_{\Sigma }$, the $U_i$’s are i.i.d. with uniform density on $[0,1]$. The sequences $\left(X_i\right),\left(U_i\right),\left({\Sigma }_i\right)$ are independent. We define projection estimators of $f$ in the two cases of indirect observations of $(X_1,\dots ,X_n)$, and we give upper bounds for their ${\mathbb{L}}^2$-risks on specific Sobolev–Laguerre or Sobolev–Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias–variance compromise.     

Optimal Berry–Esseen bound for parameter estimation of SPDE with small noise

We investigate a rate of convergence on asymptotic normality of the maximum likelihood estimator (MLE) for parameter $\theta$ appearing in parabolic SPDEs of the form

$d{u}^{?}(t,x)=(A_0+\theta A_1)u^{?}(t,x)dt+?dW(t,x),$

where $A_0$ and$A_1$ are partial differential operators, $W$ is a cylindrical Brownian motion (CBM) and $?\downarrow 0$. We find an optimal Berry–Esseen bound for central limit theorem (CLT) of the MLE. It is proved by developing techniques based on combining Malliavin calculus and Stein’s method.     

On the rate of convergence of uniform approximations for sequences of distribution functions

In this paper, we develop uniform bounds for the sequence of distribution functions of $g(V_n+{\mu }_n)$, where $g$ is some smooth function, $\{V_n,n\ge 1\}$ is a sequence of identically distributed random variables with common distribution having a bounded derivative and $\left\{{\mu }_n\right\}$ are constants such that ${\mu }_n\to \infty$. These bounds allow us to identify a suitable sequence of random variables which is asymptotically of the same type of $g(V_n+{\mu }_n)$ showing that the rate of convergence for these uniform approximations depends on the ratio of the second derivative to the first derivative of $g$. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.     

Some practical advice on polynomial regression analysis from blocked response surface designs

It is often necessary to run response surface designs in blocks. In this paper the analysis of data from such experiments, using polynomial regression models, is discussed. The definition and estimation of pure error in blocked designs are considered. It is recommended that pure error is estimated by assuming additive block and treatment effects, as this is more consistent with designs without blocking. The recovery of inter-block information using REML analysis is discussed, although it is shown that it has very little impact if the design is nearly orthogonally blocked. Finally prediction from blocked designs is considered and it is shown that prediction of many quantities of interest is much simpler than prediction of the response itself.     

劳动收入份额的决定与波动机制研究

本文通过数理经济模型的理论分析和计量经济模型的实证分析发现,决定社会产品初次分配中劳动收入份额高低的因素,除了政府的生产税税率之外,主要是劳动者的实际工资水平和生产技术水平,且二者的作用方向完全相反,都取决于要素替代弹性的大小。当资本和劳动的替代弹性小于1时,劳动者的实际工资水平越高,劳动收入份额就越高,劳动者的生产技术水平越高,劳动收入份额就越低;而当资本和劳动的替代弹性大于1时,则劳动者的实际工资水平越高,劳动收入份额就越低;劳动者的生产技术水平越高,劳动收入份额就越高。理论和实证的分析还表明,劳动者货币工资水平和实际工资水平的增长随着经济周期的波动而波动,实际工资水平增长率的波动与经济周期的波动方向相反,导致劳动收入份额也随经济周期反向波动。     

A new goodness-of-fit test for certain bivariate distributions applicable to traffic accidents

T. Cacoullos and H. Papageorgiou [On some bivariate probability models applicable to traffic accidents and fatalities, Int. Stat. Rev. 48 (1980) 345–356] studied a special class of bivariate discrete distributions appropriate for modeling traffic accidents, and fatalities resulting therefrom. The corresponding random variable may be written as $Z={(N,Y)}^{\prime }$, with $Y={\sum }_{j=1}^N{X}_j$, where ${\left\{X_j\right\}}_{j=1}^N$, are independent copies of a (discrete) random variable $X$, and $N$ is independent of ${\left\{X_j\right\}}_{j=1}^N$, and follows a Poisson law. If $X$ follows a Poisson law (resp. Binomial law), the resulting distribution is termed Poisson–Poisson (resp. Poisson–Binomial). L2-type goodness-of-fit statistics are constructed for the ‘general distribution’ of this kind, where $X$ may be an arbitrary discrete nonnegative random variable. The test statistics utilize a simple characterization involving the corresponding probability generating function, and are shown to be consistent. The proposed procedures are shown to perform satisfactorily in simulated data, while their application to accident data leads to positive conclusions regarding the modeling ability of this class of bivariate distributions.     

Locally best invariant test for outliers in a gamma type distribution

It is shown that the locally best invariant test for the existence of outliers for scale parameters of the gamma distribution is given by Bartholomew's test for exponentiality which is the ratio of the sum of squares of the data to the square of the sample mean. The optimality robustness, including null and nonnull robustness of the test is shown. A small simulation study to compare the power among the other eight competitive tests for testing exponentiality is performed. It is seen that the locally best invariant test is not always best but is reasonably good. It is slightly better than Cochran's test and suffers less from the limiting masking effect.     

A simple and asymptotically optimal test for the equality of k(≥2) exponential distributions based on type II censored samples

A test for the equality of two or more two-parameter exponential distributions is suggested. It is developed on an intuitive basis and is obtained by combining two independent tests by the Fisher method (1950, pp. 99-101). The test is simple for application and is optimal asymptotically in the sense of Bahadur efficiency (1960). A numerical example is discussed to illustrate its application in a real-world situation. The Monte Carlo simulation is used for calculating its power which is compared with that of the test suggested by Singh and Narayan (1983). The suggested test is found oftener more powerful.     

Subset selection in multiple linear regression in the presence of outlier and multicollinearity

Various subset selection methods are based on the least squares parameter estimation method. The performance of these methods is not reasonably well in the presence of outlier or multicollinearity or both. Few subset selection methods based on the $M$-estimator are available in the literature for outlier data. Very few subset selection methods account the problem of multicollinearity with ridge regression estimator.In this article, we develop a generalized version of $S_p$ statistic based on the jackknifed ridge $M$-estimator for subset selection in the presence of outlier and multicollinearity. We establish the equivalence of this statistic with the existing $C_p$, $S_p$ and $R_p$ statistics. The performance of the proposed method is illustrated through some numerical examples and the correct model selection ability is evaluated using simulation study.     

On robustness of usual confidence region under transformation misspecification

Robustness of confidence region for linear model parameters following a misspecified transformation of dependent variable is studied. It is shown that when error standard deviation is moderate to large the usual confidence region is robust against transformation misspecification. When error standard deviation is small the usual confidence region could be very conservative for structured models and slightly liberal for unstructured models. However, the conservativeness in structured case can be controlled if the transformation is selected with the help of data rather than prior information since this is the case when data is able to provide a very accurate estimate of transformation.     

Book Reviews

This article uses a Bayesian unit-root test in stochastic volatility models. The time series of interest is the volatility that is unobservable. The unit-root testing is based on the posterior odds ratio, which is approximated by Markov-chain Monte Carlo methods. Simulations show that the testing procedure is efficient for moderate sample size. The unit-root hypothesis is rejected in seven market indexes, and some evidence of nonstationarity is observed in the TWSI of Taiwan.     

SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.     

Category Distinguishability and Observer Agreement

It is common in the medical, biological, and social sciences for the categories into which an object is classified not to have a fully objective definition. Theoretically speaking the categories are therefore not completely distinguishable. The practical extent of their distinguishability can be measured when two expert observers classify the same sample of objects. It is shown, under reasonable assumptions, that the matrix of joint classification probabilities is quasi-symmetric, and that the symmetric matrix component is non-negative definite. The degree of distinguishability between two categories is defined and is used to give a measure of overall category distinguishability. It is argued that the kappa measure of observer agreement is unsatisfactory as a measure of overall category distinguishability.     

A statistical test for the hypothesis of Gaussian random function

A Gaussian random function is a functional version of the normal distribution. This paper proposes a statistical hypothesis test to test whether or not a random function is a Gaussian random function. A parameter that is equal to 0 under Gaussian random function is considered, and its unbiased estimator is given. The asymptotic distribution of the estimator is studied, which is used for constructing a test statistic and discussing its asymptotic power. The performance of the proposed test is investigated through several numerical simulations. An illustrative example is also presented.     

Sequential search using question-sets with bounded intersections

The paper is concerned with sequential search on a finite set: an unknown element of the finite set is to be found testing its subsets, and getting the information that the unknown element is or is not an element of the tested subset. The optimum of the strategy lengths is found under the condition that the intersection of any two different test-sets is bounded. This condition is generalized taking the intersection of any m different test-sets instead of two, and the general problem is also solved.     

On choosing between fixed and random block effects in some no-interaction models

The purpose of this article is to strengthen the understanding of the relationship between a fixed-blocks and random-blocks analysis in models that do not include interactions between treatments and blocks. Treating the block effects as random has been recommended in the literature for balanced incomplete block designs (BIBD) because it results in smaller variances of treatment contrasts. This reduction in variance is large if the block-to-block variation relative to the total variation is small. However, this analysis is also more complicated because it results in a subjective interpretation of results if the block variance component is non-positive. The probability of a non-positive variance component is large precisely in those situations where a random-blocks analysis is useful – that is, when the block-to-block variation, relative to the total variation, is small. In contrast, the analysis in which the block effects are fixed is computationally simpler and less subjective. The loss in power for some BIBD with a fixed effects analysis is trivial. In such cases, we recommend treating the block effects as fixed. For response surface experiments designed in blocks, however, an opposite recommendation is made. When block effects are fixed, the variance of the estimated response surface is not uniquely estimated, and in practice this variance is obtained by ignoring the block effect. It is argued that a more reasonable approach is to treat the block effects to be random than to ignore it.