Modelling and analysis of recall-based competing risks data

In this paper we consider the analysis of recall-based competing risks data. The chance of an individual recalling the exact time to event depends on the time of occurrence of the event and time of observation of the individual. In particular, it is assumed that the probability of recall depends on the time elapsed since the occurrence of an event. In this study we consider the likelihood-based inference for the analysis of recall-based competing risks data. The likelihood function is constructed by incorporating the information about the probability of recall. We consider the maximum likelihood estimation of parameters. Simulation studies are carried out to examine the performance of the estimators. The proposed estimation procedure is applied to a real life data set.     

The survival of newly founded firms: a case-study into varying-coefficient models

Summary.  The success of a newly founded firm depends on various initial risk factors or start-up conditions such as the market that the business is aiming for, the experience and the age of the founder, the preparation before the launch, the financial frame and the legal form of the firm. These risk factors determine the chance of survival for the venture. However, the effects of these risk factors may change with time. Some effects may vanish whereas others remain constant. We analyse the survival of 1123 newly founded firms in the state of Bavaria, Germany. Our focus is on the investigation of time variation in the effects of risk factors. Time variation is tackled within the framework of varying-coefficient models, where time smoothly modifies the effects of risk factors. An important issue in our analysis is the separation of risk factors which have time-varying effects from those which have time constant effects. We make use of the Akaike criterion to separate these two types of risk factor.     

Estimation of bivariate and marginal distributions with censored data

Summary. Consider a pair of random variables, both subject to random right censoring. New estimators for the bivariate and marginal distributions of these variables are proposed. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak convergence of the estimators proposed is obtained. A small simulation study suggests that the estimators of the marginal and bivariate distributions perform well relatively to respectively the Kaplan–Meier estimator for the marginal distribution and the estimators of Pruitt and van der Laan for the bivariate distribution. The use of the estimators in practice is illustrated by the analysis of a data set.     

Sufficient Conditions for the Existence of a Solution for the Log-Likelihood Equations in the Three-Parameter Gamma Distribution

The existence of maximum likelihood estimators for the three-parameter gamma distribution is still an open problem. It demands the solution of the system of the log-likelihood equations which can be solved only with the use of numerical methods. The aim of this article is to provide sufficient conditions for the existence of a solution for the system of the log-likelihood equations. The conditions are expressed in terms of the geometric, arithmetic, and harmonic mean. Also, the importance of the sign of the third central moment for the existence of a solution is revealed.     

A New Method for Statistical Multidimensional Unfolding

ABSTRACT

Consider the problem of estimating the positions of a set of targets in a multidimensional Euclidean space from distances reported by a number of observers when the observers do not know their own positions in the space. Each observer reports the distance from the observer to each target plus a random error. This statistical problem is the basic model for the various forms of what is called multidimensional unfolding in the psychometric literature. Multidimensional unfolding methodology as developed in the field of cognitive psychology is basically a statistical estimation problem where the data structure is a set of measures that are monotonic functions of Euclidean distances between a number of observers and targets in a multidimensional space. The new method presented in this article deals with estimating the target locations and the observer positions when the observations are functions of the squared distances between observers and targets observed with an additive random error in a two-dimensional space. The method provides robust estimates of the target locations in a multidimensional space for the parametric structure of the data generating model presented in the article. The method also yields estimates of the orientation of the coordinate system and the mean and variances of the observer locations. The mean and the variances are not estimated by standard unfolding methods which yield targets maps that are invariant to a rotation of the coordinate system. The data is transformed so that the nonlinearity due to the squared observer locations is removed. The sampling properties of the estimates are derived from the asymptotic variances of the additive errors of a maximum likelihood factor analysis of the sample covariance matrix of the transformed data augmented with bootstrapping. The robustness of the new method is tested using artificial data. The method is applied to a 2001 survey data set from Turkey to provide a real data example.     

带线性约束的多元线性回归模型参数估计

本文首先构造线性约束条件下的多元线性回归模型的样本似然函数,利用Lagrange法证明其合理性。其次,从似然函数的角度讨论线性约束条件对模型参数的影响,对由传统理论得出的参数估计作出贝叶斯与经验贝叶斯的改进。做贝叶斯改进时,将矩阵正态-Wishart分布作为模型参数和精度阵的联合共轭先验分布,结合构造的似然函数得出参数的后验分布,计算出参数的贝叶斯估计;做经验贝叶斯改进时,将样本分组,从方差的角度讨论由子样得出的参数估计对总样本的参数估计的影响,计算出经验贝叶斯估计。最后,利用Matlab软件生成的随机矩阵做模拟。结果表明,这两种改进后的参数估计均较由传统理论得出的参数估计更精确,拟合结果的误差比更小,可信度更高,在大数据的情况下,这种计算方法的速度更快。     

Stability of the roots of random algebraic polynomials

In this paper we consider the behavior of the roots of random algebraic polynomials. A code was developed which generates a sample of random algebraic polynomials, calculates the roots of each sample polynomial, and then calculates the averages of the roots. Finally, the roots of the deterministic algebraic polynomial whose coefficients are the averages of the sample coefficients are calculated. These data are then tabulated and graphically displayed. The relationship between the averages of the roots of the sample polynomials and the roots of the average polynomial is discussed.     

How far from identifiability? A systematic overview of the statistical matching problem in a non parametric framework

Statistical matching consists in estimating the joint characteristics of two variables observed in two distinct and independent sample surveys, respectively. In a parametric setup, ranges of estimates for non identifiable parameters are the only estimable items, unless restrictive assumptions on the probabilistic relationship between the non jointly observed variables are imposed. These ranges correspond to the uncertainty due to the absence of joint observations on the pair of variables of interest. The aim of this paper is to analyze the uncertainty in statistical matching in a non parametric setting. A measure of uncertainty is introduced, and its properties studied: this measure studies the “intrinsic” association between the pair of variables, which is constant and equal to 1/6 whatever the form of the marginal distribution functions of the two variables when knowledge on the pair of variables is the only one available in the two samples. This measure becomes useful in the context of the reduction of uncertainty due to further knowledge than data themselves, as in the case of structural zeros. In this case the proposed measure detects how the introduction of further knowledge shrinks the intrinsic uncertainty from 1/6 to smaller values, zero being the case of no uncertainty. Sampling properties of the uncertainty measure and of the bounds of the uncertainty intervals are also proved.     

Maximum likelihood estimation of variance components of heteroscedastic random anova model

the estimation of variance components of heteroscedastic random model is discussed in this paper. Maximum Likelihood (ML) is described for one-way heteroscedastic random models. The proportionality condition that cell variance is proportional to the cell sample size, is used to eliminate the efffect of heteroscedasticity. The algebraic expressions of the estimators are obtained for the model. It is seen that the algebraic expressions of the estimators depend mainly on the inverse of the variance-covariance matrix of the observation vector. So, the variance-covariance matrix is obtained and the formulae for the inversions are given. A Monte Carlo study is conducted. Five different variance patterns with different numbers of cells are considered in this study. For each variance pattern, 1000 Monte Carlo samples are drawn. Then the Monte Carlo biases and Monte Carlo MSE’s of the estimators of variance components are calculated. In respect of both bias and MSE, the Maximum Likelihood (ML) estimators of variance components are found to be sufficiently good.     

On describing multivariate skewed distributions: A directional approach

Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. These measures were designed for testing the hypothesis of distributional symmetry; their relevance for describing skewed distributions is less obvious. In this article, the authors consider the problem of characterizing the skewness of multivariate distributions. They define directional skewness as the skewness along a direction and analyze two parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of classes of distributions for particular applications. The authors use the concept of directional skewness twice in the context of Bayesian linear regression under skewed error: first in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.     

Maximum likelihood estimation of ARFIMA models with a Box-Cox transformation

In this paper we study the interaction between the estimation of the fractional differencing parameter d of ARFIMA models and the common practice of instantaneous transformation of the observed time series. At this aim, we first discuss the effect of a nonlinear transformation of the data on the identification of the process and on the estimate of d. Thus, we propose a joint estimation of the Box-Cox parameter and d by means of a modified normalized version of the Whittle likelihood. Then, the variance and covariance matrix of the parameters estimates is obtained. Finally, a Monte Carlo study is performed in order to check the behaviour of the proposed estimators in finite samples.The paper is the result of a joint research of the two authors. As far as it concerns this version of the work, A. DElia wrote Sects. 2, 3, 4, while D. Piccolo wrote Sects. 1, 5, 6.     

Change detection in the mean of a white Gaussian process by the backward standardized sum

A statistical method for detection of a change in the mean of a white Gaussian noise process is developed in this paper. The decision function of the method searches for the maximum of the backward standardized sum in a moving window to detect the change. Statistical properties of the decision function are derived to set the detection threshold. The derivation of the mean delay function and the optimal size of the moving window is also presented. The performance of the proposed method is compared, in terms of the mean delay for the detection, with that of the exponentially weighted moving average (EWMA). The mean delays of the cumulative sum control charts are also compared for benchmarking. The performance comparison is carried out by evaluating the average run length functions and by simulations. The results conclude that the mean detection delay of the proposed method is shorter than that of the standard EWMA for the same Type I error probability.     

On minimum distance estimation of location based on the kolmogorov statistic

The asymptotic distribution is derived for the minimum distance estimator of a location parameter based on the Kolmogorov goodness of fit statistic. The distribution is expressed in terms of the distribution of a functional of a Brownian bridge. An upper bound is obtained for the length of the confidence interval based on the Kolmogorov statistic. A simulation study with sample sizes 10 and 20 compares the length of the interval based on the Kolmogorov statistic to the length of the interval based on the maximum likelihood estimator. Another simulation shows the effect of model misspecification on the coverage probabilities of the interval based on the Kolmogorov statistic.     

Assessing the association between two spatial or temporal sequences

This paper deals with the codispersion coefficient for spatial and temporal series. We present some results and simulations concerning the codispersion coefficient in the context of spatial models. The results obtained are immediate consequences of the asymptotic normality of the sample codispersion coefficient and show certain limitations of the coefficient. New simulation studies provide information about the performance of the coefficient with respect to other coefficients of spatial association. The behavior of the codispersion coefficient under additively contaminated processes is also studied via Monte Carlo simulations. In the context of time series, explicit expressions for the asymptotic variance of the sample version of the coefficient are given for autoregressive and moving average processes. Resampling methods are used to compute the variance of the coefficient. A real data example is presented to explore how well the codispersion coefficient captures the comovement between two time series in practice.     

A bootstrap procedure in linear regression with nonstationary errors

In the context of linear regression with dependent and nonstationary errors, the classical moving-block bootstrap (MBB) fails to capture the nonstationarity of the errors. A new bootstrap procedure called the blocking external bootstrap (BEB) is proposed to overcome the problem. The consistency of the BEB in estimating the variance of the least-squares estimator is studied in the case of α-mixing and nonstationary sequence of errors. It is shown that the BEB only achieves partial correction if the block size is fixed. Complete consistency is achieved by the BEB when the block size is allowed to go to infinity. We also study the first-order consistency of the least squares estimator based on the BEB. A simulation study is carried out to assess the performance of the BEB versus the MBB in estimating the variance of the least-squares estimator. Finally, some open problems are discussed.     

能源消费结构统计特征研究

采用移动平均、HP滤波、VAR模型等方法对中国能源消费结构的循环及趋势特征进行研究,结果表明：化石能源消费对新能源消费影响通过GDP波动而发生作用,强化能源利用效率,维持GDP的稳定增长有助于中国能源消费结构优化,克服循环因素波动的影响。从强化提升新能源消费水平的角度看,煤炭／石油消费量的上升会拉动新能源／天然气消费量的增加,而新能源消费量上升会抑制煤炭／石油消费量的上升。因此,在中国能源消费结构上,新能源与煤炭／石油之间存在一种制衡机制,这也印证了中国产业政策取向和能源消费结构优化政策的合理性。基于此,针对性的提出了相关能源消费结构优化对策。     

Generalized monotonic functional mixed models with application to modelling normal tissue complications

Summary.  Normal tissue complications are a common side effect of radiation therapy. They are the consequence of the dose of radiation that is received by the normal tissue surrounding the site of the tumour. Within a specified organ each voxel receives a certain dose of radiation, leading to a distribution of doses over the organ. It is often not known what aspect of the dose distribution drives the presence and severity of the complications. A summary measure of the dose distribution can be obtained by integrating a weighting function of dose ( w ( d )) over the density of dose. For biological reasons the weight function should be monotonic. We propose a generalized monotonic functional mixed model to study the dose effect on a clinical outcome by estimating this weight function non-parametrically by using splines and subject to the monotonicity constraint, while allowing for overdispersion and correlation of multiple obervations within the same subject. We illustrate our method with data from a head and neck cancer study in which the irradiation of the parotid gland results in loss of saliva flow.     

A Superstructure of Process Capability Indices for Circular Specification Region

Circular specification regions can be seen in processes like hitting a target (in ballistics), drilling a hole (in manufacturing industries) and so on. However, only a few process capability indices are available in the literature to address the problem. Most of these indices, in turn, make some assumptions like equality of variance and independence of the two axes of the circular tolerance region. Since, in most of the cases, these assumptions are not practically viable, in the present article, we have proposed a few of the process capability indices which do not need the above assumptions to be valid. Also, we propose a superstructure which unifies all the proposed indices. Some properties of these indices have been studied including the threshold value and the relationship of the proportion of non-conformance with the member indices of the superstructure. These strengthen the practical utility of the superstructure. Distributional properties like expectations and variances of the member indices of the superstructure are also studied to have a better insight about the indices. A real life example has been discussed to carry out a comparative study of the performance of the existing as well as the newly developed indices.     

Statistical inference for the quintile share ratio

In recent years, the Quintile Share Ratio (or QSR) has become a very popular measure of inequality. In 2001, the European Council decided that income inequality in European Union member states should be described using two indicators: the Gini Index and the QSR. The QSR is generally defined as the ratio of the total income earned by the richest 20% of the population relative to that earned by the poorest 20%. Thus, it can be expressed using quantile shares, where a quantile share is the share of total income earned by all of the units up to a given quantile. The aim of this paper is to propose an improved methodology for the estimation and variance estimation of the QSR in a complex sampling design framework. Because the QSR is a non-linear function of interest, the estimation of its sampling variance requires advanced methodology. Moreover, a non-trivial obstacle in the estimation of quantile shares in finite populations is the non-unique definition of a quantile. Thus, two different conceptions of the quantile share are presented in the paper, leading us to two different estimators of the QSR. Regarding variance estimation, [Osier, 2006] and [Osier, 2009] proposed a variance estimator based on linearization techniques. However, his method involves Gaussian kernel smoothing of cumulative distribution functions. Our approach, also based on linearization, shows that no smoothing is needed. The construction of confidence intervals is discussed and a proposition is made to account for the skewness of the sampling distribution of the QSR. Finally, simulation studies are run to assess the relevance of our theoretical results.     

Estimating the Number of Block Boundaries from Diagonal Blockwise Matrices Without Penalization

In computational biology, numerous recent studies have been dedicated to the analysis of the chromatin structure within the cell by two‐dimensional segmentation methods. Motivated by this application, we consider the problem of retrieving the diagonal blocks in a matrix of observations. The theoretical properties of the least squares estimators of both the boundaries and the number of blocks are investigated. More precisely, the contribution of the paper is to establish the consistency of these estimators. A surprising consequence of our results is that, contrary to the one‐dimensional case, a penalty is not needed for retrieving the true number of diagonal blocks. Finally, the results are illustrated on synthetic data.