A goodness of fit test for copulas based on Rosenblatt's transformation

A goodness of fit test for copulas based on Rosenblatt's transformation is investigated. This test performs well if the marginal distribution functions are known and are used in the test statistic. If the marginal distribution functions are unknown and are replaced by their empirical estimates, then the test's properties change significantly. This is shown in detail by simulation for special cases. A bootstrap version of the test is suggested and it is shown by simulation that it performs well. An empirical application of this test to daily returns of German assets reveals that a Gaussian copula is unsuitable to describe their dependence structure. A t_ν-copula with low degrees of freedom such as ν=4 or 5 fits the data in some cases.     

A quantile estimation for massive data with generalized Pareto distribution

This paper proposes a new method of estimating extreme quantiles of heavy-tailed distributions for massive data. The method utilizes the Peak Over Threshold (POT) method with generalized Pareto distribution (GPD) that is commonly used to estimate extreme quantiles and the parameter estimation of GPD using the empirical distribution function (EDF) and nonlinear least squares (NLS). We first estimate the parameters of GPD using EDF and NLS and then, estimate multiple high quantiles for massive data based on observations over a certain threshold value using the conventional POT. The simulation results demonstrate that our parameter estimation method has a smaller Mean square error (MSE) than other common methods when the shape parameter of GPD is at least 0. The estimated quantiles also show the best performance in terms of root MSE (RMSE) and absolute relative bias (ARB) for heavy-tailed distributions.     

Likelihood inference for generalized Pareto distribution

A new methodological approach that enables the use of the maximum likelihood method in the Generalized Pareto Distribution is presented. Thus several models for the same data can be compared under Akaike and Bayesian information criteria. The view is based on a detailed theoretical study of the Generalized Pareto Distribution submodels with compact support.     

Model based bootstrap methods for interval censored data

The performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data is investigated. It is shown that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for the current status model. A model based smoothed bootstrap procedure is proposed and proved to be consistent. In fact, a general framework for proving the consistency of any model based bootstrap scheme in the current status model is established. In addition, simulation studies are conducted to illustrate the (in)-consistency of different bootstrap methods in mixed case interval censoring. The conclusions in the interval censoring model would extend more generally to estimators in regression models that exhibit non-standard rates of convergence.     

A new approach of goodness-of-fit testing for exponentiated laws applied to the generalized Rayleigh distribution

Goodness-of-fit statistics are considered which are appropriate for generalized families of distributions, resulting from exponentiation. The tests employ a variation of the data determined by the cumulative distribution function of the corresponding non-generalized distribution. The resulting test, which makes use of the Mellin transform of the transformed data, is shown to be consistent. Simulation results for the case of the generalized Rayleigh distribution show that the proposed test compares well with standard methods based on the empirical distribution function.     

Maximum likelihood-based recursive least-squares estimation for multivariable systems using the data filtering technique

For multivariable equation-error systems with an autoregressive moving average noise, this paper applies the decomposition technique to transform a multivariable model into several identification sub-models based on the number of the system outputs, and derives a data filtering and maximum likelihood-based recursive least-squares algorithm to reduce the computation complexity and improve the parameter estimation accuracy. A multivariable recursive generalised extended least-squares method and a filtering-based recursive extended least-squares method are presented to show the effectiveness of the proposed algorithm. The simulation results indicate that the proposed method is effective and can produce more accurate parameter estimates than the compared methods.     

A goodness of fit test for the Pareto distribution in the presence of Type II censoring, based on the cumulative hazard function

A goodness of fit test for the Pareto distribution, when the observations are subjected to Type II right censoring is proposed. The test statistic involves transformations of the original data and is based on the nonparametric Nelson-Aalen estimator of the cumulative hazard function. By Monte Carlo simulation, the empirical distribution of the test statistic is obtained and the power of the test is investigated for some alternative distributions. The power is compared with adaptations for Type II censored data of the Crámer-von Mises and Anderson-Darling tests, and a test based on Kullback-Leibler information. For some alternative distributions with monotone decreasing hazard function, the proposed test has higher power. The methodology is illustrated by reanalyzing two published data sets.     

A comparative study of tests for the difference of two Poisson means

We investigate different test procedures for testing the difference of two Poisson means. Asymptotic tests, tests based on an approximate p-value method, and a likelihood ratio test are considered. Size and power performance of these tests are studied by means of Monte Carlo simulation under different settings. If one wants to control the actual significance level at or below the pre-chosen nominal level, tests based on approximate p-value method are the desirable candidates. If one allows tests whose actual significance levels may occasionally exceed the pre-chosen nominal level by an acceptable margin, asymptotic tests based on an unbiased estimate and constrained maximum likelihood estimate are reasonable alternatives. We illustrate these testing procedures with a breast cancer example.     

Approximating the distribution of goodness of fit tests for discrete data

The Berry-Esseen theorem is used to study the error of the normal approximation to the distribution of several goodness of fit tests for discrete data. A practical bound is proposed and confirmed by simulations.     

Maximum likelihood estimation for generalized semi-markov processes

Parametric statistical inference for generalized semi-Markov processes is addressed. This class of processes encompasses a large number of real-world discrete-event stochastic systems. Because of its properties (e.g., consistency, asymptotic normality, etc.), maximum likelihood estimation is considered here. Under reasonable conditions on the process, we show that a maximum likelihood estimator exists, and that it converges to the true parameter at ratet ^–1/2, wheret is the length of the observation period. A related estimator, which is typically easier to compute, is also introduced. We show that the use of this estimator results in no loss of statistical efficiency. It is also shown that the estimation problem does decouple into separate subproblems when the process' transition probabilities and event distributions depend on different parameters.     

A comparison of block and semi-parametric bootstrap methods for variance estimation in spatial statistics

Efron (1979) introduced the bootstrap method for independent data but it cannot be easily applied to spatial data because of their dependency. For spatial data that are correlated in terms of their locations in the underlying space the moving block bootstrap method is usually used to estimate the precision measures of the estimators. The precision of the moving block bootstrap estimators is related to the block size which is difficult to select. In the moving block bootstrap method also the variance estimator is underestimated. In this paper, first the semi-parametric bootstrap is used to estimate the precision measures of estimators in spatial data analysis. In the semi-parametric bootstrap method, we use the estimation of the spatial correlation structure. Then, we compare the semi-parametric bootstrap with a moving block bootstrap for variance estimation of estimators in a simulation study. Finally, we use the semi-parametric bootstrap to analyze the coal-ash data.     

The random projection method in goodness of fit for functional data

The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. Simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process are presented.     

Parameter estimation of the generalized gamma distribution

This article focuses on the parameter estimation of the generalized gamma distribution. Because of many difficulties described in the literature to estimate the parameters, we propose here a new estimation method. The algorithm associated to this heuristic method is implemented in Splus. We validate the resulting routine on the particular cases of the generalized gamma distribution.     

A bootstrap method of distribution mixture proportion determination

A simulation-based methodology for distribution mixture proportion determination is presented. A Monte Carlo study using Gaussian population data and cosmic-ray-experiments data proved the reliability of the methods proposed.     

Analysis of covariance and correlation structures

The main purpose of this paper is to consider different approaches in analyzing covariance or correlation structures with parameters subject to general nonlinear constraints. A new estimation method, the two-stage constrained maximum likelihood procedure, is developed. While in large samples this procedure is shown to have similar statistical properties to the classical constrained maximum likelihood approach, computationally it requires less computer time and storage to implement. It is shown that either the sample covariance or the correlation matrix can be used in analyzing the correlation structure. Examples are given to illustrate the theory, and technical details are presented in the Appendix.     

Parameter estimation for generalized Dirichlet distributions from the sample estimates of the first and the second moments of random variables

Generalized Dirichlet distributions have a more flexible covariance structure than Dirichlet distributions, and the computation for the moments of a generalized Dirichlet distribution is still tractable. For situations under which Dirichlet distributions are inappropriate for data analysis, generalized Dirichlet distributions will generally be an applicable alternative. When the expected values and the covariance matrix of random variables can be estimated from available data, this study introduces ways to estimate the parameters of a generalized Dirichlet distribution for analyzing compositional data. Under the assumption that the sample mean of every variable must be considered for parameter estimation, we present methods for choosing the statistics from a sample covariance matrix to construct a generalized Dirichlet distribution. Some rules for removing inappropriate statistics from a sample covariance matrix to speed up the estimation process are also established. An example for Taiwan’s car market is introduced to demonstrate the applicability of the parameter estimation methods.     

A maximum likelihood approximation method for Dirichlet's parameter estimation

Dirichlet distributions are natural choices to analyse data described by frequencies or proportions since they are the simplest known distributions for such data apart from the uniform distribution. They are often used whenever proportions are involved, for example, in text-mining, image analysis, biology or as a prior of a multinomial distribution in Bayesian statistics. As the Dirichlet distribution belongs to the exponential family, its parameters can be easily inferred by maximum likelihood. Parameter estimation is usually performed with the Newton-Raphson algorithm after an initialisation step using either the moments or Ronning's methods. However this initialisation can result in parameters that lie outside the admissible region. A simple and very efficient alternative based on a maximum likelihood approximation is presented. The advantages of the presented method compared to two other methods are demonstrated on synthetic data sets as well as for a practical biological problem: the clustering of protein sequences based on their amino acid compositions.     

基于卡方拟合优度检验的序列等概性测试组

等概性是随机性中的一个重要特性,频数检验是最常见的等概性检验方法.分析了传统频数检验的片面性,利用卡方拟合优度检验提出了全面进行等概性检验的测试组,分别是码元频数检验、等长子序列频数检验和检验结果数据频数检验.使用测试组对国际上常用的随机样本的二进制展开序列进行检验,实验结果表明,该测试组具有较强的等概性检验能力.     

Maximum likelihood based identification methods for rational models

ABSTRACT

In a rational model, some terms of the information vector are correlated with the noise, which makes the traditional least squares based iterative algorithms biased. In order to overcome this shortcoming, this paper develops two recursive algorithms for estimating the rational model parameters. These two algorithms, based on the maximum likelihood principle, have three integrated key features: (1) to establish two unbiased maximum likelihood recursive algorithms, (2) to develop a maximum likelihood recursive least squares (ML-RLS) algorithm to decrease the computational efforts, (3) to update the parameter estimates by the ML-RLS based particle swarm optimisation (ML-RLS-PSO) algorithm when the noise-to-output ratio is large. Comparative studies demonstrate that (1) the ML-RLS algorithm is only valid for rational models when the noise-to-output ratio is small, (2) the ML-RLS-PSO algorithm is effective for rational models with random noise-to-output ratio, but at the cost of heavy computational efforts. Furthermore, the simulations provide cases for potential expansion and applications of the proposed algorithms.     

On the use of the bootstrap for estimating functions with functional data

The bootstrap methodology for functional data and functional estimation target is considered. A Monte Carlo study analyzing the performance of the bootstrap confidence bands (obtained with different resampling methods) of several functional estimators is presented. Some of these estimators (e.g., the trimmed functional mean) rely on the use of depth notions for functional data and do not have received yet much attention in the literature. A real data example in cardiology research is also analyzed. In a more theoretical aspect, a brief discussion is given providing some insights on the asymptotic validity of the bootstrap methodology when functional data, as well as a functional parameter, are involved.