Bootstrapping analogs of the one way MANOVA test

Abstract

Analogs of the classical one way MANOVA model have recently been suggested that do not assume that population covariance matrices are equal or that the error vector distribution is known. These tests are based on the sample mean and sample covariance matrix corresponding to each of the p populations. We show how to extend these tests using other measures of location such as the trimmed mean or coordinatewise median. These new bootstrap tests can have some outlier resistance, and can perform better than the tests based on the sample mean if the error vector distribution is heavy tailed.     

Hypothesis testing with Rao's quadratic entropy and its application to Dinosaur biodiversity

Entropy indices, such as Shannon entropy and Gini-Simpson index, have been used for analysing biological diversities. However, these entropy indices are based on abundance of the species only and they do not take differences between the species into consideration. Rao's quadratic entropy has found many applications in different fields including ecology. Further, the quadratic entropy (QE) index is the only ecological diversity index that reflects both the differences and abundances of the species. The problem of testing of hypothesis of the equality of QEs is formulated as a problem of comparing practical equivalence intervals. Simulation experiments are used to compare various equivalence intervals. Previously analyzed dinosaur data are used to illustrate the methods for determining biodiversity.     

Bootstrapping robust regression

The bootstrap principle is justified for. robust M-estimates in regression, (A short proof justifying bootstrapping the empirical process is also given.)     

Maximum entropy estimation of income share function from generalized Gini index

Following Sir Anthony and Atkinson who started thinking about the insensitivity of the Gini index to income shares of the lower and the upper income groups, a generalization of the classical Gini index was introduced by Kakwani, Donaldson, Weymark and Yitzhaki which is sensitive to both high and low incomes. In this paper, the maximum entropy method is used to estimate the underlying true income share function based on the limited information of the generalized Gini index about the income shares of a population's percentiles. The income share function is estimated through maximizing both the Shannon entropy and the second-order entropy. In the end, through parametric bootstrap and analyzing a real dataset, the results are compared with the estimator of the share function, which is obtained based on the total information. In contrast to the classic Gini index, the derived share function based on the generalized Gini index provides more accurate approximations for income shares of the lower and the upper percentiles.     

Bootstrapping regression models with BLUS residuals

To bootstrap a regression problem, pairs of response and explanatory variables or residuals can be resam‐pled, according to whether we believe that the explanatory variables are random or fixed. In the latter case, different residuals have been proposed in the literature, including the ordinary residuals (Efron 1979), standardized residuals (Bickel & Freedman 1983) and Studentized residuals (Weber 1984). Freedman (1981) has shown that the bootstrap from ordinary residuals is asymptotically valid when the number of cases increases and the number of variables is fixed. Bickel & Freedman (1983) have shown the asymptotic validity for ordinary residuals when the number of variables and the number of cases both increase, provided that the ratio of the two converges to zero at an appropriate rate. In this paper, the authors introduce the use of BLUS (Best Linear Unbiased with Scalar covariance matrix) residuals in bootstrapping regression models. The main advantage of the BLUS residuals, introduced in Theil (1965), is that they are uncorrelated. The main disadvantage is that only n —p residuals can be computed for a regression problem with n cases and p variables. The asymptotic results of Freedman (1981) and Bickel & Freedman (1983) for the ordinary (and standardized) residuals are generalized to the BLUS residuals. A small simulation study shows that even though only n — p residuals are available, in small samples bootstrapping BLUS residuals can be as good as, and sometimes better than, bootstrapping from standardized or Studentized residuals.     

谱检验的Wild Bootstrap方法

 本文考察了基于谱估计的拟合优度检验在判断时间序列鞅差性质方面的应用，提出了利用Wild Bootstrap来近似该类检验统计量渐近分布的方法，通过模拟试验揭示出新的方法改进了检验的水平并提高了其功效，最后比较了不同方法在分析实际金融数据时的不同效果。     

On entropy goodness-of-fit test based on integrated distribution function

In this study, we consider an entropy-type goodness-of-fit (GOF) test based on integrated distribution functions. We first construct the test for the simple vs. simple hypothesis and then extend it to the composite hypothesis case. It is shown that under regularity conditions, the null limiting distribution of the proposed test is a function of a Brownian bridge. A bootstrap method is also considered and is shown to be weakly consistent. A simulation study and real data analysis are conducted for illustration.     

Bootstrapping the latent roots of certain random matrices

Eigenvalues and functions of eigenvalues play an important role in the reduction of the dimensionality of data in multivariate analysis. However, even under the usual normal model context, the associated distributional theory is extremely complicated. In this paper, bootstrap algorithms for ap-proximating the distributions of functions of certain eigenvalues are given, with applications to confidence interval construction for population param-eters. Extensive Monte Carlo simulation results demonstrate the small sample performance of the bootstrap simultaneous confidence sets     

Bootstrapping in least absolute value regression: an application to hypothesis testing

A Monte Carlo simulation is used to study the performance of hypothesis tests for regression coefficients when least absolute value regression methods are used. In small samples, the results of the simulation suggest that using the bootstrap method to compute standard errors will provide improved test performance     

On Parametric Bootstrapping and Bayesian Prediction

Abstract.  We investigate bootstrapping and Bayesian methods for prediction. The observations and the variable being predicted are distributed according to different distributions. Many important problems can be formulated in this setting. This type of prediction problem appears when we deal with a Poisson process. Regression problems can also be formulated in this setting. First, we show that bootstrap predictive distributions are equivalent to Bayesian predictive distributions in the second-order expansion when some conditions are satisfied. Next, the performance of predictive distributions is compared with that of a plug-in distribution with an estimator. The accuracy of prediction is evaluated by using the Kullback–Leibler divergence. Finally, we give some examples.     

On weighted generalized entropy

In the literature of information theory, the concept of generalized entropy has been proposed and the length-based shift dependent information measure has been studied. In this paper, the concept of weighted generalized entropy has been introduced. The properties of weighted generalized residual entropy and weighted generalized past entropy are also discussed.     

Bootstrapping a time series model: some empirical results

The bootstrap is a methodology for estimating standard errors. The idea is to use a Monte Carlo simulation experiment based on a nonparametric estimate of the error distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order autoregressive model fitted by least squares and maximum likelihood estimation. Additionally, a comparison of the bootstrap and the conventional methodology is made. As it turns out, the conventional asymptotic formulae (both the least squares and maximum likelihood estimates) for estimating standard errors appear to overestimate the true standard errors. But there are two problems:i. The first two observations y₁ and y₂ have been fixed, and ii. The residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional maximum likelihood and bootstrap estimates of the standard errors appear to be performing quite well.     

Bootstrapping stochastic regression models under homoskedasticity: wild bootstrap vs. pairs bootstrap

We investigate by simulation how the wild bootstrap and pairs bootstrap perform in t and F tests of regression parameters in the stochastic regression model, where explanatory variables are stochastic and not given and there exists no heteroskedasticity. The wild bootstrap procedure due to Davidson and Flachaire [The wild bootstrap, tamed at last, Working paper, IER#1000, Queen's University, 2001] with restricted residuals works best but its dominance is not strong compared to the result of Flachaire [Bootstrapping heteroskedastic regression models: wild bootstrap vs. pairs bootstrap, Comput. Statist. Data Anal. 49 (2005), pp. 361–376] in the fixed regression model where explanatory variables are fixed and there exists heteroskedasticity.     

Bootstrapping the renewal spacings processes

We define the generalized bootstrapped version of the empirical and quantile renewal spacing processes. We show that the asymptotic theory of the renewal spacings processes holds for the bootstrap version.     

Bootstrap entropy test for general location-scale time series models with heteroscedasticity

This study considers a goodness-of-fit test for location-scale time series models with heteroscedasticity, including a broad class of generalized autoregressive conditional heteroscedastic-type models. In financial time series analysis, the correct identification of model innovations is crucial for further inferences in diverse applications such as risk management analysis. To implement a goodness-of-fit test, we employ the residual-based entropy test generated from the residual empirical process. Since this test often shows size distortions and is affected by parameter estimation, its bootstrap version is considered. It is shown that the bootstrap entropy test is weakly consistent, and thereby its usage is justified. A simulation study and data analysis are conducted by way of an illustration.     

Bootstrapping Malmquist Indices for Danish Seiners in the North Sea and Skagerrak

In connection with assessing how an ongoing development in fisheries management may change fishing activity, evaluation of Total Factor Productivity (TFP) change over a period, including efficiency, scale and technology changes, is an important tool. The Malmquist index, based on distance functions evaluated with Data Envelopment Analysis (DEA), is often employed to estimate TFP changes. DEA is generally gaining attention for evaluating efficiency and capacity in fisheries. One main criticism of DEA is that it does not have any statistical foundation, i.e. that it is not possible to make inference about DEA scores or related parameters. The bootstrap method for estimating confidence intervals of deterministic parameters can however be applied to estimate confidence intervals for DEA scores. This method is applied in the present paper for assessing TFP changes between 1987 and 1999 for the fleet of Danish seiners operating in the North Sea and the Skagerrak.     

Residual Renyi entropy of k-record values

ABSTRACT

In this article, the residual Renyi entropy (RRE) of k-record values arising from an absolutely continuous distribution is considered. A representation of RRE of k-records arising from an arbitrary distribution in terms of RRE of k-record values arising from uniform distribution is given. Some properties for RRE of k-records are also discussed.     

An efficient integrated nonparametric entropy estimator of serial dependence

ABSTRACT

We propose an efficient numerical integration-based nonparametric entropy estimator for serial dependence and show that the new entropy estimator has a smaller asymptotic variance than Hong and White’s (2005 Hong, Y., White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901.[Crossref], [Web of Science ®] , [Google Scholar]) sample average-based estimator. This delivers an asymptotically more efficient test for serial dependence. In particular, the uniform kernel gives the smallest asymptotic variance for the numerical integration-based entropy estimator over a class of positive kernel functions. Moreover, the naive bootstrap can be used to obtain accurate inferences for our test, whereas it is not applicable to Hong and White’s (2005 Hong, Y., White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901.[Crossref], [Web of Science ®] , [Google Scholar]) sample averaging approach. A simulation study confirms the merits of our approach.