Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Probability plots are often used to estimate the parameters of distributions. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, and the Gumbel distribution. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators totally, especially in small samples.     

Asymptotic properties of the MAMSE adaptive likelihood weights

The weighted likelihood is a generalization of the likelihood designed to borrow strength from similar populations while making minimal assumptions. If the weights are properly chosen, the maximum weighted likelihood estimate may perform better than the maximum likelihood estimate (MLE). In a previous article, the minimum averaged mean squared error (MAMSE) weights are proposed and simulations show that they allow to outperform the MLE in many cases. In this paper, we study the asymptotic properties of the MAMSE weights. In particular, we prove that the MAMSE-weighted mixture of empirical distribution functions converges uniformly to the target distribution and that the maximum weighted likelihood estimate is strongly consistent. A short simulation illustrates the use of bootstrap in this context.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.     

APPROXIMATIONS TO SOME TEST STATISTICS FOR PERMUTATION TESTS IN A COMPLETELY RANDOMIZED DESIGN1

Under a randomization model for a completely randomized design permutation tests are considered based on the usual F statistic and on a multi-response permutation procedure statistic. For the first statistic the first two moments are obtained so a comparision with the distribution under the normal theory model can be made. The second statistic is shown to converge in distribution to an infinite weighted sum of chi-squared variates, the weights being the limits of the eigenvalues of a matrix depending on the distance measure used and the order statistics of the observations.     

Weighted L1-estimates for the First-order Bifurcating Autoregressive Model

We developed robust estimators that minimize a weighted L₁ norm for the first-order bifurcating autoregressive model. When all of the weights are fixed, our estimate is an L₁ estimate that is robust against outlying points in the response space and more efficient than the least squares estimate for heavy-tailed error distributions. When the weights are random and depend on the points in the factor space, the weighted L₁ estimate is robust against outlying points in the factor space. Simulated and artificial examples are presented. The behavior of the proposed estimate is modeled through a Monte Carlo study.     

A class of local likelihood methods and near-parametric asymptotics

The local maximum likelihood estimate θ^ _t of a parameter in a statistical model f ( x , θ) is defined by maximizing a weighted version of the likelihood function which gives more weight to observations in the neighbourhood of t . The paper studies the sense in which f ( t , θ^ _t ) is closer to the true distribution g ( t ) than the usual estimate f ( t , θ^) is. Asymptotic results are presented for the case in which the model misspecification becomes vanishingly small as the sample size tends to ∞. In this setting, the relative entropy risk of the local method is better than that of maximum likelihood. The form of optimum weights for the local likelihood is obtained and illustrated for the normal distribution.     

Stable prediction in high-dimensional linear models

We propose a Random Splitting Model Averaging procedure, RSMA, to achieve stable predictions in high-dimensional linear models. The idea is to use split training data to construct and estimate candidate models and use test data to form a second-level data. The second-level data is used to estimate optimal weights for candidate models by quadratic optimization under non-negative constraints. This procedure has three appealing features: (1) RSMA avoids model overfitting, as a result, gives improved prediction accuracy. (2) By adaptively choosing optimal weights, we obtain more stable predictions via averaging over several candidate models. (3) Based on RSMA, a weighted importance index is proposed to rank the predictors to discriminate relevant predictors from irrelevant ones. Simulation studies and a real data analysis demonstrate that RSMA procedure has excellent predictive performance and the associated weighted importance index could well rank the predictors.     

Combined Exponentially Weighted Moving Average Charts for the Mean and Variance Based on the Predictive Distribution

This article develops combined exponentially weighted moving average (EWMA) charts for the mean and variance of a normal distribution. A Bayesian approach is used to incorporate parameter uncertainty. We first use a Bayesian predictive distribution to construct the control chart, and we then use a sampling theory approach to evaluate it under various hypothetical specifications for the data generation model. Simulations are used to compare the proposed charts for different values of both the weighing constant for the exponentially weighted moving averages and for the size of the calibration sample that is used to estimate the in-statistical-control process parameters. We also examine the separate performance of the EWMA chart for the variance.     

Four Correlation Coefficients with a Third Blocking Variable: Their Efficacy,Relative Efficiency,and Test Statistics

Abstract

The efficacy and the asymptotic relative efficiency (ARE) of a weighted sum of Kendall's taus, a weighted sum of Spearman's rhos, a weighted sum of Pearson's r's, and a weighted sum of z-transformation of the Fisher–Yates correlation coefficients, in the presence of a blocking variable, are discussed. The method of selecting the weighting constants that maximize the efficacy of these four correlation coefficients is proposed. The estimate, test statistics and confidence interval of the four correlation coefficients with weights are also developed. To compare the small-sample properties of the four tests, a simulation study is performed. The theoretical and simulated results all prefer the weighted sum of the Pearson correlation coefficients with the optimal weights, as well as the weighted sum of z-transformation of the Fisher–Yates correlation coefficients with the optimal weights.     

A weighted dispersion function for estimation in linear models

Robust estimates for the parameters in the general linear model are proposed which are based on weighted rank statistics. The method is based on the minimization of a dispersion function defined by a weighted Gini's mean difference. The asymptotic distribution of the estimate is derived with an asymptotic linearity result. An influence function is determined to measure how the weights can reduce the influence of high-leverage points. The weights can also be used to base the ranking on a restricted set of comparisons. This is illustrated in several examples with stratified samples, treatment vs control groups and ordered alternatives.     

Bayesian design for estimating the turning point of a quadratic regression

The problem of designing an experiment to estimate the point at which a quadratic regression is a maximum, or minimum. is studied. The efficiency of a design depends on the value of the unknown parameters and sequential design is, therefore, more efficient than non-sequential design. We use a Bayesian criterion which is a weighted trace of the inverse of the information matrix with the weights depending on a prior distribution. If design occurs sequentially the weights can be updated. Both sequential and non-sequential Bayesian designs are compared to non-Bayesian sequential designs. The comparison is both theoretical and by simulation.     

Adaptive Lasso Variable Selection for the Accelerated Failure Models

This article considers the adaptive lasso procedure for the accelerated failure time model with multiple covariates based on weighted least squares method, which uses Kaplan-Meier weights to account for censoring. The adaptive lasso method can complete the variable selection and model estimation simultaneously. Under some mild conditions, the estimator is shown to have sparse and oracle properties. We use Bayesian Information Criterion (BIC) for tuning parameter selection, and a bootstrap variance approach for standard error. Simulation studies and two real data examples are carried out to investigate the performance of the proposed method.     

Bootstrapping in a high dimensional but very low-sample size problem

This article is concerned with testing multiple hypotheses, one for each of a large number of small data sets. Such data are sometimes referred to as high-dimensional, low-sample size data. Our model assumes that each observation within a randomly selected small data set follows a mixture of C shifted and rescaled versions of an arbitrary density f. A novel kernel density estimation scheme, in conjunction with clustering methods, is applied to estimate f. Bayes information criterion and a new criterion weighted mean of within-cluster variances are used to estimate C, which is the number of mixture components or clusters. These results are applied to the multiple testing problem. The null sampling distribution of each test statistic is determined by f, and hence a bootstrap procedure that resamples from an estimate of f is used to approximate this null distribution.     

Computation of the asymptotic null distribution of goodness-of-fit tests for multi-state models

We develop an improved approximation to the asymptotic null distribution of the goodness-of-fit tests for panel observed multi-state Markov models (Aguirre-Hernandez and Farewell, Stat Med 21:1899–1911, 2002) and hidden Markov models (Titman and Sharples, Stat Med 27:2177–2195, 2008). By considering the joint distribution of the grouped observed transition counts and the maximum likelihood estimate of the parameter vector it is shown that the distribution can be expressed as a weighted sum of independent c²₁{\chi^2_1} random variables, where the weights are dependent on the true parameters. The performance of this approximation for finite sample sizes and where the weights are calculated using the maximum likelihood estimates of the parameters is considered through simulation. In the scenarios considered, the approximation performs well and is a substantial improvement over the simple χ ² approximation.     

Goodness‐of‐fit testing based on a weighted bootstrap: A fast large‐sample alternative to the parametric bootstrap

The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness‐of‐fit testing. The simplest way to carry out such goodness‐of‐fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data increase. An alternative resampling technique based on a fast weighted bootstrap is proposed in this paper, and is studied both theoretically and empirically. The outcome of this work is a generic and computationally efficient multiplier goodness‐of‐fit procedure that can be used as a large‐sample alternative to the parametric bootstrap. In order to approximately determine how large the sample size needs to be for the parametric and weighted bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments are carried out in dimension one, two and three, and for models containing up to nine parameters. The computational gains resulting from the use of the proposed multiplier goodness‐of‐fit procedure are illustrated on trivariate financial data. A by‐product of this work is a fast large‐sample goodness‐of‐fit procedure for the bivariate and trivariate t distribution whose degrees of freedom are fixed. The Canadian Journal of Statistics 40: 480–500; 2012 © 2012 Statistical Society of Canada     

Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

A Bayesian approach to zero-inflated data in extremes

Abstract

The generalized extreme value (GEV) distribution is known as the limiting result for the modeling of maxima blocks of size n, which is used in the modeling of extreme events. However, it is possible for the data to present an excessive number of zeros when dealing with extreme data, making it difficult to analyze and estimate these events by using the usual GEV distribution. The Zero-Inflated Distribution (ZID) is widely known in literature for modeling data with inflated zeros, where the inflator parameter w is inserted. The present work aims to create a new approach to analyze zero-inflated extreme values, that will be applied in data of monthly maximum precipitation, that can occur during months where there was no precipitation, being these computed as zero. An inference was made on the Bayesian paradigm, and the parameter estimation was made by numerical approximations of the posterior distribution using Markov Chain Monte Carlo (MCMC) methods. Time series of some cities in the northeastern region of Brazil were analyzed, some of them with predominance of non-rainy months. The results of these applications showed the need to use this approach to obtain more accurate and with better adjustment measures results when compared to the standard distribution of extreme value analysis.     

Cluster-based multivariate outlier identification and re-weighted regression in linear models

A cluster methodology, motivated by a robust similarity matrix is proposed for identifying likely multivariate outlier structure and to estimate weighted least-square (WLS) regression parameters in linear models. The proposed method is an agglomeration of procedures that begins from clustering the n-observations through a test of ‘no-outlier hypothesis’ (TONH) to a weighted least-square regression estimation. The cluster phase partition the n-observations into h-set called main cluster and a minor cluster of size n?h. A robust distance emerge from the main cluster upon which a test of no outlier hypothesis’ is conducted. An initial WLS regression estimation is computed from the robust distance obtained from the main cluster. Until convergence, a re-weighted least-squares (RLS) regression estimate is updated with weights based on the normalized residuals. The proposed procedure blends an agglomerative hierarchical cluster analysis of a complete linkage through the TONH to the Re-weighted regression estimation phase. Hence, we propose to call it cluster-based re-weighted regression (CBRR). The CBRR is compared with three existing procedures using two data sets known to exhibit masking and swamping. The performance of CBRR is further examined through simulation experiment. The results obtained from the data set illustration and the Monte Carlo study shows that the CBRR is effective in detecting multivariate outliers where other methods are susceptible to it. The CBRR does not require enormous computation and is substantially not susceptible to masking and swamping.