A comparison of estimators for the proportion in the tail of a normal distribution

Three estimators of the proportion in a tail of the normal distribution are compared using the criteria of mean squared error and mean absolute error. The estimators that we compare are the maximum likelihood estimator, the minimum variance unbiased estimator, and an intuitive estimator that is frequently used in practice. The intuitive estimator is similar to the MLE but uses the usual unbiased estimator of σ² rather than the MLE of σ². We show that the intuitive estimator has low efficiency, and for this reason it is not recommended. For very smallp and for largep the MVUE has the highest efficiency. The MLE is best for moderate values ofp.     

Deviation of order p for estimators of the variance in first-order stochastic differential equation (SDE)

In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in L ^p norm (p≥; 1) between the variance and this estimator. The method of the proof consists in writing the L ^p norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for m-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov's formula.     

Robustness of weighted L p–depth and L p–median

Summary: L ^p –norm weighted depth functions are introduced and the local and global robustness of these weighted L ^p –depth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted L ^p –depth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted L ^p –depth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted L ^p –medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted L ^p depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted L ^p depth and medians, respectively.*The author thanks the referees for their very insightful and constructive comments and suggestions which led to corrections and substantial improvements. Supported in part by NSF Grants DMS-0071976 and DMS-0134628.     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

EMPIRICAL LIKELIHOOD-BASED KERNEL DENSITY ESTIMATION

This paper considers the estimation of a probability density function when extra distributional information is available (e.g. the mean of the distribution is known or the variance is a known function of the mean). The standard kernel method cannot exploit such extra information systematically as it uses an equal probability weight n^-1 at each data point. The paper suggests using empirical likelihood to choose the probability weights under constraints formulated from the extra distributional information. An empirical likelihood-based kernel density estimator is given by replacing n^-1 by the empirical likelihood weights, and has these advantages: it makes systematic use of the extra information, it is able to reflect the extra characteristics of the density function, and its variance is smaller than that of the standard kernel density estimator.     

Simultaneous robust estimation of location and scale parameters: A minimum-distance approach

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum-distance criterion function. The criterion function measures the squared distance between the pth power (p > 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, is asymptotically bivariate normal and for p > 0.3 has positive breakdown. If the scale parameter is known, when p = 0.9 the asymptotic variance (1.0436) of the location estimator for the normal model is smaller than the asymptotic variance of the Hodges-Lehmann (HL)estimator (1.0472). Efficiencies with respect to HL and maximum-likelihood estimators (MLE) are 1.0034 and 0.9582, respectively. Similarly, if the location parameter is known, when p = 0.97 the asymptotic variance (0.6158) of the scale estimator is minimum. The efficiency with respect to the MLE is 0.8119. We show that the estimator can tolerate more corrupted observations at oo than at – for p < 1, and vice versa for p > 1.     

Hill's estimator for the tail index of an ARMA model

This paper investigates Hill's estimator for the tail index of an ARMA model with i.i.d. residuals. Based on the estimated residuals, it is shown that Hill's estimator is asymptotically normal. This method can achieve a smaller asymptotic variance than applying Hill's estimator to the original data. These results are the same as those in Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) for an AR model. However, Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) imposed one more condition on the choice of sample fraction than the i.i.d. case. This condition is removed in this paper so that data-driven methods for choosing optimal sample fraction based on i.i.d. data can be applied to our case. As an auxiliary theorem, we establish the weak convergence of the tail empirical process of the estimated residuals, which may be of independent interest.     

Inference on finite population categorical response: nonparametric regression-based predictive approach

Suppose that a finite population consists of N distinct units. Associated with the ith unit is a polychotomous response vector, d _i , and a vector of auxiliary variable x _i . The values x _i ’s are known for the entire population but d _i ’s are known only for the units selected in the sample. The problem is to estimate the finite population proportion vector P. One of the fundamental questions in finite population sampling is how to make use of the complete auxiliary information effectively at the estimation stage. In this article a predictive estimator is proposed which incorporates the auxiliary information at the estimation stage by invoking a superpopulation model. However, the use of such estimators is often criticized since the working superpopulation model may not be correct. To protect the predictive estimator from the possible model failure, a nonparametric regression model is considered in the superpopulation. The asymptotic properties of the proposed estimator are derived and also a bootstrap-based hybrid re-sampling method for estimating the variance of the proposed estimator is developed. Results of a simulation study are reported on the performances of the predictive estimator and its re-sampling-based variance estimator from the model-based viewpoint. Finally, a data survey related to the opinions of 686 individuals on the cause of addiction is used for an empirical study to investigate the performance of the nonparametric predictive estimator from the design-based viewpoint.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

Simultaneous Equation Systems With Heteroscedasticity: Identification,Estimation, and Stock Price Elasticities

We give a set of identifying conditions for p-dimensional (p ≥ 2) simultaneous equation systems (SES) with heteroscedasticity in the framework of Gaussian quasi-maximum likelihood (QML). Our conditions rely on the presence of heteroscedasticity in the data rather than identifying restrictions traditionally employed in the literature. The QML estimator is shown to be consistent for the true parameter point and asymptotically normal. Monte Carlo experiments indicate that the QML estimator performs well in comparison to the generalized method of moments (GMM) estimator in finite samples, even when the conditional variance is mildly misspecified. We analyze the relationship between traded stock prices and volumes in the setting of SES. Based on a sample of the Russell 3000 stocks, our findings provide new evidence against perfectly elastic demand and supply schedules for equities.     

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.     

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern ⁻¹). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.     

Using a Truncated C p Statistic for Variable Selection in Multiple Linear Regression

In multiple linear regression analysis each lower-dimensional subspace L of a known linear subspace M of ? ⁿ corresponds to a non empty subset of the columns of the regressor matrix. For a fixed subspace L, the C _p statistic is an unbiased estimator of the mean square error if the projection of the response vector onto L is used to estimate the expected response. In this article, we consider two truncated versions of the C _p statistic that can also be used to estimate this mean square error. The C _p statistic and its truncated versions are compared in two example data sets, illustrating that use of the truncated versions may result in models different from those selected by standard C _p .     

Divergence based robust estimation of the tail index through an exponential regression model

The extreme value theory is very popular in applied sciences including finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction characterized by its tail index; there are three broad classes of tails—the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (Stat Sin 13:853–880, 2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.     

On tail index estimation based on multivariate data

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, that is, of which Pareto-like marginals share the same tail index. A multivariate central limit theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. We introduce the concept of (standard) heavy-tailed random vector of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.     

A study on the chain ratio-ratio-type exponential estimator for finite population variance

This paper considers the problem of estimating the population variance S²_y of the study variable y using the auxiliary information in sample surveys. We have suggested the (i) chain ratio-type estimator (on the lines of Kadilar and Cingi (2003)), (ii) chain ratio-ratio-type exponential estimator and their generalized version [on the lines of Singh and Pal (2015)] and studied their properties under large sample approximation. Conditions are obtained under which the proposed estimators are more efficient than usual unbiased estimator s²_y and Isaki (1893) ratio estimator. Improved version of the suggested class of estimators is also given along with its properties. An empirical study is carried out in support of the present study.     

Improved estimators of common variance of <Emphasis Type="Italic">p</Emphasis>-populations when Kurtosis is known

The pooled variance of p samples presumed to have been obtained from p populations having common variance σ², has invariably been adopted as the default estimator for σ². In this paper, alternative estimators of the common population variance are developed. These estimators are biased and have lower mean-squared error values than . The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (a ratio of mean-squared error values).     

Optimal choice of sample fraction in univariate financial tail index estimation

This study introduces a technique to estimate the Pareto distribution of the stock exchange index by using the maximum-likelihood Hill estimator. Recursive procedures based on the goodness-of-fit statistics are used to determine the optimal threshold fraction of extreme values to be included in tail estimation. These procedures are applied to three indices in the Malaysian stock market which included the consideration of a drastic economic event such as the Asian financial crisis. The empirical results evidenced alternating varying behavior of heavy-tailed distributions in the regimes for both upper and lower tails.     

Inference for probability of selection with dependently truncated data using a Cox model

A truncated sample consists of realizations of two variables L and T subject to the constraint L < T. One simple solution to dependently truncated data is to take L as a covariate of T in the Cox model. We aimed at studying the probability of selection, P(L < T), in this framework. We proposed the point estimator and derived its asymptotic distribution. Both truncated-only data and censored and truncated data were generated in the simulation study. The proposed point and variance estimators showed good performance in various simulated settings. The bone marrow transplant registry data were analyzed as the illustrative example.     

Moment-based tail index estimation

A general method of tail index estimation for heavy-tailed time series, based on examining the growth rate of the logged sample second moment of the data was proposed and studied in Meerschaert and Scheffler (1998. A simple robust estimator for the thickness of heavy tails. J. Statist. Plann. Inference 71, 19–34) as well as Politis (2002. A new approach on estimation of the tail index. C. R. Acad. Sci. Paris, Ser. I 335, 279–282). To improve upon the basic estimator, we introduce a scale-invariant estimator that is computed over subsets of the whole data set. We show that the new estimator, under some stronger conditions on the data, has a polynomial rate of consistency for the tail index. Empirical studies explore how the new method compares with the Hill, Pickands, and DEdH estimators.