Robustness of three power transformation estimation procedures

Maximum likelihood, goodness-of-fit, and symmetric percentile estimators of the power transformation parameterp, are considered. The comparative robustness of each estimation procedure is evaluated when the transformed data can be made symmetric, but may not necessarily be normal. Seven types of symmetric distributions are considered as well as four contaminated normal distributions over a range of six p values for samples of size 25, 50, and 100. The results indicate that the maximum likelihood estimator was slightly better than the goodness-of-fit estimator, but both were greatly superior to the percentile estimator. In general, the procedures were robust to distributional symmetric departures from normality, but increasing kurtosis caused appreciable increases in variation for estimated p values. The variability of p was found to decrease more than exponentially with decreases in the underlying normal distribution coefficient of variation. The standard likelihood ratio confidence interval procedure was found not to be generally useful.     

Symmetric maximum kernel likelihood estimation

We introduce an estimator for the population mean based on maximizing likelihoods formed from a symmetric kernel density estimate. Due to these origins, we have dubbed the estimator the symmetric maximum kernel likelihood estimate (smkle). A speedy computational method to compute the smkle based on binning is implemented in a simulation study which shows that the smkle at an optimal bandwidth is decidedly superior in terms of efficiency to the sample mean and other measures of location for heavy-tailed symmetric distributions. An empirical rule and a computational method to estimate this optimal bandwidth are developed and used to construct bootstrap confidence intervals for the population mean. We show that the intervals have approximately nominal coverage and have significantly smaller average width than the corresponding intervals for other measures of location.     

Robust joint estimation of location and scale parameters in ranked set samples

A robust estimator is developed for the location and scale parameters of a location-scale family. The estimator is defined as the minimizer of a minimum distance function that measures the distance between the ranked set sample empirical cumulative distribution function and a possibly misspecified target model. We show that the estimator is asymptotically normal, robust, and has high efficiency with respect to its competitors in literature. It is also shown that the location estimator is consistent within the class of all symmetric distributions whereas the scale estimator is Fisher consistent at the true target model. The paper also considers an optimal allocation procedure that does not introduce any bias due to judgment error classification. It is shown that this allocation procedure is equivalent to Neyman allocation. A numerical efficiency comparison is provided.     

L Ranked Set Sampling: A Generalization Procedure for Robust Visual Sampling

In this article, a robust ranked set sampling (LRSS) scheme for estimating population mean is introduced. The proposed method is a generalization for many types of ranked set sampling that introduced in the literature for estimating the population mean. It is shown that the LRSS method gives unbiased estimator for the population mean with minimum variance providing that the underlying distribution is symmetric. However, for skewed distributions a weighted mean is given, where the optimal weights is computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo comparisons for detecting outliers are made with the traditional simple random sample and the ranked set sampling for some distributions. The results indicate that the LRSS estimator is superior alternative to the existing methods.     

A Note on the Use of Some Functions of Spacings in the Estimation of Common Scale Parameter of Several Symmetric Distributions

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the common scale parameter of several symmetric distributions using some functions of spacings of all observations taken from individual samples. We also proved a sufficient condition for the non negativity of the common scale estimator obtained by the above method. Furthermore, we obtained necessary and sufficient conditions for the derived estimators to be constant multiple of the sum of first and last spacings of the pooled sample.     

Adaptive estimation of the center of symmetry based on the empirical characteristic function

A procedure based on the empirical characteristic function is proposed for the estimation of the center of symmetric distributions. The method is an adaptive modification of the procedure proposed by Koutrouvelis (1985). The asymptotic behavior of the resulting estimator is investigated and finite sample comparisons are made with the previous nonadaptive estimator and an adaptive trimmed mean proposed by Hogg (1974).     

Optimum linear unbiased estimation of the scale parameter by absolute values of order statistics in the double exponential and the double weibull distributions

The first two moments and product moments of absolute values of order statistics are obtained for the double exponential and the double Weibull distributions. In both of the distributions an optimum linear unbiased estimator of the scale parameter, by absolute values of the order statistics, is obtained from complete and censored samples of size n=3(1)10. It is found that the new estimator is generally more efficient than the best linear unbiased estimator (BLUE) of the scale parameter by order statistcs in both of the distributions.     

Normal tests for unit roots based on instrumental variable estimators

For estimating unit roots of autoregressive processes, we introduce a new instrumental variable (IV) method which discounts large values of regressors corresponding to the unit roots. Based on the IV estimator, we propose new unit root tests whose limiting null distributions are standard normal. Observation at time t is adjusted for mean recursively by the sample mean of observations up to the time t. The powers of the proposed tests are better than those of the Dickey–Fuller tests and are comparable to those of the tests based on the weighted symmetric estimator, which are known to have the best power against stationary alternatives.     

Adaptive Estimation Using Weighted Least Squares

This paper proposes an adaptive estimator that is more precise than the ordinary least squares estimator if the distribution of random errors is skewed or has long tails. The adaptive estimates are computed using a weighted least squares approach with weights based on the lengths of the tails of the distribution of residuals. Smaller weights are assigned to those observations that have residuals in the tails of long-tailed distributions and larger weights are assigned to observations having residuals in the tails of short-tailed distributions. Monte Carlo methods are used to compare the performance of the proposed estimator and the performance of the ordinary least squares estimator. The estimates that were studied in this simulation include the difference between the means of two populations, the mean of a symmetric distribution, and the slope of a regression line. The adaptive estimators are shown to have lower mean squared errors than those for the ordinary least squares estimators for short-tailed, long-tailed, and skewed distributions, provided the sample size is at least 20. The ordinary least squares estimator has slightly lower mean squared error for normally distributed errors. The adaptive estimator is recommended for general use for studies having sample sizes of at least 20 observations unless the random errors are known to be normally distributed.     

Nonnegative Unbiased Estimation of Scale Parameters and Associated Quantiles Based on a Ranked Set Sample

In this article, we are interested in estimating the scale parameter in location and scale families. It is well known that the best linear unbiased estimator (BLUE) of scale parameter based on a simple random sample (SRS) is nonnegative. However, the BLUE of scale parameter based on a ranked set sample (RSS) can assume negative values. We suggest various modifications of BLUE of scale parameter based on RSS so that the resulting estimators are unbiased as well as nonnegative. Their performances in terms of relative efficiencies are compared and some recommendations are made for normal, logistic, double exponential, two-parameter exponential and Weibull distributions. We also briefly discuss an application of the proposed nonnegative BLUE of scale parameter for quantile estimation for the above populations.     

A Note on the Hodges–Lehmann Estimator and the Logistic Distribution

ABSTRACT

It is well known that the Hodges–Lehmann estimator is asymptotically efficient for the location parameter of the logistic distribution. In this article we give a simple and direct proof that this property also characterizes the logistic between all the symmetric location distributions under mild conditions. Using pseudolikelihood, we also show how to find from the Hodges–Lehmann estimator an asymptotically efficient estimator of the scale parameter of the logistic distribution.     

Optimum linear unbiased estimation of scale parameter by absolute values of order statistics in symmetric distributions

A new estimator of the scale parameter by the optimum linear combination of absolute values of order statistics in symmetric location-scale families with known location parameter (without loss of generality assumed to be zero) from complete and Type II censored samples is introduced and is termed as optimum unbiased absolute estimator of the scale parameter. The new estimator of the scale parameter is compared with the corresponding best linear unbiased estimator (BLUE) in the rectangular and normal distributions. Generally it is found that the new estimator is more efficient than the BLUE.     

Improving the best linear unbiased estimator for the scale parameter of symmetric distributions by using the absolute value of ranked set samples

Ranked set sampling is a cost efficient sampling technique when actually measuring sampling units is difficult but ranking them is relatively easy. For a family of symmetric location-scale distributions with known location parameter, we consider a best linear unbiased estimator for the scale parameter. Instead of using original ranked set samples, we propose to use the absolute deviations of the ranked set samples from the location parameter. We demonstrate that this new estimator has smaller variance than the best linear unbiased estimator using original ranked set samples. Optimal allocation in the absolute value of ranked set samples is also discussed for the estimation of the scale parameter when the location parameter is known. Finally, we perform some sensitivity analyses for this new estimator when the location parameter is unknown but estimated using ranked set samples and when the ranking of sampling units is imperfect.     

A Robust and Almost Fully Efficient M-Estimator

Simultaneous robust estimates of location and scale parameters are derived from a class of M-estimating equations. A coefficient p ( p > 0), which plays a role similar to that of a tuning constant in the theory of M-estimation, determines the estimating equations. These estimating equations may be obtained as the gradient of a strictly convex criterion function. This article shows that the estimators are uniquely defined, asymptotically bi-variate normal and have positive breakdown for some choices of p . When p = 0.12 and p = 0.3, the estimators are almost fully efficient for normal and exponential distributions: efficiencies with respect to the maximum likelihood estimators are 1.00 and 0.99, respectively. It is shown that the location estimator for known scale has the maximum breakdown point 0.5 independent of p , when the target model is symmetric. Also it is shown that the scale estimator has a positive breakdown point which depends on the choice of p . A simulation study finds that the proposed location estimator has smaller variance than the Hodges–Lehmann estimator, Huber's minimax and bisquare M-estimators.     

Adaptive m estimation of symmetric distribution location

An adaptive M esitmation procedure for using a random sample to estimate the location parameter of an unknown symmetric distribution is developed. The procedure may be applied to samples from distributions with tail lenghts at least as heavy as normal distribution tails. Simulation studies demonstrate the potential of the new estimator for producing good location estimates.     

On the efficiencies of some common quick estimators

The quick estimators of location and scale have broad applications and are widely used. For a variety of symmetric populations we obtain the quantiles and the weights for which the asymptotic variances of the quick estimators are minimum. These optimal quick estimators are then used to obtain the asymptotic relative efficiencies of the commonly used estimators such as trimean. gastwirth. median, midrange. and interquartile range with respect to the optimal quick estimators in order to determine a choice among them and to check whether they are unacceptably poor. In the process it is seen that the interquartile range is the optimal quick estimator of scale for Cauchy populations; but the interdecile range is in general preferable. Also the optimal estimator of the location for the logistic distribution puts weights 0.3 on each of the two quartiles and 0.4 on the median. It is shown that for the symmetric distributions, such as the beta and Tukey- lambda with [d] > 0, which have finite support and short tails, i.e. the tail exponents (Parzen, 1979) satisfy [d] < 1, the midrange and the range are the optimal quick estimators of location and scale respectively if [d] < 1/2. The class of such distributions Include the distributions with high discontinuous tails, e.g. Tukey-lambda with [d] > 1, as well as some distributions with p.d.f.'s going to zero at the ends of the finite support, such as Tukey-lambda with 1/2 < [d] < 1. As a byproduct an interesting tail correspondence between beta and Tukey-lambda distributions is seen.     

Robust extreme ranked set sampling

In this paper, a robust extreme ranked set sampling (RERSS) procedure for estimating the population mean is introduced. It is shown that the proposed method gives an unbiased estimator with smaller variance, provided the underlying distribution is symmetric. However, for asymmetric distributions a weighted mean is given, where the optimal weights are computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo simulations are used to demonstrate the performance of the RERSS estimator relative to the simple random sample (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) estimators. The results indicate that the proposed estimator is more efficient than the estimators based on the traditional sampling methods.     

Strong Consistency of the Maximum Likelihood Estimator for Finite Mixtures of Location–Scale Distributions When Penalty is Imposed on the Ratios of the Scale Parameters

Abstract.  In finite mixtures of location–scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyse the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves.     

Improved inference for the shape-scale family of distributions under type-II censoring

The presence of a nuisance parameter may often perturb the quality of the likelihood-based inference for a parameter of interest under small to moderate sample sizes. The article proposes a maximal scale invariant transformation for likelihood-based inference for the shape in a shape-scale family to circumvent the effect of the nuisance scale parameter. The transformation can be used under complete or type-II censored samples. Simulation-based performance evaluation of the proposed estimator for the popular Weibull, Gamma and Generalized exponential distribution exhibits markedly improved performance in all types of likelihood-based inference for the shape under complete and type-II censored samples. The simulation study leads to a linear relation between the bias of the classical maximum likelihood estimator (MLE) and the transformation-based MLE for the popular Weibull and Gamma distributions. The linearity is exploited to suggest an almost unbiased estimator of the shape parameter for these distributions. Allied estimation of scale is also discussed.     

Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and their superior efficiency over the traditional normal-based control charts when applied to processes with moderate- or heavy-tailed underlying distributions, such as the double exponential or the Cauchy distributions.