Robust Estimators for the Parameters of Multivariate Lognormal Distribution

Abstract

In this paper, we introduce a class of location and scale estimators for the p-variate lognormal distribution. These estimators are obtained by applying a log transform to the data, computing robust Fisher consistent estimators for the obtained Gaussian data and transforming those estimators for the lognormal using the relationship between the parameters of both distributions. We prove some of the properties of these estimators, such as Fisher consistency, robustness and asymptotic normality.     

An extension of Franklin's randomized response technique for multi-proportions

ABSTRACT

In this article, we introduce six estimators, three based on row averages and the remaining three on column averages of population proportions for trichotomous population when randomized response sampling with a normal randomizing distribution is used. The estimators have been obtained using the method of moments. All the proposed estimators are shown to be unbiased and their variances have been worked out. The percent relative efficiencies of the column total based estimators with respect to row total based estimators are investigated through empirical study.     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

Influence diagnostics in the capital asset pricing model under elliptical distributions

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.     

A new extended generalized Gompertz distribution with statistical properties and simulations

Abstract

Statistical distributions are very useful in describing and predicting real world phenomena. In many applied areas there is a clear need for the extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. The choice of the best-suited statistical distribution for modeling data is very important.

In this article, we proposed an extended generalized Gompertz (EGGo) family of EGGo. Certain statistical properties of EGGo family including distribution shapes, hazard function, skewness, limit behavior, moments and order statistics are discussed. The flexibility of this family is assessed by its application to real data sets and comparison with other competing distributions. The maximum likelihood equations for estimating the parameters based on real data are given. The performances of the estimators such as maximum likelihood estimators, least squares estimators, weighted least squares estimators, Cramer-von-Mises estimators, Anderson-Darling estimators and right tailed Anderson-Darling estimators are discussed. The likelihood ratio test is derived to illustrate that the EGGo distribution is better than other nested models in fitting data set or not. We use R software for simulation in order to perform applications and test the validity of this model.     

The LAD estimation of the change-point linear model with randomly censored data

Abstract

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT

Let {y_t } be a Poisson-like process with the mean μ _t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.     

On minimum Lp-distance estimation for inhomogeneous Poisson processes

ABSTRACT

We consider the problem of parameter estimation by the observations of the inhomogeneous Poisson processes. We suppose that the intensity function of these processes is a smooth function of the unknown parameter and as a method of estimation we take the minimum distance approach. We are interested by the behavior of estimators in non Hilbertian situation and we define the minimum distance estimation (MDE) with the help of the L^p metrics. We show that (under regularity conditions) the MDE is consistent and we describe its limit distribution.     

Moderate deviations for the moment estimators in Rayleigh distribution with two parameters

ABSTRACT

We study the moderate deviations of the moment estimators in Rayleigh distribution with two parameters. The moderate deviations are obtained by the delta method in large deviation principle.     

An almost sure central limit theorem for self-normalized partial sums of weakly dependent random variables

Abstract

We give here an almost sure central limit theorem for self-normalized partial sums of a strictly stationary φ-mixing sequences which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Our result substantially extend a result on the almost sure central limit theorem previously obtained by Huang and Pang (2010).     

Using logistic regression for semiparametric comparison of population means and variances

Abstract

We propose to compare population means and variances under a semiparametric density ratio model. The proposed method is easy to implement by employing logistic regression procedures in many statistical software, and it often works very well when data are not normal. In this paper, we construct semiparametric estimators of the differences of two population means and variances, and derive their asymptotic distributions. We prove that the proposed semiparametric estimators are asymptotically more efficient than the corresponding non parametric ones. In addition, a simulation study and the analysis of two real data sets are presented. Finally, a short discussion is provided.     

Partly linear models on Riemannian manifolds

In partly linear models, the dependence of the response y on (x ^T, t) is modeled through the relationship y=x ^T β+g(t)+?, where ? is independent of (x ^T, t). We are interested in developing an estimation procedure that allows us to combine the flexibility of the partly linear models, studied by several authors, but including some variables that belong to a non-Euclidean space. The motivating application of this paper deals with the explanation of the atmospheric SO₂ pollution incidents using these models when some of the predictive variables belong in a cylinder. In this paper, the estimators of β and g are constructed when the explanatory variables t take values on a Riemannian manifold and the asymptotic properties of the proposed estimators are obtained under suitable conditions. We illustrate the use of this estimation approach using an environmental data set and we explore the performance of the estimators through a simulation study.     

On the effect of long-range dependence on extreme value copula estimation with fixed marginals

ABSTRACT

We establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.     

Response-based multiple imputation method for minimizing the impact of covariate detection limit in logistic regression

Abstract

Presence of detection limit (DL) in covariates causes inflated bias and inaccurate mean squared error to the estimators of the regression parameters. This paper suggests a response-driven multiple imputation method to correct the deleterious impact introduced by the covariate DL in the estimators of the parameters of simple logistic regression model. The performance of the method has been thoroughly investigated, and found to outperform the existing competing methods. The proposed method is computationally simple and easily implementable by using three existing R libraries. The method is robust to the violation of distributional assumption for the covariate of interest.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

Improved estimation of quantiles of two normal populations with common mean and ordered variances

Abstract

Estimation of quantiles from two normal populations is considered under the assumption of common mean and ordered variances. Several new estimators have been proposed using certain estimators of the common mean, including the plug-in type restricted MLE. A sufficient condition for improving equivariant estimators is proved and as a result improved estimators are derived. The percentage of risk improvements for each of the improved estimators have been computed numerically, which are quite significant. All the improved estimators have been compared numerically using Monte-Carlo simulation method. Finally, recommendations have been made for the use of estimators in practice.     

Bayesian estimation of mean and square of mean of normal distribution using linex loss function

In this paper, the Bayes estimators for mean and square of mean ol a normal distribution with mean μ and vaiiance σ r2 (known), relative to LINEX loss function are obtained Comparisons in terms of risk functions and Bayes risks of those under LINEX loss and squared error loss functions with their respective alternative estimators viz, UMVUE and Bayes estimators relative to squared error loss function, are made. It is found that Bayes estimators relative to LINEX loss function dominate the alternative estimators m terms of risk function snd Bayes risk. It is also found that if t2 is unknown the Bayes estimators are still preferable over alternative estimators.     

Inference on the Common Variance of Correlated Normal Random Variables

ABSTRACT

Scale equivariant estimators of the common variance σ², of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ² which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ²: a Bayesian estimator for invariant prior as well as two non-Bayesian estimators. We then generalized these three estimators for the case of several variables with multiple unknown correlations. In addition, we developed a system of confidence intervals which produce the desired coverage probability while being efficient in terms of expected length.     

The Cornish-Fisher expansion for a class of statistics in first order autoregression

ABSTRACT

This article is concerned with the derivation and study of the Cornish-Fisher expansion for a wide class of estimators of the parameter in the first order autoregressive process. Second and third order Cornish-Fisher approximations to the quantile of the distribution of the corresponding asymptotically normal standardized statistic are stated explicitly and their accuracy is examined, both theoretically and numerically, by comparing them with the exact value of the quantile obtained by Monte Carlo simulation.     

Asymptotics in the β-model for networks with a differentially private degree sequence

Abstract

The β-model is a natural model for characterizing the degree heterogeneity that widely exists in the network data. The estimators of the model parameters in the differentially private β-model with the denoised process have been shown to be consistent and asymptotically normal. In this paper, we show that the moment estimators of the parameters based on the differentially private degree sequence without the denoised process is consistent and asymptotically normal.