Affine Equivariant Estimator Research Articles

Estimating a new stress–strength index for several exponential populations with a common location

In design and development of products in various industries, a key characteristic is stress–strength reliability. In this article, we consider estimation of a new stress–strength index for several exponential populations with a common location. We derive various estimators such as the maximum likelihood, the uniformly minimum variance unbiased (UMVU), and Bayes estimators. We additionally apply Brewster–Zidek technique for improving upon estimators based on UMVU or best affine equivariant estimators of scale parameters. We derive the asymptotic distribution of the ML estimator and prove that the Bayes estimators' limit under a suitable prior distribution is a generalized Bayes estimator. We then evaluate the risk performance of the obtained estimators in an extensive simulation study. Two applications are given on real data sets to illustrate the new methods. One example relates to the duration analysis and the other to a problem of comparing strengths of different fibres in jute industry.

Equivariant estimation following selection from two normal populations having common unknown variance

For estimating the mean of the selected normal population (the one corresponding to the larger sample mean), we prove some admissibility and minimaxity results for the estimators belonging to the class of linear, affine and permutation equivariant estimators under the criteria of scaled mean squared error and scaled absolute bias. We derive a sufficient condition for inadmissibility of any affine and permutation equivariant estimator under the scaled mean squared error criterion. Consequently, all linear, affine and permutation equivariant estimators (except the naive estimator, larger of the two sample means) and some estimators proposed in Hsieh [On estimating the mean of the selected population with unknown variance. Commun Stat-Theor Methods. 1981;10(18):1869–1878] become inadmissible under scaled mean squared error criterion. We further show that under the scaled mean squared error criterion, the naive estimator is minimax, admissible and generalized Bayes with respect to the Jeffreys' prior. We analyse a real data set for illustration purpose and provide a simulation study to compare the performances of various competing estimators.

Componentwise estimation of ordered scale parameters of two exponential distributions under a general class of loss function

In many real life situations, prior information about the parameters is available, such as the ordering of the parameters. Incorporating this prior information about the order restrictions on parameters leads to more efficient estimators. In the present communication, we investigate estimation of the ordered scale parameters of two shifted exponential distributions with unknown location parameters under a class of bowl-shaped loss functions. We have proved that the best affine equivariant estimator (BAEE) is inadmissible. Various non smooth and smooth estimators has been obtained which improve upon the BAEE. In particular we have derived the improved estimators for some well known loss functions. Finally numerical comparison is carried out to compare the risk performance of the proposed estimators.

Improved estimators for functions of scale parameters in mixture models

Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions.

Improved estimators of the entropy in scale mixture of exponential distributions

In the present communication, the problem of estimating entropy of a scale mixture of exponential distributions is considered under the squared error loss. Inadmissibility of the best affine equivariant estimator(BAEE) is established by deriving an improved estimator which is not smooth. Using the integral expression of risk difference (IERD) approach of Kubokawa (The Annals of Statistics 22 (1994) 290–299), classes of estimators are obtained which improve upon the BAEE. The boundary estimator of this class is the Brewster and Zidek-type estimator and this estimator is smooth. We have shown that the Brewster and Zidek-type estimator is a generalized Bayes estimator. As an application of these results, we have obtained improved estimators for the entropy of a multivariate Lomax distribution. Finally, percentage risk reduction of the improved estimators for the entropy of a multivariate Lomax distribution is plotted to compare the risk performance of the improved estimators.

Measuring Uncertainty Under Prior Information

There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: $(i)$ i.i.d. sampling, $(ii)$ record values, $(iii)$ Type-II censoring and $(iv)$ progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.

Inference on stress–strength reliability for exponential distributions with a common scale parameter

ABSTRACT This paper considers estimation of the stress–strength reliability when the stress and strength follow two-parameter exponential distributions having different location parameters and a common scale parameter. All parameters are assumed to be unknown. We derive the uniformly minimum variance unbiased estimator, Bayes estimators and an affine equivariant estimator of the stress–strength reliability. We propose confidence intervals of the stress–strength reliability based on the generalized variable approach and percentile bootstrap method. We also derive an approximate confidence interval and Bayesian intervals of the reliability parameter. Numerical comparisons among the proposed estimators are carried out using intensive simulations. Illustrative examples have been given using real data sets.

Quantile estimation for a progressively censored exponential distribution

In this paper, we consider the problem of estimating the quantile of a two-parameter exponential distribution with respect to an arbitrary strictly convex loss function under progressive type II censoring. Inadmissibility of the best affine equivariant (BAE) estimator is established through a conditional risk analysis. In particular we provide dominance results for quadratic, linex and absolute value loss functions. Further, a class of dominating estimators is derived using the IERD (integral expression of risk difference) approach of Kubokawa (1994). In sequel the generalized Bayes estimator is shown to improve the BAE estimator.

Minimax estimation of the common variance and precision of two normal populations with ordered restricted means

Consider two independent normal populations with a common variance and ordered means. For this model, we study the problem of estimating a common variance and a common precision with respect to a general class of scale invariant loss functions. A general minimaxity result is established for estimating the common variance. It is shown that the best affine equivariant estimator and the restricted maximum likelihood estimator are inadmissible. In this direction, we derive a Stein-type improved estimator. We further derive a smooth estimator which improves upon the best affine equivariant estimator. In particular, various scale invariant loss functions are considered and several improved estimators are presented. Furthermore, a simulation study is performed to find the performance of the improved estimators developed in this paper. Similar results are obtained for the problem of estimating a common precision for the stated model under a general class of scale invariant loss functions.

Improved estimation of a function of scale parameter of a doubly censored exponential distribution

In the present article, we have studied the estimation of the reciprocal of scale parameter , that is, hazard rate of a two parameter exponential distribution based on a doubly censored sample. This estimation problem has been investigated under a general class of bowl-shaped scale invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of scale equivariant estimator is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for three special loss functions. A simulation study is conducted to compare the risk performance of the proposed estimators. Finally, we analyze a real data set.

Estimating the common hazard rate of several exponential distributions

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.

Estimating a function of scale parameter of an exponential population with unknown location under general loss function

In the present study, we consider the problem of estimating a function of scale parameter $$\ln \sigma $$ under an arbitrary location invariant bowl-shaped loss function, when location parameter $$\mu $$ is unknown. Various improved estimators are proposed. Inadmissibility of the best affine equivariant estimator (BAEE) of $$\ln \sigma $$ is established by deriving a Stein-type estimator. This improved estimator is not smooth. We derive a smooth estimator improving upon the BAEE. Further, the integral expression of risk difference (IERD) approach of Kubokawa is used to derive a class of improved estimators. To illustrate these results, we consider two specific loss functions: squared error and linex loss functions, and derive various estimators improving upon the BAEE. Finally, a simulation study has been carried out to numerically compare the risk performance of the improved estimators.

Estimating the common hazard rate of two exponential distributions with ordered location parameters

ABSTRACTThis paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.

Estimating a linear parametric function of a doubly censored exponential distribution

ABSTRACTFor an arbitrary strictly convex loss function, we study the problem of estimating a linear parametric function is a known constant, when a doubly censored sample is available from a two-parameter exponential population. We establish the inadmissibility of the best affine equivariant (BAE) estimator by deriving an improved estimator. We provide various implications for quadratic and linex loss functions in detail. Improvements are obtained for the absolute value loss function as well. Further a new class of estimators improving upon the BAE estimator is derived using the Kubokawa method. This class is shown to include some benchmark estimators from the literature.

Depth‐weighted robust multivariate regression with application to sparse data

AbstractA robust method for multivariate regression is developed based on robust estimators of the joint location and scatter matrix of the explanatory and response variables using the notion of data depth. The multivariate regression estimator possesses desirable affine equivariance properties, achieves the best breakdown point of any affine equivariant estimator, and has an influence function which is bounded in both the response as well as the predictor variable. To increase the efficiency of this estimator, a re‐weighted estimator based on robust Mahalanobis distances of the residual vectors is proposed. In practice, the method is more stable than existing methods that are constructed using subsamples of the data. The resulting multivariate regression technique is computationally feasible, and turns out to perform better than several popular robust multivariate regression methods when applied to various simulated data as well as a real benchmark data set. When the data dimension is quite high compared to the sample size it is still possible to use meaningful notions of data depth along with the corresponding depth values to construct a robust estimator in a sparse setting. The Canadian Journal of Statistics 45: 164–184; 2017 © 2017 Statistical Society of Canada

English

Minimum vector variance (MVV) is one of the latest contributions in the study of multivariate robust estimators.MVV estimators possess three important properties of a good robust estimator, namely, high breakdown point, affine equivariance and computational efficiency.However, highly robust affine equivariant estimators with the best breakdown point commonly have to compensate with low statistical efficiency.In order to cater this drawback, a reweighted minimum vector variance (RMVV) which is capable of increasing the efficiency while retaining the highest breakdown point is proposed in this paper.A simulation study was conducted to investigate the asymptotic relative efficiency and finite-sample behavior of the estimators for several types of distributions. The numerical results revealed that the reweighed scheme is able to attain higher efficiency compared to MVV estimators.

Estimation of ordered scale parameters of two exponential distributions with a common guarantee time

This paper deals with the estimation of the scale parameters of two exponential distributions having a common location parameter. It is assumed that the failure rates are unknown, but an ordering among the failure rates is known a priori. We derive the UMVUE. The restricted MLE is shown to be better than the MLE. We also derive admissible classes of estimators of scale parameters. Improved estimators are derived in classes of scale and affine equivariant estimators. A Monte Carlo simulation study is then performed to compare new estimators to the existing estimators. The percentage risk improvements over the UMVUE are presented to show the performance of new estimators.

Estimating the Renyi entropy of several exponential populations

Suppose independent random samples are drawn from $k$ shifted exponential populations with a common location but unequal scale parameters. The problem of estimating the Renyi entropy is considered. The uniformly minimum variance unbiased estimator (UMVUE) is derived. Sufficient conditions for improvement over affine and scale equivariant estimators are obtained. As a consequence, improved estimators over the UMVUE and the maximum likelihood estimator (MLE) are obtained. Further, for the case $k=1$, an estimator that dominates the best affine equivariant estimator is derived. Cases when the location parameter is constrained are also investigated in detail.

Outlier detection for high dimensional data using the Comedian approach

The process of detection of outliers is an interesting and important aspect in the analysis of data, as it could impact the inference. There are various methods available in the literature for detection of outliers in multivariate data [V. Barnett and T. Lewis, Outliers in Statistical Data, John Wiley & Sons, Chichester, 1994] using the Mahalanobis distance measure. An attempt is made to propose an alternate method of outlier detection based on the comedian introduced by Falk [On MAD and Comedians, Ann. Inst. Statist. Math. 49 (1997), pp. 615–644]. The proposed method is computationally efficient with high breakdown value and low computation time. Further, important properties, namely, success rates (SR) and false detection rates (FDR) are studied and compared with some of the well-known outlier detection methods through a simulation study. The Comedian method has high SR and low FDR for all combination of parameters. On removal of the detected outliers or down weighing, the same, highly robust and approximately affine equivariant estimators of multivariate location and scatter can be obtained. Finally, the method is applied to well-known real data sets to evaluate its performance.