On estimating the mean of the selected normal population under the LINEX loss function

Following Parsian and Farsipour (1999), we consider the problem of estimating the mean of the selected normal population, from two normal populations with unknown means and common known variance, under the LINEX loss function. Some admissibility results for a subclass of equivariant estimators are derived and a sufficient condition for the inadmissibility of an arbitrary equivariant estimator is provided. As a consequence, several of the estimators proposed by Parsian and Farsipour (1999) are shown to be inadmissible and better estimators are obtained. Received January 2001/Revised May 2002     

Sequential point estimation of normal mean under LINEX loss function

A sequential point estimation of the mean of a normal distribution is considered under LINEX loss function. The regret of sequential procedures are obtained. Furthermore, it is shown that a sequential procedure with the sample mean as an estimate is asymptotically inadmissible. An accerelated stopping time is also considered. Received: December 1999     

Sequential estimation of normal mean under asymmetric loss function with a shrinkage stopping rule

The problem of estimating a normal mean with unknown variance is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, a sequential stopping rule and two sequential estimators of the mean are proposed and second-order asymptotic expansions of their risk functions are derived. It is demonstrated that the sample mean becomes asymptotically inadmissible, being dominated by a shrinkage-type estimator. Also a shrinkage factor is incorporated in the stopping rule and similar inadmissibility results are established. Received September 1997     

Sequential estimation of a linear function of normal means under asymmetric loss function

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions. Received: August 1999     

Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

Prediction of <Emphasis Type="Italic">k</Emphasis>-records from a general class of distributions under balanced type loss functions

We study the problem of predicting future k-records based on k-record data for a large class of distributions, which includes several well-known distributions such as: Exponential, Weibull (one parameter), Pareto, Burr type XII, among others. With both Bayesian and non-Bayesian approaches being investigated here, we pay more attention to Bayesian predictors under balanced type loss functions as introduced by Jafari Jozani et al. (Stat Probab Lett 76:773–780, 2006a). The results are presented under the balanced versions of some well-known loss functions, namely squared error loss, Varian’s linear-exponential loss and absolute error loss or L ₁ loss functions. Some of the previous results in the literatures such as Ahmadi et al. (Commun Stat Theory Methods 34:795–805, 2005), and Raqab et al. (Statistics 41:105–108, 2007) can be achieved as special cases of our results. Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged by J. Ahmadi. M. J. Jozani’s research supported partially by a grant of Statistical Research and Training Center. é. Marchand’s research supported by NSERC of Canada. A. Parsian’s research supported by a grant of the Research Council of the University of Tehran.     

Properties of optimal forecasts under asymmetric loss and nonlinearity

Evaluation of forecast optimality in economics and finance has almost exclusively been conducted under the assumption of mean squared error loss. Under this loss function optimal forecasts should be unbiased and forecast errors serially uncorrelated at the single period horizon with increasing variance as the forecast horizon grows. Using analytical results we show that standard properties of optimal forecasts can be invalid under asymmetric loss and nonlinear data generating processes and thus may be very misleading as a benchmark for an optimal forecast. We establish instead that a suitable transformation of the forecast error—known as the generalized forecast error—possesses an equivalent set of properties. The paper also provides empirical examples to illustrate the significance in practice of asymmetric loss and nonlinearities and discusses the effect of parameter estimation error on optimal forecasts.     

Estimation of the location parameter under LINEX loss function: multivariate case

The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, ${\beta\in\Re^r}The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, b ? ?^r{\beta\in\Re^r} may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing is known about the mean vector θ).     

Credibility estimators with dependence structure over risks and time under balanced loss function

In this paper, we study the Bühlmann credibility model with constant interest rate and equal dependence structure over risks and time under balanced loss function. By means of orthogonal projection, the inhomogeneous and homogeneous credibility premium estimators are derived, which extend those for the existing models to slightly more general versions. Finally, we investigate the estimation of the structure parameters and present a numerical example to show the effectiveness of the inhomogeneous estimator.     

Estimation of order-restricted means of two normal populations under the LINEX loss function

In this paper, the estimation of order-restricted means of two normal distributions is studied under the LINEX loss function, when the variances are unknown and possibly unequal. Under certain sufficient conditions to be described in this paper, the proposed plug-in estimators uniformly perform better than the existing unrestricted maximum likelihood estimators. Further, the restricted maximum likelihood estimators are compared with the unrestricted maximum likelihood estimators under the Pitman nearness criterion. A simulation study is conducted and it is shown that our proposed plug-in estimators perform better than the unrestricted maximum likelihood estimators. An illustrative example of real data analysis is also given to compare the estimators.     

Testing forecast accuracy of expectiles and quantiles with the extremal consistent loss functions

Forecast evaluations aim to choose an accurate forecast for making decisions by using loss functions. However, different loss functions often generate different ranking results for forecasts, which complicates the task of comparisons. In this paper, we develop statistical tests for comparing performances of forecasting expectiles and quantiles of a random variable under consistent loss functions. The test statistics are constructed with the extremal consistent loss functions of Ehm et al. (2016). The null hypothesis of the tests is that a benchmark forecast at least performs equally well as a competing one under all extremal consistent loss functions. It can be shown that if such a null holds, the benchmark will also perform at least equally well as the competitor under all consistent loss functions. Thus under the null, when different consistent loss functions are used, the result that the competitor does not outperform the benchmark will not be altered. We establish asymptotic properties of the proposed test statistics and propose to use the re-centered bootstrap to construct their empirical distributions. Through simulations, we show that the proposed test statistics perform reasonably well. We then apply the proposed method to evaluations of several different forecast methods.     

Estimation of the mean in ranked set sampling with non responses

The estimation of the population mean when ranked set sampling [rss] is used for selecting the sample and non responses [nr] are present, is studied. The nr stratum is sub sampled using simple random sampling with replacement. Two strategies are analyzed. One of them is based on the selection of a sub sample from the nr in each cycle. The other uses sub samples selected among the nr in each rank.  The accuracy of the proposed estimators is characterized by the corresponding expected variances. Simulations and real life data are used for analyzing the behavior of them. Acknowledgements: This paper was developed partially during the visit of the author to Université des Antilles et Gouyane. The author gratefully acknowledges the helpful suggestions of the referees and thanks the support of DAAD for visiting Humboldt University where a version of the paper version was made.     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

Estimation of the scale matrix of a multivariate t-model under entropy loss

This paper deals with the estimation of the scale matrix of a multivariatet-model with unknown location vector and scale matrix to improve upon the usual estimators based on the sample sum of product matrix. The well-known results of the estimation of the scale matrix of the multivariate normal model under the assumption of entropy loss function have been generalized to that of a multivariatet-model. The paper is based on the first author’s unpublished Ph.D. dissertation ‘Estimation of the Scale Matrix of a Multivariate T-model’, University of Western Ontario, Canada. Present address: School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia.     

Asymmetric loss in the Greenbook and the Survey of Professional Forecasters

This paper examines the forecast rationality of the Greenbook and the Survey of Professional Forecasters (SPF) under asymmetric loss functions, using the method proposed by Elliott, Komunjer, and Timmermann (2005) with a rolling window strategy. Over rolling periods, the degree and direction of the asymmetry in forecast loss functions are time-varying. While rationality under symmetric loss is often rejected, forecast rationality under asymmetric loss fails to be rejected over nearly all rolling periods. Besides, real output growth is consistently under-predicted in the 1990s, and the inflation rate is consistently over-predicted in the 1980s and 1990s. In general, inflation forecasts, especially for long horizons, exhibit greater levels of loss asymmetry in magnitude and frequency. The loss asymmetry of real output growth forecasts is more pronounced when the last revised vintage data are used than when the real-time vintage is used. All of these results hold for both the Greenbook and SPF forecasts. The results are also similar with the use of different sets of instrumental variables for estimating the asymmetric loss and testing for forecast rationality.     

On the completeness of certain classes of two stage sampling strategies

Summary In this paper the problem of estimation of the population mean, for a finite two statge population, under the assumptions of non-informativeness of labels of distinguishable units has been studied. Completeness of certain classes of strategies, for estimating the population mean, under a two stage random permutation model, has been established.     

Estimation of the characteristic roots of the scale matrix

The simultaneous estimation of the characteristic roots of the scale matrix of the multivariatet-model is considered. The improved estimation strategies are developed in the light of a quadratic loss function. It is demonstrated analytically and numerically that the class of proposed estimators outperforms the class of usual estimators in the sense of having smaller risk.     

基于非首都功能疏解的保定养老产业发展研究

《京津冀协同发展纲要》明确提出要疏解北京非首都功能,这为保定养老产业发展提供了难得的机遇。依据对北京老人的养老需求及对来保定养老的调研,并根据美国学者E.S.Lee的推拉理论,提出在保定养老产业发展中,推进与北京的同城化、聚集社会各方力量、增强对北京老人吸引力、满足个性化养老需求等建议。     

引导京津冀地区人口合理布局的思路与对策研究

近年来,北京城市人口规模持续高位攀升,造成了交通拥堵、就业紧张、住房困难、水资源紧缺、环境污染等城市问题。从城市发展战略出发,采取引导和疏散人口的有效措施和政策,是首都实现可持续发展的重要环节。本文基于"京津冀一体化"国家战略,探索通过产业发展和功能提升增强首都周边区域对人口的吸引力,引导京津冀区域人口合理分布和有序流动,从而促进京津冀地区人口与经济社会、资源环境全面协调可持续发展。