Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

RANKED SET SAMPLING FROM LOCATION-SCALE FAMILIES OF SYMMETRIC DISTRIBUTIONS

Statistical inference based on ranked set sampling has primarily been motivated by nonparametric problems. However, the sampling procedure can provide an improved estimator of the population mean when the population is partially known. In this article, we consider estimation of the population mean and variance for the location-scale families of distributions. We derive and compare different unbiased estimators of these parameters based on rindependent replications of a ranked set sample of size n.Large sample properties, along with asymptotic relative efficiencies, help identify which estimators are best suited for different location-scale distributions.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.     

Generalized estimator for the estimation of rare and clustered population variance in adaptive cluster sampling

Adaptive cluster sampling (ACS) is considered to be the most suitable sampling design for the estimation of rare, hidden, clustered and hard-to-reach population units. The main characteristic of this design is that it may select more meaningful samples and provide more efficient estimates for the field investigator as compare to the other conventional sampling designs. In this paper, we proposed a generalized estimator with a single auxiliary variable for the estimation of rare, hidden and highly clustered population variance under ACS design. The expressions of approximate bias and mean square error are derived and the efficiency comparisons have been made with other existing estimators. A numerical study is carried out on a real population of aquatic birds together with an artificial population generated by Poisson cluster process. Related results of numerical study show that the proposed generalized variance estimator is able to provide considerably better results over the competing estimators.     

Bias-calibrated estimation from sample surveys containing outliers

We discuss the problem of estimating finite population parameters on the basis of a sample containing representative outliers. We clarify the motivation for Chambers's bias-calibrated estimator of the population total and show that bias calibration is a key idea in constructing estimators of finite population parameters. We then link the problem of estimating the population total to distribution function or quantile estimation and explore a methodology based on the use of Chambers's estimator. We also propose methodology based on the use of robust estimates and a bias-calibrated form of the Chambers and Dunstan estimator of the population distribution function. This proposal leads to a bias-calibrated estimator of the population total which is an alternative to that of Chambers. We present a small simulation study to illustrate the utility of these estimators.     

The combined dynamically weighted modified maximum likelihood estimators of the location and scale parameters

In this study, two new types of estimators of the location and scale parameters are proposed having high efficiency and robustness; the dynamically weighted modified maximum likelihood (DWMML) and the combined dynamically weighted modified maximum likelihood (CDWMML) estimators. Three pairs of the DWMML and two pairs of the CDWMML estimators of the location and scale parameters are produced, namely, the DWMML1, the DWMML2 and the DWMML3, and the CDWMML1 and the CDWMML2 estimators, respectively. Based on the simulation results, the DWMML1 estimators of the location and scale parameters are almost fully efficient (under normality) and robust at the same time. The DWMML3 estimators are asymptotically fully efficient and more robust than the M-estimators. The DWMML2 estimators are a compromise between efficiency and robustness. The CDWMML1 and CDWMML2 estimators are jointly very efficient and robust. Particularly, the CDWMML1 and CDWMML2 estimators of the scale parameter are superior compared to the other estimators of the scale parameter.     

Shrinkage averaging estimation

We discuss the impact of tuning parameter selection uncertainty in the context of shrinkage estimation and propose a methodology to account for problems arising from this issue: Transferring established concepts from model averaging to shrinkage estimation yields the concept of shrinkage averaging estimation (SAE) which reflects the idea of using weighted combinations of shrinkage estimators with different tuning parameters to improve overall stability, predictive performance and standard errors of shrinkage estimators. Two distinct approaches for an appropriate weight choice, both of which are inspired by concepts from the recent literature of model averaging, are presented: The first approach relates to an optimal weight choice with regard to the predictive performance of the final weighted estimator and its implementation can be realized via quadratic programming. The second approach has a fairly different motivation and considers the construction of weights via a resampling experiment. Focusing on Ridge, Lasso and Random Lasso estimators, the properties of the proposed shrinkage averaging estimators resulting from these strategies are explored by means of Monte-Carlo studies and are compared to traditional approaches where the tuning parameter is simply selected via cross validation criteria. The results show that the proposed SAE methodology can improve an estimators’ overall performance and reveal and incorporate tuning parameter uncertainty. As an illustration, selected methods are applied to some recent data from a study on leadership behavior in life science companies.     

Loss functions in restricted parameter spaces and their Bayesian applications

ABSTRACT

Squared error loss remains the most commonly used loss function for constructing a Bayes estimator of the parameter of interest. However, it can lead to suboptimal solutions when a parameter is defined on a restricted space. It can also be an inappropriate choice in the context when an extreme overestimation and/or underestimation results in severe consequences and a more conservative estimator is preferred. We advocate a class of loss functions for parameters defined on restricted spaces which infinitely penalize boundary decisions like the squared error loss does on the real line. We also recall several properties of loss functions such as symmetry, convexity and invariance. We propose generalizations of the squared error loss function for parameters defined on the positive real line and on an interval. We provide explicit solutions for corresponding Bayes estimators and discuss multivariate extensions. Four well-known Bayesian estimation problems are used to demonstrate inferential benefits the novel Bayes estimators can provide in the context of restricted estimation.     

Rare and clustered population estimation using the adaptive cluster sampling with some robust measures

The use of robust measures helps to increase the precision of the estimators, especially for the estimation of extremely skewed distributions. In this article, a generalized ratio estimator is proposed by using some robust measures with single auxiliary variable under the adaptive cluster sampling (ACS) design. We have incorporated tri-mean (TM), mid-range (MR) and Hodges-Lehman (HL) of the auxiliary variable as robust measures together with some conventional measures. The expressions of bias and mean square error (MSE) of the proposed generalized ratio estimator are derived. Two types of numerical study have been conducted using artificial clustered population and real data application to examine the performance of the proposed estimator over the usual mean per unit estimator under simple random sampling (SRS). Related results of the simulation study show that the proposed estimators provide better estimation results on both real and artificial population over the competing estimators.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

Semiparametric inference for the recurrent events process by means of a single-index model

In this paper, we introduce new parametric and semiparametric regression techniques for a recurrent event process subject to random right censoring. We develop models for the cumulative mean function and provide asymptotically normal estimators. Our semiparametric model which relies on a single-index assumption can be seen as a dimension reduction technique that, contrary to a fully nonparametric approach, is not stroke by the curse of dimensionality when the number of covariates is high. We discuss data-driven techniques to choose the parameters involved in the estimation procedures and provide a simulation study to support our theoretical results.     

Mean square error comparison among variance estimators with known coefficient of variation

The improved estimators for the population parameters were considered by several statisticians under various conditions. Recently Laheetharan and Wijekoon (Improved estimation of the population parameters when some additional information is available. Stat Papers doi:, 2008) demonstrated a generalized procedure for obtaining optimal shrunken estimators, and derived such estimators for both population mean and variance when coefficient of variation is known. In this article the mean square errors of those estimators were compared, and a numerical illustration was done using the scaled mean square error loss as used by Kanefuji and Iwase (Stat Papers 39:377–388, 1998) to understand the efficiency of the estimators with increasing sample size.     

Sampling a finite population in the presence of trend and correlation: Estimation of total 305-day lactation production in cattle

This paper studies the estimation of a finite population total in the presence of trend. A practical problem of dairy science is to estimate a cow's total 305-day milk production given a number of test-day records. We analyze this problem as one of estimating the total of a discrete population when the population values are correlated and exhibit a trend over time. Linear prediction estimators that are BLUE for known covariance and trend function linear in unknown parameters were applied to the estimation of the milk yield total. An empirical study compares BLUE with the expansion estimator and the procedure currently used by the Canadian Record of Performance for Dairy Cattle.     

Estimation and prediction for a Burr type-III distribution with progressive censoring

We consider estimation of unknown parameters and reliability characteristics of a Burr type-III distribution under progressive censoring. Predictive estimates for censored observations and the associated prediction intervals are also obtained. We derive maximum-likelihood estimators of unknown quantities using the EM algorithm and then also obtain the observed Fisher information matrix. We provide various Bayes estimators for unknown parameters under the squared error loss function. Highest posterior density and asymptotic intervals are also constructed. We evaluate performance of proposed methods using simulations. Finally, an illustrative example is presented in support of the methods discussed.     

Shrinkage and pretest estimators for longitudinal data analysis under partially linear models

In this paper, we develop marginal analysis methods for longitudinal data under partially linear models. We employ the pretest and shrinkage estimation procedures to estimate the mean response parameters as well as the association parameters, which may be subject to certain restrictions. We provide the analytic expressions for the asymptotic biases and risks of the proposed estimators, and investigate their relative performance to the unrestricted semiparametric least-squares estimator (USLSE). We show that if the dimension of association parameters exceeds two, the risk of the shrinkage estimators is strictly less than that of the USLSE in most of the parameter space. On the other hand, the risk of the pretest estimator depends on the validity of the restrictions of association parameters. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the USLSE. A real data example is applied to illustrate the practical usefulness of the proposed estimation procedures.     

Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics

In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.     

Estimating the fundamental frequency of a periodic function

In this paper we consider the problem of estimation of the fundamental frequency of a periodic function, which has several applications in Speech Signal Processing. The problem was originally proposed by Hannan (1974) and later on Quinn and Thomson (1991) provided an estimation procedure of the unknown parameters. It is observed that the estimation procedure of Quinn and Thomson (1991) is quite involved numerically. In this paper we propose to use two simple estimators and it is observed that their performance are quite satisfactory. Asymptotic properties of the proposed estimators are obtained. The large sample properties of the estimators are compared theoretically. We present some simulation results to compare their small sample performance. One speech data is analyzed using this particular model.     

A consistent parameter estimation in the three-parameter lognormal distribution

In this work, we propose a consistent method of estimation for the parameters of the three-parameter lognormal distribution. We then discuss some properties of these estimators and show by means of a Monte Carlo simulation study that the proposed estimators perform better than some other prominent estimators in terms of bias and root mean squared error. Finally, we present two real-life examples to illustrate the method of estimation proposed.