Second-order Efficiency Research Articles

Purely Sequential Point Estimation of a Function of the Mean in an Exponential Distribution

A purely sequential minimum risk point estimation methodology with an associated stopping time N is designed to come up with a unified strategy in an exponential distribution (Expo). We work under a weighted squared error loss due to the estimator $$g(\overline{X}_{N})$$ of $$g(\beta ),$$ a function of the unknown mean parameter $$\beta (>0)$$ , plus linear cost of sampling from an Expo $$(\beta )$$ population. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, $$\rightarrow 0$$ ) results are laid out including (1) the first- and second-order efficiency properties, (2) the risk efficiency property, and (3) the second-order regret expansion under suitable conditions on g(.).

Estimation of location parameter within pre-specified error bound with second-order efficient two-stage procedure

This paper develops a general approach for constructing a confidence interval for a parameter of interest with a specified confidence coefficient and a specified width. This is done assuming known a positive lower bound for the unknown nuisance parameter and independence of suitable statistics. Under mild conditions, we develop a modified two-stage procedure which enjoys attractive optimality properties including a second-order efficiency property and asymptotic consistency property. We extend this work for finding a confidence interval for the location parameter of the inverse Gaussian distribution. As an illustration, we developed a modified mean absolute deviation-based procedure in the supplementary section for finding a fixed-width confidence interval for the normal mean.

High-confidence nonparametric fixed-width uncertainty intervals and applications to projected high-dimensional data and common mean estimation

Nonparametric two-stage procedures to construct fixed-width confidence intervals are studied to quantify uncertainty. It is shown that the validity of the random central limit theorem (RCLT) accompanied by a consistent and asymptotically unbiased estimator of the asymptotic variance already guarantees consistency and first-order as well as second-order efficiency of the two-stage procedures. This holds under the common asymptotics where the length of the confidence interval tends toward 0 as well as under the novel proposed high-confidence asymptotics where the confidence level tends toward 1. The approach is motivated by and applicable to data analysis from distributed big data with nonnegligible costs of data queries. The following problems are discussed: Fixed-width intervals for the mean, for a projection when observing high-dimensional data, and for the common mean when using nonlinear common mean estimators under order constraints. The procedures are investigated by simulations and illustrated by a real data analysis.

Multistage Estimation of the Rayleigh Distribution Variance

In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results.

A broader class of modified two-stage minimum risk point estimation procedures for a normal mean

In this paper, we design an innovative and general class of modified two-stage sampling schemes to enhance double sampling and modified double sampling procedures. Under the squared error loss plus linear cost of sampling, we revisit the classic problem of minimum risk point estimation (MRPE) for an unknown normal mean μ () when the population variance () also remains unknown. With stopping variables constructed based on an arbitrary general estimator Wm for σ, which satisfies a set of certain conditions, our procedures are proved to enjoy asymptotic first- and second-order efficiency as well as asymptotic first-order risk efficiency. For illustrative purposes, we further investigate specific modified two-stage MRPE procedures, where we substitute appropriate multiples of sample standard deviation, Gini’s mean difference (GMD), and mean absolute deviation (MAD) in the place of Wm , respectively. Extensive simulation studies are utilized to validate our theoretical findings. A real-life data set of weight change from female anorexic patients is then analyzed to demonstrate the practical applicability of these modified two-stage MRPE procedures. Comparing them in the case where there exist suspect outliers in the pilot sample, we are empirically confident that the GMD- and MAD-based procedures appear more robust than the sample-standard-deviation-based procedures.

Second-order approximations of a robust sequential procedure in Bayes sequential estimation

The problem of sequential estimation of the mean with quadratic loss and fixed cost per observation is considered within the Bayesian framework. A robust sequential procedure, not depending on the distributions of outcome variables and the prior, in the Bayes sequential estimation is investigated. In the present article, the second-order approximations to the expected sample size and the Bayes risk of the robust sequential procedure are obtained for the arbitrary distributions of outcome variables and the prior. The second-order efficiency is further discussed in an example.

On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) strategies for a normal mean and optimality

We develop a generalized class of purely sequential sampling strategies associated with both fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) problems for the unknown mean $$\mu $$ of a normally distributed population having its variance $$\sigma ^{2}$$ also unknown. Under this newly proposed general class of associated estimation strategies, we develop a variety of asymptotic first-order and asymptotic second-order properties such as asymptotic consistency, first-order efficiency, first-order risk efficiency, second-order efficiency, and second-order regret analysis. Next, we proceed to locate an optimal strategy within our newly built large class of possibilities. Such optimality is defined as having been associated with the minimal second-order asymptotic variance of a stopping time within the general class of proposed strategies. We follow through by exploring both the FWCI and MRPE problems with the help of data analysis from simulations.

Minimum risk point estimation (MRPE) of the mean in an exponential distribution under powered absolute error loss (PAEL) due to estimation plus cost of sampling

We begin with a review of asymptotic properties of a purely sequential minimum risk point estimation (MRPE) methodology for an unknown mean in a one-parameter exponential distribution under a class of generalized loss functions. This class of powered absolute error loss (PAEL) includes both squared error loss (SEL) and absolute error loss (AEL) plus cost of sampling. We prove the asymptotic second-order efficiency property and asymptotic first-order risk efficiency property associated with the purely sequential MRPE problem. For operational convenience, we then move to implement an accelerated sequential MRPE methodology and prove the analogous asymptotic second-order efficiency property and asymptotic first-order risk efficiency property. We follow up with extensive data analysis from simulations and provide illustrations using cancer data.

A general theory of purely sequential minimum risk point estimation (MRPE) of a function of the mean in a normal distribution

A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of a function of μ, with plus linear cost of sampling from a population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, ) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size.

Robust estimation for longitudinal data based upon minimum Hellinger distance

ABSTRACTGeneralized linear mixed models have been widely used in the analysis of correlated data in a lot of research areas. The linear mixed model with normal errors has been a popular model for the analysis of repeated measures and longitudinal data. Outliers, however, can severely have an wrong influence on the linear mixed model. The aforementioned model has not fully taken those severe outliers into consideration. One of the popular robust estimation methods, M-estimator attains robustness at the expense of first-order or second-order efficiency whereas minimum Hellinger distance estimator is efficient and robust. In this paper, we propose more robust Bayesian version of parameter estimation via pseudo posterior distribution based on minimum Hellinger distance. It accommodates an appropriate nonparametric kernel density estimation for longitudinal data to require the proposed cross-validation estimator. We conduct simulation study and real data study with the orthodontic study data and the Alzheimers Disease (AD) study data. In simulation study, the proposed method shows smaller biases, mean squared errors, and standard errors than the (residual) maximum likelihood method (REML) in the presence of outliers or missing values. In real data analysis, standard errors and variance-covariance components for the proposed method in two data sets are shown to be lower than those for REML method.

Second-order efficiency of fully sequential designs for estimating the product of two means with application in reliability estimation

Second-order efficiency of fully sequential designs for estimating the product of two means with application in reliability estimation

Second-order efficiency conditions for $C^{1,1}$-vector equilibrium problems in terms of contingent derivatives and applications

Second-order efficiency conditions for $C^{1,1}$-vector equilibrium problems in terms of contingent derivatives and applications

Plug-In Two-Stage and Sequential Normal Density Estimation Under MISE Loss: Both Mean and Variance are Unknown

Consider independent observations having a common normal probability density function with x ∈ R, unknown mean μ(∈R), and unknown variance σ2(∈R +). We propose estimating f(x; μ, σ2) with both two-stage and purely sequential methodologies under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a preassigned positive number c, referred to as the risk bound. No fixed-sample-size methodology would handle this estimation problem. We show that both density estimation methodologies satisfy an asymptotic (a) first-order efficiency property and a (b) first-order risk-efficiency property. Interestingly, purely sequential density estimation methodology has a better second-order efficiency property than that associated with two-stage methodology. Some robustness issues have been addressed. Small, moderate, and large sample performances are examined with the help of simulations. Illustrations are given with real data sets.

Higher-order expansions and efficiencies of tests based on spacings

Statistics based on spacings, or the gaps between points, have been widely used in many contexts, primarily in testing goodness of fit. This paper derives Edgeworth-type asymptotic expansions for the sum of functions of s-step spacings where s, the order of spacings, may increase together with the sample size n. When s is fixed, it is known that only the Greenwood test, based on the sum of squares of these spacings, is first-order asymptotically efficient. In contrast, it is shown here that if s goes to infinity, there exist many other tests which are first-order efficient. We introduce and study the second-order efficiency of such tests and show that if s is sufficiently large relative to n, the Greenwood test is no longer second-order efficient. Interestingly, we see that the common phenomenon of first-order efficiency implying second-order efficiency does not hold true in this situation.

Two-Stage Fixed-Width Confidence Intervals for a Normal Mean in the Presence of Suspect Outliers

We revisit fixed-width (= 2d) confidence interval procedures with a preassigned confidence coefficient (≥1 − α) for the mean μ of a normal distribution when its variance σ2 is unknown. Had σ been known, the required optimal fixed sample size would be C ≡ a 2σ2/d 2 where a ≡ a α is the upper 50α% point of N(0, 1). In his fundamental two-stage procedure, Stein (1945, 1949) estimated C by replacing σ2 with a sample variance from pilot data of size m(≥2) and a with t m−1, α/2. Mukhopadhyay (1982) opened the possibility of incorporating less traditional estimators of σ2. We focus on estimating σ2 by a statistic defined via Gini's mean difference (GMD), mean absolute deviation (MAD), and range. Such modifications are warranted especially when we suspect one or more outliers but normality of the data may not be questioned by a standard test for normality assumption. Obviously, then, t m−1, α/2 must be replaced by the upper 50α% point corresponding to a pivotal distribution of the sample mean standardized appropriately by U m . This way, we explore the role of Mukhopadhyay (1982) two-stage confidence interval procedure when the requisite sample size is determined through GMD, MAD, or range. Associated exact and some asymptotic first-order properties are developed first. Next, we revisit Mukhopadhyay and Duggan (1997) updated two-stage methodology that was proposed when a known positive lower bound was available for σ2 in the present light. We highlight associated (i) first-order efficiency properties for all proposed fixed-width confidence interval procedures and (ii) second-order efficiency property of the GMD-based procedure. These are accompanied with extensive data analyses via simulations as well as real data.

Plug-In Sequential Normal Density Estimation Under Unknown Variance: Mise Loss

Consider independent observations X1, X2,… having a common normal probability density function [Formula: see text] and unknown variance σ2(> 0). We propose estimating f( x; σ2) with a purely sequential methodology under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a preassigned positive number c, referred to as the risk-bound. Since no fixed-sample-size methodology would be able to handle this estimation problem, Mukhopadhyay and Pepe (2009) first gave a two-stage sampling method. We show that our purely sequential density estimation methodology satisfies asymptotic (i) first-order efficiency property (Theorem 2.1) and (ii) first-order risk-efficiency property (Theorem 2.2) just like the Mukhopadhyay and Pepe two-stage methodology did. But, the present purely sequential density estimation methodology has better second-order efficiency property (Theorems 2.3 and 3.1) than those associated with the Mukhopadhyay and Pepe two-stage methodology. Small, moderate, and large sample performances are investigated with the help of simulations. Illustrations are included with the help of a real dataset.

Dividend and Corporate Taxation in an Agency Model of the Firm

Recent evidence on the effect of dividend taxes on firm behavior is inconsistent with neoclassical theories of dividend and corporate taxation. We develop a simple agency model in which managers and shareholders have conflicting interests to explain the evidence. In this model, dividend taxation induces managers to undertake unproductive investments by retaining earnings, and creates a first-order deadweight cost. In contrast, corporate taxes do not distort the manager's payout decision and may only create second-order efficiency costs. Corporate income taxation may therefore be a more efficient way to generate revenue than dividend taxation, challenging existing intuitions based on neoclassical models. (JEL D21, G35, H25, H32)

The First- and Second-Order Large-Deviation Efficiency for an Exponential Family and Certain Curved Exponential Models

The first- and second-order large-deviation efficiency is discussed for an exponential family of distributions. The lower bound for the tail probability of asymptotically median unbiased estimators is directly derived up to the second order by use of the saddlepoint approximation. The maximum likelihood estimator (MLE) is also shown to be second-order large-deviation efficient in the sense that the MLE attains the lower bound. Further, in certain curved exponential models, the first- and second-order lower bounds are obtained, and the MLE is shown not to be first-order large-deviation efficient.

A Three-Stage Selection Procedure With an Exact Consistency

This article considers the problem of selecting the best population by using Bechhofer's indifference zone approach. Stein's two-stage procedure meets the probability requirement (exact consistency). However, it is shown that the procedure does not become asymptotically second-order efficient whatever the first sample sizes are chosen. Hall's three-stage procedure is asymptotically second-order efficient, but it loses the exact consistency. We propose a three-stage procedure that has an exact consistency and show that the procedure enjoys the asymptotic second-order efficiency.

Selecting the Best Component of a Multivariate Normal Population

This article considers the problem of selecting the best component of a mean vector of a multivariate normal distribution. Using Bechhofer's indifference-zone approach, Clark and Yang (1986) proposed a two-stage selection procedure to solve the problem when the covariance matrix is totally unknown. We are interested in the asymptotic performances of their sample size and show that their procedure becomes asymptotically first-order efficient, but is not asymptotically second-order efficient. We propose a three-stage procedure which enjoys an asymptotically second-order efficiency.