Order Asymptotic Efficiency Research Articles

Estimation of Gini Index within Pre-Specified Error Bound

Gini index is a widely used measure of economic inequality. This article develops a theory and methodology for constructing a confidence interval for Gini index with a specified confidence coefficient and a specified width without assuming any specific distribution of the data. Fixed sample size methods cannot simultaneously achieve both specified confidence coefficient and fixed width. We develop a purely sequential procedure for interval estimation of Gini index with a specified confidence coefficient and a specified margin of error. Optimality properties of the proposed method, namely first order asymptotic efficiency and asymptotic consistency properties are proved under mild moment assumptions of the distribution of the data.

EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF k-STEP AND EXTREMUM STATISTICS

It is well known that a one-step scoring estimator that starts from any N1/2-consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k ≥ 1, higher order asymptotic efficiency, and general extremum estimators and test statistics.The paper shows that a k-step estimator has the same higher order asymptotic efficiency, to any given order, as the extremum estimator toward which it is stepping, provided (i) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds.For example, for the Newton–Raphson k-step estimator based on an initial estimator in a wide class, we obtain asymptotic equivalence to integer order s provided 2k ≥ s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders, respectively. This means that the maximum differences between the probabilities that the (N1/2-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N−3/2), and o(N−3), respectively.

Asymptotic Expansions for Perturbed Systems on Wiener Space: Maximum Likelihood Estimators

By means of the Malliavin Calculus, we derive asymptotic expansion of the probability distributions of statistics for systems perturbed by small noises. These results are applied to the problem of the second order asymptotic efficiency of the maximum likelihood estimator.

Second order asymptotic efficiency in a partial linear model

In this paper, the authors discuss the second order asymptotic efficiency of estimators of β based on MLE for Y= Xβ+ g( T)+ ϵ, where X,T, ϵ are independent, g is unknown, ϵ ≈ ϕ(·) is known with mean 0 and variance σ 2.

The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases

In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.

Three-Stage Procedures for Selecting the Largest Normal Mean

SYNOPTIC ABSTRACTWe select, from k(≥2) independent normal populations, the one with the largest mean, where the common variance is unknown. Under the “indifference zone” approach of Bechhofer (1954), we consider three–stage selection procedures. Asymptotic second order expansions for the average sample size and the probability of correct selection are provided here for arbitrary k. The second order expansions show that our three–stage procedure enjoys the second–order asymptotic efficiency property, and gives us the asymptotic number of extra samples necessary over the sequential procedure of Robbins, Sobel and Starr (1968), and over the fixed sample approach, had the variance been known, of Bechhofer (1954). Computer simulations are used to explore moderate sample size performance.

Asymptotic optimality of tests in a hypothesis testing problem for recurrent jump Markov processes

A family of jump Markov processes, depending on a real parameter, is considered. For this parameter, on the basis of an increasing number of observations, one tests the simple hypothesis against a one-sided alternative. A test is suggested for such a problem and its second order asymptotic efficiency is proved. This proof does not require the construction of the asymptotic expansions of the corresponding power functions.

Third order asymptotic efficiency of the sequential maximum likelihood estimation procedure

Under suitable regularity conditions, the third order asymptotic bounds for distributions of regular estimators are obtained. It is shown that the modified maximum likelihood estimation procedure combined with appropriate stopping rule is uniformly third order asymptotically efficient in the sense that its asymptotic distribution attains the bound uniformly in stopping rules up to the third order.

On the Second Order Asymptotic Efficiency of Estimators of Gaussian ARMA Processes

In this paper we investigate an optimal property of maximum likelihood and quasi-maximum likelihood estimators of Gaussian autoregressive moving average processes by the second order approximation of the sampling distribution. It is shown that appropriate modifications of these estimators for Gaussian ARMA processes are second order asymptotically efficient if efficiency is measured by the degree of concentration of the sampling distribution up to second order. This concept of efficiency was introduced by Akahira and Takeuchi (1981).

Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency.

Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency.

Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency (Lecture Notes in Statistics, Vol. 7).

Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency (Lecture Notes in Statistics, Vol. 7).

Asymptotic optimality of the generalized bayes estimator in multiparameter cases

The higher order asymptotic efficiency of the generalized Bayes estimator is discussed in multiparameter cases. For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the classA 2 of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted classD of estimators.