Problem Of Point Estimation Research Articles

On guaranteed estimation of the mean of an autoregressive process

This paper considers the problem of sequential point estimation of the mean of a stable autoregressive process with unknown scale and autoregressive parameters. The construction of a sequential procedure makes use of special stopping rules and some modifications of least-squares estimates. The procedure enables estimating the mean with prescribed mean-square accuracy uniformly in nuisance parameters. The uniform asymptotic normality and the asymptotic minimaxity of the sequential estimate are established. The asymptotic formula for the mean sample size is obtained.

On the Efficacy of Bayesian Inference for Nonidentifiable Models

Although classical statistical methods are inapplicable in point estimation problems involving nonidentifiable parameters, a Bayesian analysis using proper priors can produce a closed form, interpretable point estimate in such problems. The question of whether, and when, the Bayesian approach produces worthwhile answers is investigated. In contrast to the preposterior analysis of this question offered by Kadane, we examine the question conditionally, given the information provided by the experiment. An important initial insight on the matter is that posterior estimates of a nonidentifiable parameter can actually be inferior to the prior (no-data) estimate of that parameter, even as the sample size grows to infinity. In general, our goal is to characterize, within the space of prior distributions, classes of priors that lead to posterior estimates that are superior, in some reasonable sense, to one's prior estimate. This goal is shown to be feasible through a detailed examination of a particular two-parameter Binomial model. Our results support the proposition that a Bayesian analysis tends to be efficacious in the estimation problem studied, and suggest that Bayesian updating can produce useful answers in the presence of nonidentifiability.

A note on sequential point estimation of the mean of a gamma population

The problem of sequential point estimation of the mean of a gamma population, considered by Woodroofe (Second-order approximations for sequential point and interval estimation. Ann. Statist., 1977, 5, 984–995), is further analyzed. Much simpler techniques are provided for obtaining the second-order approximations for the risk, corresponding to the sequential procedure and the asymptotic distribution of the stopping time.

A bayesian approach to sequential estimation without using the

The problem of Bayes sequential point estimation without using the prior information is considered. Two sequential procedures of the type of Bickel and Yahav (1967, 1968), for the scale exponential and location normal families respectively, are proposed. It is shown in the present paper that the proposed procedures are asymptotically pointwise optimal (A.P.O.) in the sense of Bickel and Yahav (1967) and the second order approximation of the Bayes risk is established for a large class of prior distributions.

Multi-stage point estimation of the common location parameter of several exponential distributions

The problem of point estimation of the common location parameter of several exponential distributions is considered. The scale parameters are assumed to be unknown and unequal. Consideration is given to a family of loss functions. Non-existence of the fixed sample size procedure is established to deal with the present estimation problem. Purely sequential, accelerated sequential, three-stage and two-stage procedures are developed to handle the situation. Relative advantage and disadvantage of these estimation approaches are established.

A Maximum Decisional Efficiency Estimation Principle

This paper proposes a systematic approach to certain parameter estimation problems relevant to applied optimization models. A context-specific measure of decision maker performance called decisional efficiency is defined as a function of the unknown parameter vector. The values of this measure are considered to be distributed according to a performance density. Then a principle of Maximum Decisional Efficiency (MDE) is proposed and its relationship to the Maximum Likelihood principle of statistics is discussed. The principle is first illustrated on a problem of group ideal point estimation. Next, estimation of the ratio of ordering and holding costs in inventories managed by the EOQ model is considered. The results in both these examples are intuitively appealing, suggesting face validity of the principle. A more detailed illustration is developed for the problem of deriving a consensus scoring model for credit applicants which combines the data of more than one loan officer.

Memorial: Nikolai Nikolaevitch Chentsov

Memorial: Nikolai Nikolaevitch Chentsov

Linking the estimation and ranking and selection problems through sequential procedures: The normal case

Taking into consideration the relationship among the ‘optimal’ fixed sample size solutions of estimation (point, as well as, interval) and ranking and selection problems related to normal populations, a class of sequential procedures is developed. The confidence interval and ranking and selection problems are linked with the bounded risk point estimation problems under some suitable loss functions. Exploiting the common functional form of the risks associated with these problems, second-order approximations are obtained for the ‘regret’. It is illustrated that the results obtained under this general set-up provide second-order approximations to estimation and ranking and selection problems and, as such, no separate dealing is required. As an additional contribution, a much simple technique of obtaining the asymptotic distribution of stopping time is provided.

Sequential estimation for the parameters of a stationary auto regressive model

This paper considers the problem of sequential point estimation and fixed accuracy confidence set procedures of autoregressive parameters in a ρ-th order stationary autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be risk efficient as the cost of estimation error tends to infinity. Furthermore, the proposed procedure for fixed-width confidence set is shown to be both asymptotically consistent and asymptotically efficient as the width approaches zero.

Sequential estimation of a parameter of an exponential distribution

We consider the problem of minimum risk point estimation for the parameter θ=aμ+bσ of the exponential distribution with unknown location parameter μ and scale parameter σ when the loss function is squared error plus linear cost. In this paper, we propose a sequential estimator of θ and show that the associated risk is asymptotically one cost less than that given by Ghosh and Mukhopadhyay (1989,South African Statist. J.,23, 251–268).

Sequential estimation via replicated piecewise stopping number in a tow—parameter exponential family of distributions

In a certain class of two—parameter exponential distributions, we consider minimum risk point estimation problems for one of the parameters. We propose to implement the sequential procedure of Bose and Boukai (1993) in smaller “pieces” along the lines of Mukhopadhyay and Sen (1993). Unlike the fully sequential procedure, one can obtain an unbiased estimator for the variance of the storpping number in a piecewise methodology. On top of this, the asymptotic second—order expansions of the regret functions for both the piecewise and fully sequential estimators are same up to o(1) term and the piecewise sampling scheme is operationally more covenient. The effect on cost due to parallel processing is also discussed

Single and twin-heaps as natural data structures for percentile point simulation algorithms

Sometimes percentile points cannot be determined analytically. In such cases one has to resort to Monte Carlo techniques. In order to provide reliable and accurate results it is usually necessary to generate rather large samples. Thus the proper organization of the relevant data is of crucial importance. In this paper we investigate the appropriateness of heap-based data structures for the percentile point estimation problem. Theoretical considerations and empirical results give evidence of the good performance of these structures regarding their time and space complexity.

Further developments in estimation of the largest mean ofK normal populations

We revisit the bounded maximal risk point estimation problem as well as the fixed-width confidence interval estimation problem for the largest mean amongk(≥2) independent normal populations having unknown means and unknown but equal variance. In the point estimation setup, we devise appropriate two-stage and modified two-stage methodologies so that the associatedmaximal risk can bebounded from aboveexactly by a preassigned positive number. Kuo and Mukhopadhyay (1990), however, emphasized only the asymptotics in this context. We have also introduced, in both point and interval estimation problems,accelerated sequential methodologies thereby saving sampling operations tremendously over the purely sequential schemes considered in Kuo and Mukhopadhyay (1990), but enjoying at the same time asymptotic second-order characteristics, fairly similar to those of the purely sequential ones.

Estimating the Probability of Toxicity at the Recommended Dose Following a Phase I Clinical Trial in Cancer

The problem of point and interval estimation following a Phase I trial, carried out according to the scheme outlined by O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33-48), is investigated. A reparametrization of the model suggested in this earlier work can be seen to be advantageous in some circumstances. Maximum likelihood estimators, Bayesian estimators, and one-step estimators are considered. The continual reassessment method imposes restrictions on the sample space such that it is not possible for confidence intervals to achieve exact coverage properties, however large a sample is taken. Nonetheless, our simulations, based on a small finite sample of 20, not atypical in studies of this type, indicate that the calculated intervals are useful in most practical cases and achieve coverage very close to nominal levels in a very wide range of situations. The relative merits of the different estimators and their associated confidence intervals, viewed from a frequentist perspective, are discussed.

A Noninformative Bayesian Approach to Interval Estimation in Finite Population Sampling

Abstract A noninformative Bayesian approach to interval estimation in finite population sampling is discussed. Given the sample, this method introduces the Polya distribution as a pseudo posterior distribution over the unobserved members of the population. In many cases this distribution yields interval estimates similar to those of standard frequentist theory. In addition, it can be used in situations where the standard methods are difficult to apply, for example, in producing an interval estimate for the ratio of two medians. We also consider related point estimation problems and observe that estimators derived from the pseudo posterior often perform better than classical alternatives.

A Noninformative Bayesian Approach to Interval Estimation in Finite Population Sampling

Abstract A noninformative Bayesian approach to interval estimation in finite population sampling is discussed. Given the sample, this method introduces the Polya distribution as a pseudo posterior distribution over the unobserved members of the population. In many cases this distribution yields interval estimates similar to those of standard frequentist theory. In addition, it can be used in situations where the standard methods are difficult to apply, for example, in producing an interval estimate for the ratio of two medians. We also consider related point estimation problems and observe that estimators derived from the pseudo posterior often perform better than classical alternatives.

Bayesian analyses for a multiple capture-recapture model

SUMMARY In multiple capture-recapture surveys, the probability of capture can vary between sampling occasions. The model accounting for this variation is known as At. Bayes, empirical Bayes, and Bayes empirical Bayes solutions are given to the problems of interval estimation, decision making, and point estimation of the population size N. When the number of sampling occasions is small to moderate and the number of recaptured units observed on each sampling occasion is moderate, estimates obtained from empirical Bayes and Bayes empirical Bayes methods compare closely to Bayesian methods using a reference prior distribution for the capture probabilities. However, when the number of sampling occasions is large and the number of recaptured units observed on each sampling occasion is small, inferences obtained using different reference priors can differ considerably.

On a Class of Asymptotically Risk-Efficient Sequential Procedures

A class of sequential procedures is developed for the point estimation of the parameter(s) of a population under a family of loss functions plus cost function of the general form. Condition on the initial sample size is determined which ensures the asymptotic ‘risk-efficiency’ of the proposed class. By means of various examples, it is shown that many sequential point estimation problems can be handled with the help of the proposed class.

Multi-stage point and interval estimation of the largest mean ofK normal populations and the associated second-order properties

We havek independent normal populations with unknown meansμ1, …,μk and a common unknown varianceσ2. Both point and interval estimation procedures for the largest mean are proposed by means of sequential and three-stage procedures. For the point estimation problem, we require that the maximal risk be at mostW, a preassigned positive number. For the other problem, we wish to construct a fixed-width confidence interval having the confidence coefficient at least 1-α, a preassigned number between zero and one. Asymptotic second order expansions are provided for various characteristics, such as average sample size, associated risks etc., for the suggested multi-stage estimation procedures.

On Truncatd sequential estimation of the drifting parametermean in the first order autoregressive models

This paper considers the problem of sequential point estimation of the drifting parameter mean in the first order autoregression process. The truncated sequential procedure proposed here is based on the least squares estimator and is shown to ensure the preassigned mean square accuracy of the estimates. The uniform in parameter asymptotic normality of the sequential estimator is established.