Asymptotically bounded regret sequential estimation of the mean

This paper provides a two-stage rule and a sequential rule for estimating the mean of an unspecified distribution with a given constant risk. Under appropriate regularity conditions, the rules are shown to have asymptotically bounded regrets.     

Sequential estimation of the autoregressive parameter in a first order autoregressive process

This paper considers the problem of sequential point estimation of the autoregressive parameter in a first order autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be asymtotically risk efficient as the cost of estimation error tends to infinity, under certain regularity conditions. Furthermore, nonlinear renewal theory is used to obtain a second order approximation to the expected stopping time. The asymptotic normality and uniform integrability of the standardized stopping time are also established.     

Further remarks on sequential point estimation of the mean of a multinormal population

Asymptotic behaviour of the regret is studied for the sequential procedure developed by Wang (1980) for the point estimation of the mean of a multinormal population. A condition on the initial sample size is also provided for which the regret is bounded.     

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.     

An accelerated bias-corrected sequential estimation for the mean of NEF-PVF distributions

An accelerated sequential procedure for estimating the mean p in the class of the natural exponential family of distributions having power variance function (NEF-PVF), is proposed. The accelerated procedure is conducted under a combined loss of weighted squared estimation error and sarnpling cost. A class of bias-corrected estimators, which are natural for this suggested accelerated sampling scheme is proposed. The asymptotic properties of the suggested estimators are provided. In particular, one realizes the impact of acceleration on the regret and the tradeoff between bias reduction and regret reduction.     

Improved sequential and accelerated sequential Procedures for estimating the scale parameter in a aniform distribution

Ghosh and Mukhopadhyay (1975a) and more recently Mukhopadhyay et al.(1983) have consideredsequential minimum risk point estimation of θ in a uniform (0,θ)population.The loss function has been squared error plus linear cost and θ has beencustomarily estimated by the sample maximum.In this paper,we consider instead the best scalarmultiple of the sample aximum as the estimator of θ.The percentage saving in the fictitious optimal fixed sample size as well as the percentage reduction in the corresponding minimum risk ar seen to be about 20.6X when compared with the Mukhopadhyay et al.(1983) cedure.Since thisamount of saving is quite substantial,we set out to develop     

Asymptotically pointwise optimal rules of sequential estimation of mean vector when an information matrix has some structure in a multivariate normal population

The exact formulas of Bayes stopping times are often difficult to derive. Bickel and Yahav (1965) had provided the large sample approximation known as the "Asymptotically pointwise optimal" (A.P.O.) rule. The A.P.O. rule for the problem of the mean of a multivariate normal distribution, for a completely unknown covariance matrix , has been developed by the present author. This paper gives the A.P.O. rule of the mean of a multivariate normal ditribution for a covariance matrix with some structure. Also the result is shown to be asymptotically" non-deficient" in the sence of Woodroofe.     

Unbiased estimation of the Weibull scale parameter using linear least squares analysis

Unbiased estimation of the Weibull scale parameter using unweighted linear least squares (LLS) analysis was investigated in this work as a function of specimen number, N. Mean, median and mode values of the Weibull scale parameter were estimated using a Monte Carlo procedure and general probability estimator of F = (i – a)/(N + b) with a and b varying between zero and unity. Unlike the mean and mode cases, the median value of Weibull scale parameter was essentially independent of Weibull modulus, m, with the coefficient of variation decreasing with b and being a factor of five smaller compared to that of m. Optimum values of a as a function of N were obtained using the median of the Weibull scale parameter with a being found to vary between 0.4796 for N = 10 and 0.2698 for N = 200 with b = 0.     

Maximum quasi-likelihood estimation for the near(2) model

Abstract.  Maximum quasi-likelihood estimation is investigated for the NEAR(2) model, an autoregressive time series model with marginal exponential distributions. In certain regions of the parameter space, simulations indicate that maximum quasi-likelihood estimators perform better than two-stage conditional least squares estimators in terms of the per cent of estimates falling in the parameter space. The problem of out-of-range estimates is shown to be caused by the lack of information in the data rather than the characteristics of the method of estimation.     

On sequential R estimation of location in the general behrens fisher model

For a general Behrens-Fisher model (relating to a shift in location between two symmetric distributions with possibly different shapes), asymptotic theory of sequential R-estimation is studied. Sequential versions of the usual two sample R-estimators and some alternative ones (based on the difference of one sample R-estimators) are considered and asymptotic risk efficiency results for these estimators are presented. In this context, under the general Behrens-Fisher model, for R-estimators, uniform integrability and moment convergence results are established.     

Second order asymptotic efficlency in terms of the asymptotic variance of seqrential estimation procedures in the presence of nrisance parameters

In the presence of nuisance parameters, the Bhattacharyya type bound for the asymptotic variance of estimation procedures is obtained. It is shown that the modified maximum likelihood (ML) estimation procedures together with any stopping rule does not attain the bound. Further it is shown that the modified ML estimation procedure with the appropriate stopping rule is second order asymptotically efficient in some class of estimation procedures in the sense that it attains the lower bound for the asymptotic variance in the class.     

sequential estimation of the hgarginal density function for a strongly mixing process

Suppose that {X_n} is a strongly mixing process with unknow marginal density f(x) and that we estimate f(x) by a kernel estimator [fcirc]_n(x|h_n)and want to achive the MISE no larger than some preassigned postive number w. However,the appropriate sample size n^*depends on a functional of the unknow density function. Therefore some sequential procedure is required and we adopt a fully sequential procedure. In this paper we investigate the asymptotic properties of the procedure and show that the producure is asymptotically efficient in a certain sense as w→0. The results are almost the same in the i.i.d. setting. our result extend a class of models to which the methodology can be applied. For example economic variable,experiments on a single subject in which obervation are not indepent, and so on.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.     

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 and two-stage point estimation for the range in a power family distribution

Sequential and two-stage point estimation procedures for the range of a power family distribution are discussed using a loss function from a fairly general class.

Some asymptotic characteristics of the sequential procedure are presented, including approximate expressions for the distribution of the sample size. For the two stage procedure, we present asymptotic properties and exact expressions for the risk of the procedure, and distribution and mean of the stopping rule. The results of numerical simulations are presented to demonstrate some practical merits of our procedures for moderate sample sizes.     

Sequential experimental design for estimation and analysis of thermal parameters in a fixed bed

The analysis of the effective radial thermal conductivity and the film heat transfer coefficient were carried out in a fixed bed. The temperature profiles were described by two‐dimensional pseudo‐homogeneous model. The thermal parameters were estimated using a sequential experimental design technique. The minimum volume criterion was used to design the next point for temperature measurement in the bed. The utilization of T = T₀ (constant) as the boundary condition at the bed inlet resulted in an axial variation of thermal parameters, which was the factor responsible for the inadequacy of the model in fitting experimental data of different bed heights simultaneously. Using T = T(r) as the boundary condition makes the thermal parameters independent of the axial position and the model statiscally adequate to describe the axial and radial temperature profiles throughout the bed.     

Explicit solutions of the Bayes sequential estimation problem for a time-transformed exponential distribution

A problem of Bayesian time-sequential estimation of an unknown parameter of a time-transformed exponential distribution is considered. It is supposed that the cost of observing the process is the sum of an increasing function of time and a linear function of the number of observations. Under some general assumptions concerning the loss function associated with the error of estimation, the optimal stopping time is derived using the free-boundary method. As an example, a solution of the problem considered is given in the case of a weighted precautionary loss function.     

Two-stage procedures for the bounded risk point estimation of the parameter and hazard rate in two families of distributions

Two families of distributions are considered that cover a large number of probability distributions useful in investigations involving reliability studies and survival analyses. The problem of bounded risk point estimation of the parameter and hazard rate function of the two families of distribution is handled. Motivated by Mukhopadhyay and Pepe (2006 Mukhopadhyay, N. and Pepe, W. (2006). Exact Bounded Risk Estimation When the Terminal Sample Size and Estimator Are Dependent: The Exponential Case, Sequential Analysis 25: 85–101.[Taylor &; Francis Online] , [Google Scholar]), Roughani and Mahmoudi (2015 Roughani, G. and Mahmoudi, E. (2015). Exact Risk Evaluation of the Two-Stage Estimation of the Gamma Scale Parameter under Bounded Risk Constraint, Sequential Analysis 34: 387–405.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Mahmoudi and Lalehzari (2017 Mahmoudi, E. and Lalehzari, R. (2017). Bounded Risk Estimation of the Hazard Rate Function of the Exponential Distribution: Two-Stage Procedure, Sequential Analysis 36: 38–54.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), two-stage procedures are developed based on the maximum likelihood estimator (MLE) as well as uniformly minimum variance unbiased estimator (UMVUE). The estimation problem based on the minimum mean square estimator (MMSE) is also considered. We establish that the MMSE of the parameter and hazard rate provides a smaller risk.