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

The problem of constructing a confidence interval of ‘preassigned width and coverage probability’ considered by Costanza/ Hamdy and Son(1986) is further analyzed. Several multi-stage estimation procedures [ like, purely sequential, accelerated sequential and three-stage procedures ] are utilized to deal with the same estimation problem. The relative advantages and disadvantages of these procedures are discussed.     

Bounded risk estimation of a linear combination of location parameters in negative exponential distributions via three-stage sampling

The paper deals with the problem of bounded risk point estimation for a linear combination of location parameters of two negative exponential distributions. Isogai and Futschik considered the situation when the location and scale parameters are all unknown. They proposed purely sequential procedures and gave second order expansions of the average sample sizes and risks. In this paper we propose three-stage procedures and derive second order expansions of the average sample sizes and risks. Further, we compare the results with those from previous work.     

Sequential estimation of exponential lacation parameter using an asymmetric loss function

For a two-parameter negative exponential population with both parameters unknown, the bounded risk sequential estimation problem of the location parameter is considered under an asymmetric linex loss funmction. Asymptotic second-order expansion of the risk function is derived for a general class of stopping variables. Some examples are include involving purely scquential and accelerated sequential sampling methodologies. A Monte-Carlo study is carried out to support the asymptotic results and to compare the performance of the different sampling methodologies.     

Asymptotic optimality of a two-stage 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. Instead of fully sequential sampling, a two-stage sampling technique is introduced to solve the problem. The proposed two-stage procedure is robust in the sense that it does not depend on the distribution of outcome variables and the prior. It is shown to be asymptotically not worse than the optimal fixed-sample-size procedures for the arbitrary distributions, and to be asymptotically Bayes for the distributions of one-parameter exponential family.     

On accelerating sequential procedures for estimating exponential populations

In this study, we accelerate the purely sequential procedure due to Anscombe(1953), Chow and Robbins(1965) to reduce the number of sampling operations required to carry out the estimation process. The method is proposed while estimating the location parameter(s) of the exponential distribution(s). We also develop theory for the asymptotic characteristic of the associated stopping variables. Our findings are applicable to both point as well as confidence interval estimation problems. Other interesting results are also given.     

Three-stage and accelerated sequential point estimation of the normal mean using LINEX loss function

Consider a normal population with unknown mean μ and unknown variance σ². We estimate μ under an asymmetric LINEX loss function such that the associated risk is bounded from above by a known quantity w. This necessitates the use of a random number (N) of observations. Under a fairly broad set of assumptions on N, we derive the asymptotic second-order expansion of the associated risk function. Some examples have been included involving accelerated sequential and three-stage sampling techniques. Performance comparisons of these procedures are considered using a Monte-Carlo study.     

Sequential Estimation of a Continuous Distribution Function from Delayed Observations

The problem of sequentially estimating a continuous distribution function is considered in the case when the observations become available at random times. A certain class of sequential estimation procedures which are composed of optimal stopping time and sequential minimum risk invariant estimator of a continuous distribution function is obtained under a nonparametric invariant loss function and with the observation cost determined by a convex function of the moment of stopping and the number of observations up to this moment.     

Three-stage designs for monitoring clinical trials

Optimal three-stage designs with equal sample sizes at each stage are presented and compared to fixed sample designs, fully sequential designs, designs restricted to use the fixed sample critical value at the final stage, and to modifications of other group sequential designs previously proposed in the literature. Typically, the greatest savings realized with interim analyses are obtained by the first interim look. More than 50% of the savings possible with a fully sequential design can be realized with a simple two-stage design. Three-stage designs can realize as much as 75% of the possible savings. Without much loss in efficiency, the designs can be modified so that the critical value at the final stage equals the usual fixed sample value while maintaining the overall level of significance, alleviating some potential confusion should a final stage be necessary. Some common group sequential designs, modified to allow early acceptance of the null hypothesis, are shown to be nearly optimal in some settings while performing poorly in others. An example is given to illustrate the use of several three-stage plans in the design of clinical trials.     

COMPARISON OF COSTS FOR MULTI-STAGE GROUP TESTING METHODS IN THE PHARMACEUTICAL INDUSTRY

The issue of whether to test components individually, or alternatively in groups, in order to detect certain chemical properties remains an important issue in the pharmaceutical industry. Economic viability is of paramount importance since, for multi-stage procedures, the cost of additional stages must be taken into consideration, along with the cost of testing mixtures of components. Optimum groups sizes are calculated for the two-stage, three-stage (both members of Li's family of algorithms) and the row-and-column procedures. The γ-two-stage design is investigated, which involves using a γ-separating design at the first stage, followed (if necessary) by a strongly separating design at the second stage. Finally, comparisons are made between the costs of single and multi-stage procedures, for both optimum and standard groups sizes, through the use of two different cost functions.     

Risk efficient estimation of fully dependent random coefficient autoregressive models of general order

We consider a stochastic dynamic model with autoregressive progression. The drift coefficients of the autoregressive model are random where the randomness in the coefficients can have any dependence structure. We propose a two-step sequential estimator and study the asymptotic behavior of few important properties. Paradigm of sequential estimation has its own advantage in reducing sample size and plugging estimates of nuisance parameters while inferring about the main parameters. Our proposed estimator is asymptotically optimal as the predictive risk of the proposed estimator attains the risk of the oracle that assumes known nuisance parameters. Extensive simulation confirms our results.     

Seemingly unrelated regression on the autoregressive (Ar(p)) singular equation system

This paper studies the estimation of seemingly unrelated regressions (SUR) of singular equation systems with an autoregressive error process (AR(p)) for each equation.Parameter estimates of the autoregressive singular equation system are not generally invariant to the equation deleted. Under the model specification restriction on the autoregressive parameters, the invariance property is preserved, and this paper shows that a single equation generalized least squares (GLS) estimation for a general autoregressive error process is equivalent to the SURGLS estimation of the AR(p) singular equation system.     

Second-order approximations for a multivariate analog of Behrens-Fisher problem through three-stage procedure

Abstract

In this article, we have proposed a three-stage procedure for the estimation of the difference of the means of two multivariate normal populations having unknown and unequal variances. Point as well as confidence region estimation is done for the same. Here, we have used the concept of classical Behrens-Fisher problem. Second-order approximations are obtained in both the cases, i.e., point estimation and confidence region estimation.     

Triple sampling procedure for estimating the scale parameters of pareto distributions

This paper sheds light on the large sample performance of the three stage sam- pling procedure, as it pertains to estimating the scale parameter(s) of the Pareto distribution(s). This group sampling procedure merges the efficiency of the purely sequential procedure of Anscombe (1953) and Chow and Robbins (1965) with substan-tial savings in the number of sampling operations, as noted by Hall (1981). Both its simplicity and its economical features provide visible advantages over the one-by-one sampling as an alternative. In this paper we develop some asymptotic properties for the final stage sample size of the triple stage sampling originated by Hall (1981). These results are used to study both the point and the interval estimation problems for the scale parameters of the Pareto distributions. Since our results are asymptotic in nature, a simulation study is given to discuss the moderate sample size peformance of the proposed procedures.     

On kernel density estimation near endpoints

In this paper, we consider the estimation problem of f(0), the value of density f at the left endpoint 0. Nonparametric estimation of f(0) is rather formidable due to boundary effects that occur in nonparametric curve estimation. It is well known that the usual kernel density estimates require modifications when estimating the density near endpoints of the support. Here we investigate the local polynomial smoothing technique as a possible alternative method for the problem. It is observed that our density estimator also possesses desirable properties such as automatic adaptability for boundary effects near endpoints. We also obtain an ‘optimal kernel’ in order to estimate the density at endpoints as a solution of a variational problem. Two bandwidth variation schemes are discussed and investigated in a Monte Carlo study.     

Efficiency and Validity Analyses of Two-Stage Estimation Procedures and Derived Testing Procedures in Quantitative Linear Models with AR(1) Errors

Abstract

In a quantitative linear model with errors following a stationary Gaussian, first-order autoregressive or AR(1) process, Generalized Least Squares (GLS) on raw data and Ordinary Least Squares (OLS) on prewhitened data are efficient methods of estimation of the slope parameters when the autocorrelation parameter of the error AR(1) process, ρ, is known. In practice, ρ is generally unknown. In the so-called two-stage estimation procedures, ρ is then estimated first before using the estimate of ρ to transform the data and estimate the slope parameters by OLS on the transformed data. Different estimators of ρ have been considered in previous studies. In this article, we study nine two-stage estimation procedures for their efficiency in estimating the slope parameters. Six of them (i.e., three noniterative, three iterative) are based on three estimators of ρ that have been considered previously. Two more (i.e., one noniterative, one iterative) are based on a new estimator of ρ that we propose: it is provided by the sample autocorrelation coefficient of the OLS residuals at lag 1, denoted r(1). Lastly, REstricted Maximum Likelihood (REML) represents a different type of two-stage estimation procedure whose efficiency has not been compared to the others yet. We also study the validity of the testing procedures derived from GLS and the nine two-stage estimation procedures. Efficiency and validity are analyzed in a Monte Carlo study. Three types of explanatory variable x in a simple quantitative linear model with AR(1) errors are considered in the time domain: Case 1, x is fixed; Case 2, x is purely random; and Case 3, x follows an AR(1) process with the same autocorrelation parameter value as the error AR(1) process. In a preliminary step, the number of inadmissible estimates and the efficiency of the different estimators of ρ are compared empirically, whereas their approximate expected value in finite samples and their asymptotic variance are derived theoretically. Thereafter, the efficiency of the estimation procedures and the validity of the derived testing procedures are discussed in terms of the sample size and the magnitude and sign of ρ. The noniterative two-stage estimation procedure based on the new estimator of ρ is shown to be more efficient for moderate values of ρ at small sample sizes. With the exception of small sample sizes, REML and its derived F-test perform the best overall. The asymptotic equivalence of two-stage estimation procedures, besides REML, is observed empirically. Differences related to the nature, fixed or random (uncorrelated or autocorrelated), of the explanatory variable are also discussed.     

Experimental designs for binary data in switching measurements on superconducting Josephson junctions

Summary.  We study the optimal design of switching measurements of small Josephsonjunction circuits which operate in the macroscopic quantum tunnelling regime. In the experiment, sequences of current pulses are applied to the Josephson junction sample, while the voltage over the structure is monitored. The appearance of a voltage pulse to a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Starting from the D -optimality criterion we derive the optimal design for the estimation of the unknown parameters of the underlying Gumbel-type distribution. As a practical method for the measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The design presented has immediate applications in the area of superconducting electronics, implying faster data acquisition. The experimental results presented confirm the usefulness of the method.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

On estimating the difference of location parameters of two negative exponential distributions

Fixed-width confidence intervals for the difference of location parameters of two negative-exponential distributions have been constructed through two-stage and purely sequential schemes. The two cases when the scale parameters are equal but unknown, and unequal but unknown, have been dealt with separately. Our two-stage procedures guarantee the exact confidence coefficient to be at least the nominal prescribed level. Various second-order expansions are also considered when sequential procedures are proposed. It is noted that no new tables are needed to implement these procedures in practice.     

Data skeletons: simultaneous estimation of multiple quantiles for massive streaming datasets with applications to density estimation

We consider the problem of density estimation when the data is in the form of a continuous stream with no fixed length. In this setting, implementations of the usual methods of density estimation such as kernel density estimation are problematic. We propose a method of density estimation for massive datasets that is based upon taking the derivative of a smooth curve that has been fit through a set of quantile estimates. To achieve this, a low-storage, single-pass, sequential method is proposed for simultaneous estimation of multiple quantiles for massive datasets that form the basis of this method of density estimation. For comparison, we also consider a sequential kernel density estimator. The proposed methods are shown through simulation study to perform well and to have several distinct advantages over existing methods.     

Higher than second order approximations for sequential point estimation by a two-stage procedure

We consider the sequential point estimation problem of the mean of a normal distribution N(μ, σ²) when the loss function is squared error plus linear cost. It is shown that a two-stage procedure has the asymptotic efficiency of which the order is higher than second order, provided the standard deviation has a known lower bound. We also give a higher than second-order approximation to the risk.