Sequential fixed-width confidence bands for distribution functions under random censoring

Summary Sequential fixed-width confidence bands for a distribution function are derived in the case, when the data are censored from the right. The Breslow-Crowley invariance principle for the Kaplan Meier estimate is extended to the random sample size situation. Also some simulation results are reported, which illustrate the behavior of the stopping times.     

Sequential estimation of means of linear processes

First the minimum risk point estimation as well as the fixed-width confidence interval problems for the mean parameter of a linear process are addressed under the framework of Fakhre-Zakeri and Lee (1992). The accelerated versions of their full sequential methodologies are introduced in order to achieve operational savings. Next, multi-sample analogs are discussed along the lines of Mukhopadhyay and Sriram (1992) both under full sequential as well as accelerated sequential sampling. In either setup, the first-order asymptotic characteristics are highlighted.     

Multivariate multistage methodologies for simultaneous all pairwise comparisons

We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences μ₁−μ _J , in the presence ofk(≥2) independent populationsN _p (μ₁,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ _pxp is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is a known multiple of one of the variances or Σ=σ² H where σ(>0) is unknown, butH _pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay and Aoshima (1997).     

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.     

Sequential estimation of a linear function of normal means under asymmetric loss function

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions. Received: August 1999     

Remarks on iteration.

1. Iteration is an operation with many aspects; we shall here only occupy ourselves with iteration as a means of numerical determination of a real root of an equation. Nor shall we go into the whole of the theory of this subject ² See, for instance, Whittaker & Robinson: The Calculus of Observations, or Runge & König: Vorlesungen über numerisches Rechnen. , our principal object being to show how the process of iteration which is often futile in its primitive form may be improved by a suitable combination of three consecutive values. But before doing this, it will be convenient to recall briefly some of the principles involved in iteration.