Two-stage Sequential Procedure Research Articles

Sequential nonparametric estimation of controlled multivariate regression

The article considers an adaptive sequential nonparametric estimation of a multivariate regression with assigned mean integrated squared error (MISE) and minimax mean stopping time when the estimator matches performance of an oracle knowing all nuisance parameters and functions. It is known that the problem has no solution if regression belongs to a Sobolev class of differentiable functions. What if an underlying regression is smoother, say, analytic? It is shown that in this case it is possible to match performance of the oracle. Furthermore, similar to the classical Stein solution for a parameter estimation, a two-stage sequential procedure solves the problem. The proposed regression estimator for the first stage, based on a sample with fixed sample size, is of interest on its own, and a thought-provoking environmental example of reducing potent greenhouse gas emission by an anaerobic digestion system is used to discuss a number of important topics for small samples.

Bounded risk per unit cost index constraint for sequential estimation of the mean in a two-parameter exponential distribution

In this article, the two-stage sequential sampling procedure is proposed to estimate the mean of an exponential distribution under the modified square error loss function. The main aim of the article is to consider the associated risk per unit cost function by bounding it from above with a fixed preassigned positive number, ω. We provide the exact distribution of the total sample size, explicit formulas for the mathematical expectation of the rth power of the total sample size, the expected value and mean squared error of the maximum likelihood estimate of the scale, and mean parameters of the exponential distribution under the two-stage sequential procedure. The performances of the proposed methodologies are investigated with the help of simulations. In the end, we supplement with extensive sets of data analysis via computer simulations, validating that the two-stage sequential method performs very well. Finally, using a real data set, the procedure is clearly illustrated.

An extended Two-Stage Sequential Optimization approach: Properties and performance

This paper looks into nonlinear non convex stochastic unconstrained optimization with finite simulation budget. Our work builds upon the Two-Stage Sequential Optimization (TSSO) algorithm that addresses the class of problems of interest by using the modified nugget effect kriging (MNEK) meta-model and proposing a budget allocation followed by a two-stage sequential procedure. Despite its efficiency and performance, we have observed that, given a finite budget, the choice of the number of replications per iteration, currently left to the user, is particularly critical for the algorithm performance. A fixed a-priori assignment can affect the ability to control the algorithm making it particularly sensitive to the initial settings. In this paper, we propose the extended TSSO (eTSSO). Specifically, a general simulation budget allocation scheme is proposed with the objective to balance the need of accurate function estimations to improve the selection in the search stage, with the need to explore the solution space. The new scheme adaptively, and recursively, increases the simulation budget based upon information iteratively returned by the optimizer itself. We analyze the asymptotic properties of eTSSO. Subsequently, we propose four alternative variants of the general allocation that we empirically analyze by comparing the quality of the estimated optimum input combination and the corresponding estimated optimum output against TSSO and other state of the art algorithms.

Sequential fixed-accuracy confidence intervals for the stress–strength reliability parameter for the exponential distribution: two-stage sampling procedure

In this paper, a two-stage sequential estimation procedure is considered to construct fixed-accuracy confidence intervals of the reliability parameter R under the stress–strength model when the stress and strength are independent exponential random variables with different scale parameters. The exact distribution of the total sample size, 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. Performances of the proposed methodology are investigated with the help of simulations. Finally, using three pairs of real datasets, the procedure is clearly illustrated. We used MATLAB in order to implement computations, simulation studies, and real data analysis.

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

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.

On the optimal computing budget allocation problem for large scale simulation optimization

Selecting a set that contains the best simulated systems is an important area of research. When the number of alternative systems is large, then it becomes impossible to simulate all alternatives, so one needs to relax the problem in order to find a good enough simulated system rather than simulating each alternative. One way for solving this problem is to use two-stage sequential procedure. In the first stage the ordinal optimization is used to select a subset that overlaps with the actual best systems with high probability. Then in the second stage an optimization procedure can be applied on the smaller set to select the best alternatives in it. In this paper, we consider the optimal computing budget allocation (OCBA) in the second stage that distribute available computational budget on the alternative systems in order to get a correct selection with high probability. We also discuss the effect of the simulation parameters on the performance of the procedure by implementing the procedure on three different examples. The numerical results indeed indicate that the choice of these parameters affect its performance.

Two-stage sequential procedure for the estimation of the powers of the ratio of two exponential scale parameters

Two-stage sequential procedure for the estimation of the powers of the ratio of two exponential scale parameters

Two-Stage Nonparametric Sequential Estimation of the Directional Density with Complete and Missing Observations

Stein (1945) proposed a two-stage sequential methodology of inference that influenced numerous areas of statistics. In this article, the Stein's methodology is used and expands upon nonparametric estimation of the directional probability density. The aim is to propose a data-driven nonparametric sequential procedure that mimics performance of an oracle that knows smoothness of an estimated density and minimizes the mean stopping time given an assigned mean integrated squared error. For such a setting, using a sequential estimator is the must because smoothness of the density is unknown. It is known that for a general random variable the stated problem has no solution because an estimator cannot perform as well as the minimax oracle. At the same time, this article shows that for the case of directional density, under a mild assumption, there exists a data-driven two-stage sequential procedure that is minimax and adapts to unknown smoothness of an underlying density. Furthermore, we are able to solve the same problem for a setting where some observations in a sample may be lost due to a stochastic missing mechanism. This is a practically important and theoretically interesting problem because for missing data even the above-mentioned oracle cannot solve it without knowing the missing mechanism.

Likelihood based frequentist inference when data are missing at random

One of the most often quoted results from the original work of Rubin and Little on the classification of missing value processes is the validity of likelihood based inferences under missing at random (MAR) mechanisms. Although the sense in which this result holds was precisely defined by Rubin, and explored by him in later work, it appears to be now used by some authors in a general and rather imprecise way, particularly with respect to the use of frequentist modes of inference. In this paper an exposition is given of likelihood based frequentist inference under an MAR mechanism that shows in particular which aspects of such inference cannot be separated from consideration of the missing value mechanism. The development is illustrated with three simple setups: a bivariate binary outcome, a bivariate Gaussian outcome and a two-stage sequential procedure with Gaussian outcome and with real longitudinal examples, involving both categorical and continuous outcomes. In particular, it is shown that the classical expected information matrix is biased and the use of the observed information matrix is recommended.

Asymptotic expansions for sequential confidence levels in inverse linear regression

Non-linear renewal theory is used to derive second order asymptotic expansions for the coverage probability of a fixed-width sequential confidence interval for an unknown parameter xin the inverse linear regression model. These expansions are obtained for a two-stage sequential procedure, proposed by Perng and Tong (1974) for the construction of a confidence interval for x.

A Sequential Solution to the Inverse Linear Regression Problem

In this note we apply the sequential theory developed by Chow and Robbins [1] and Gleser [3], [4] to the inverse linear regression problem. A two-stage sequential procedure has been proposed for the construction of a fixed-width confidence interval for $x$ (an unknown parameter). It is shown that the limiting probabilities of correct decision are equal to $P^\ast$ (pre-assigned).

Two-Stage Sequential Estimation in the Uniform Density

Abstract The main problem to be solved here may be described as follows: let X 1, X2, …, be independent random variables, each with density f 8(x) =1/θ over (0, θ) and zero elsewhere. It is desired to use a two-stage sequential procedure to estimate the unknown parameter θ by an interval of length at most d units and with confidence at least I—α, for some specified d>0 and α in (0, 1). The procedure proposed in this article satisfies an admissibility criterion which may be stated in terms of the maximum possible number of observations or, alternatively, the expected number of observations.

Two-Stage Sequential Estimation in the Uniform Density

Abstract The main problem to be solved here may be described as follows: let X 1, X2, …, be independent random variables, each with density f 8(x) =1/θ over (0, θ) and zero elsewhere. It is desired to use a two-stage sequential procedure to estimate the unknown parameter θ by an interval of length at most d units and with confidence at least I—α, for some specified d>0 and α in (0, 1). The procedure proposed in this article satisfies an admissibility criterion which may be stated in terms of the maximum possible number of observations or, alternatively, the expected number of observations.