Mun S. Son

Triple stage point estimation for the exponential location parameter

This paper deals with the problem of estimating the minimum lifetime (guarantee time) of the two parameter exponential distribution through a three-stage sampling procedure. Several forms of loss functions are considered. The regret associated with each loss function is determined. The results in this paper generalize the basic results of Hall (1981, Ann. Statist., 9, 1229–1238).

A New Two-Stage Sampling Design for Estimating the Maximum Average Time To Flower

A horticulturist was considering the number of days (X) each variety took from planting seeds to reach the stage when the first bud appeared for three local marigold varieties. The data X could be recorded with accuracy of one-half day. The primary interest was to estimate the maximum waiting time between “seeding” and “first budding” among three varieties under consideration. It was thought that a 99% confidence interval with width one day would suffice. The horticulturist felt comfortable to assume a normal distribution for the response variable. He provided positive lower bounds for the variances which forced pilot sample sizes to become unequal. We are not aware of any existing methodology with unequal pilot sample sizes that would readily apply here. Accordingly, a new two-stage sampling design was developed and implemented. The gathered data validated all assumptions made during the course of this investigation. Important exact as well as large-sample properties of the proposed methodology are highlighted (Theorem 1). This methodology is proven to be theoretically superior (Theorem 2) to the existing methodology for large sample sizes provided that the pilot sizes could be “chosen” equal. For the experimental data on hand, the superiority of the new methodology has also been indicated (Section 4.3). The solution to the primary estimation problem ultimately led to a natural and yet nontraditional selection problem involving identification of the “worst” marigold variety. For this selection problem, a practical approach is developed (Section 4.4) for evaluating the associated probability of correctly selecting the worst marigold variety.

Two stage fixed width confidence intervals for the common location parameter of several exponential distributins

Two stage sampling schemes are introduced for use in estimating the common location parameter (guarantee time) of two or more exponential distributions with a confidence interval of prespecified width whose coverage probability is at least a given nominal value. Exact expressions for all moments of order r ≥ 1 of the associated two stage sample sizes and for the actual coverage probabilities are derived. The performance of the procedures in a variety of two population, moderate fixed sample size cases is examined via numerical studies involving both exact calculations and Monte Carlo simulations. No new tables are needed to implement any of the proposed methods. A modified two stage procedure is recommended for practical use

Some properties of, and relationships among, several uncorrelated and homoscedastic residual vectors

In this paper we examine the properties of four types of residual vectors, arising from fitting a linear regression model to a set of data by least squares. The four types of residuals are (i) the Stepwise residuals (Hedayat and Robson, 1970), (ii) the Recursive residuals (Brown, Durbin, and Evans, 1975), (iii) the Sequentially Adjusted residuals (to be defined herein), and (iv) the BLUS residuals (Theil, 1965, 1971). We also study the relationships among the four residual vectors. It is found that, for any given sequence of observations, (i) the first three sets of residuals are identical, (ii) each of the first three sets, being identical, is a member of Thei’rs (1965, 1971) family of residuals; specifically, they are Linear Unbiased with a Scalar covariance matrix (LUS) but not Best Linear Unbiased with a Scalar covariance matrix (BLUS). We find the explicit form of the transformation matrix and show that the first three sets of residual vectors can be written as an orthogonal transformation of the BLU...

The classification problem with autoregressive process

We consider the problem of assigning a realization into one of several autoregressive soursces that share a common known order and unknown error variance. The approach is to use an informal Bayesian inference based on the marginal posterior distribution of the classification vector. A realization is assigned to that actoregessive process with the largest posterior probability, and an example demomtrates the classification technique behaves in a reasonable way. A generalization is developed.

Controlling Type II Error While Constructing Triple Sampling Fixed Precision Confidence Intervals for the Normal Mean

The rationale and methodology for estimating a mean with a fixed width confidence interval through sampling in three stages are extended to cover the additional problem of testing hypotheses concerning shifts in the mean with controlled Type II error. The coverage probability and operating characteristic function of the confidence interval based on the integrated approach are derived and compared with those of the usual triple sampling confidence interval. The extended methodology leads to better coverage probability and uniformly better Type II error probabilities. Achieving the additional objective of controlling Type II error inevitably implies a two- to threefold increase in the required optimal sample size. Some suggestions for dealing with this apparent limitation are discussed from a practical viewpoint. It is recommended that an integrated approach to estimation and testing based on confidence intervals be incorporated in the design stage for credible inferences.

Sensitivity Analysis of Multistage Sampling to Departure of an Underlying Distribution from Normality with Computer Simulations

Abstract The current study examines empirically the impact of using normal distribution–based theory and methodology of multistage sampling procedures when the population distribution moves away from normality. We focus on some relevant kinds of departures and illustrate the impact of such departures on the quality of multistage inference. We also address the quality of inference due to shifts in parameters and investigate the extent of sensitivity of both coverage probability and the type II error probability. We do so by examining the capabilities of a fixed-width confidence interval to detect possible shifts in the true parameters occurring outside the confidence interval. Extensive sets of Monte Carlo simulations are reported in a number of interesting situations to highlight small to moderate to large-sample-size performances due to change(s) in the underlying distribution or shifts in the population parameters.

On the Covariance Between the Sample Mean and Variance

Dodge and Rousson (1999) and Zhang (2007) gave an expression for Cov( , S 2) based upon independent and identically distributed (i.i.d.) observations not necessarily normal. In this note, we derive the expression (Theorem 2.1) for Cov( , S 2) based upon non-independent and/or nonidentically distributed observations and move away from the i.i.d. scenario. Some illustrations are included.

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.

The validity of least squares estimation in a time series model using the bootstrap methodology

Abstract The boostrap methodology may be used for estimating standard errors of the estimated parameters in a time series model. The idea is to approximate the theoretical error distribution by the residual distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order auto-regressive model fitted by least squares estimation. A comparison of the conventional and bootstrap methodology is made. A numerical result shows that the traditional least squares asymptotic formula for estimating standard errors appear to overestimate the true standard errors. But there are two problems in the simulation world of bootstrap for the autoregressive model of order two: (1) the first two observations y1 and y2 have been fixed, and (2) the residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional least squares and bootstrap estimates of the standard errors appear to be performing quite well.