Fixed-width Confidence Interval Estimation Research Articles

Fixed width confidence interval estimation for general continuous responses under a response adaptive design

. A fixed width confidence interval for the treatment difference is obtained under heteroscedastic normal responses following a response adaptive allocation targeting Neyman optimum allocation. The developed procedure is further extended for general location-scale model based responses under a compound response adaptive allocation. The developed procedures are studied theoretically and are supported empirically assuming various response models. A real data study is also provided following the proposed procedure for the purpose of illustration.

Second-order (s.o.) multi-stage fixed-width confidence interval (FWCI) estimation strategies for comparing location parameters from two negative exponential (NE) populations: illustrations with cancer data

Second-order (s.o.) multi-stage fixed-width confidence interval (FWCI) estimation strategies for comparing location parameters from two negative exponential (NE) populations: illustrations with cancer data

An optimal purely sequential strategy with asymptotic second-order properties: Applications from statistical inference and data analysis

We develop a new class of purely sequential methodologies under an assumption that the population distribution belongs to a location-scale family. Both asymptotic first-order and second-order theories are put forward with substantial generality under a big and unified tent that successfully lead to a broad set of illustrations. After we identify an appropriately defined optimal strategy under this unified structure, we introduce applications that handle a variety of interesting inference problems. These are associated with, but not limited to, the following areas: (a) the fixed-width confidence interval (FWCI) estimation, (b) the minimum risk point estimation (MRPE), (c) the fixed-size confidence region (FSCR) estimation, (d) multiple comparisons, and (e) selecting the best normal treatment (StBNT). In illustrations (a)–(d), we have highlighted a number of choices of population distributions. Some illustrations are accompanied with data analyses.

A new formulation of minimum risk fixed-width confidence interval (MRFWCI) estimation problems for a normal mean with illustrations and simulations: Applications to air quality data

Research on classical fixed-width confidence interval (FWCI) estimation problems for a normal mean when the variance remains unknown have steadily moved along under a zero-one loss function. On the other hand, minimum risk point estimation (MRPE) problems have grown largely under a squared error loss function plus sampling cost. However, the FWCI problems customarily do not take into account any sampling cost in their formulations. This fundamental difference between the two treatments has led the literature on the FWCI and MRPE problems to grow in multiple directions in their own separate ways from one another. In this article, we introduce a new formulation combining both MRPE and FWCI methodologies with desired asymptotic first-order (Theorems 2.1–2.2) and asymptotic second-order characteristics (Theorem 2.3) under a single unified structure, allowing us to develop a genuine minimum risk fixed-width confidence interval (MRFWCI) estimation strategy. Fruitful ideas are proposed by incorporating illustrations from purely sequential and other multistage MRFWCI problems with an explicit presence of a cost function incurred due to sampling. We supplement the general theory and methodology by means of illustrations and analyses from simulated data along with applications to air quality data.

Multistage Estimation of the Rayleigh Distribution Variance

In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results.

A double-sequential sampling scheme

In this article, we have developed a new sequential sampling scheme, called the double-sequential sampling scheme, in which sampling is proceeded k-at-a-time () part of way, and then one-at-a-time sequentially. Requiring the same projected sample size as the Anscombe-Chow-Robbins (ACR) scheme, it further helps to save sampling operations. We include the statistical inference problems of fixed-width confidence interval estimation and minimum risk point estimation for a normal mean when the population variance also remains unknown as two illustrations. These are followed by extensive sets of simulations and real-data illustrations using horticulture data, showing great practical applicability of this new sequential sampling scheme.

Purely sequential estimation problems for the mean of a normal population by sampling in groups under permutations within each group and illustrations

Purely sequential estimation for unknown mean () in a normal population having an unknown variance () when observations are gathered in groups has been recently discussed in Mukhopadhyay and Wang (2020). In this article, we briefly revisit two fundamental problems on sequential estimation: (i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem. However, we substitute the estimators defining the stopping boundaries with newly constructed unbiased and consistent estimators under permutations within each group. These new estimators incorporated in the definition of the stopping boundaries have led to tighter estimation of requisite optimal fixed sample sizes. We have analyzed the first-order and second-order asymptotic properties under appropriate requirements on the pilot size. Large-scale computer simulations and substantial data analysis have validated such first-order and second-order results. The methodologies are illustrated with the help of time series data on offshore wind energy.

On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) strategies for a normal mean and optimality

We develop a generalized class of purely sequential sampling strategies associated with both fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) problems for the unknown mean $$\mu $$ of a normally distributed population having its variance $$\sigma ^{2}$$ also unknown. Under this newly proposed general class of associated estimation strategies, we develop a variety of asymptotic first-order and asymptotic second-order properties such as asymptotic consistency, first-order efficiency, first-order risk efficiency, second-order efficiency, and second-order regret analysis. Next, we proceed to locate an optimal strategy within our newly built large class of possibilities. Such optimality is defined as having been associated with the minimal second-order asymptotic variance of a stopping time within the general class of proposed strategies. We follow through by exploring both the FWCI and MRPE problems with the help of data analysis from simulations.

Improving Hall’s Accelerated Sequential Procedure: Generalized Multistage Fixed-Width Confidence Intervals for a Normal Mean

We have developed an innovative generalized multistage fixed-width confidence interval (FWCI) estimation procedure for an unknown normal mean when the variance also remains unknown, which can be viewed as a mixture of sequential sampling and batch sampling procedures. Requiring the same expected sample size as Hall’s accelerated sequential sampling procedure, it enables the number of sampling operations to be further reduced by a predetermined factor. Extensive sets of simulations are provided to highlight the remarkable performance of the new multistage FWCI estimation procedure, and real data illustrations are included to demonstrate the practical applicability.

Purely sequential FWCI and MRPE problems for the mean of a normal population by sampling in groups with illustrations using breast cancer data

Two fundamental problems on purely sequential estimation are revisited—(i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem—in the context of estimating an unknown mean (μ) in a normal population having an unknown variance (). We begin by laying down general frameworks for the second-order asymptotic analyses, in both problems, under sequential sampling of one observation at a time. Then, instead of gathering one observation at a time, we consider sequentially sampling k observations at a time in defining our proposed estimation strategies. We replace the customary sample standard deviation as an estimator for σ with a number of other pertinent estimators to come up with new and more appropriate stopping rules to suit the occasion. We do so because in real life we know that packaged items purchased in bulk often cost less per unit sample than the cost of an individual item. This article builds the whole array of estimation methodologies in order to address both FWCI and MRPE problems with appropriate first-order and second-order asymptotic analyses. These are followed by extensive sets of carefully laid out data analyses assisted via large-scale computer simulations. These are wrapped up with illustrations using breast cancer data.

Two-stage fixed-width and bounded-width confidence interval estimation methodologies for the common correlation in an equi-correlated multivariate normal distribution

In this article, two-stage and fixed sample size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. Two different approaches for estimation are considered, namely, fixed-width and bounded-width interval estimation. In the fixed-width confidence interval estimation problem, a two-stage procedure is developed and the exact distribution of the corresponding stopping variable and the exact coverage probability of the interval estimator are derived. Asymptotic optimality properties such as asymptotic first-order efficiency and asymptotic consistency properties of the two-stage procedure are established. Bounded-width confidence intervals are obtained by applying fixed-accuracy estimation methodologies of Mukhopadhyay and Banerjee (2014, 2015a,b) and Banerjee and Mukhopadhyay (2016) using different transformations on ρ. Both fixed sample size and two-stage sampling methodologies for bounded-width confidence interval estimation of are developed incorporating such transformations. Finally, performances of all the procedures are compared via extensive sets of simulation studies.

Three-stage confidence intervals for a linear combination of locations of two negative exponential distributions

Mukhopadhyay and Padmanabhan (Metrika 40:121–128, 1993) considered the construction of fixed-width confidence intervals for the difference of location parameters of two negative exponential distributions via triple sampling when the scale parameters are unknown and unequal. Under the same setting, this paper deals with the problem of fixed-width confidence interval estimation for a linear combination of location parameters, using the above mentioned three-stage procedure.

Confidence intervals and point estimators for a normal mean under purely sequential strategies involving Gini’s mean difference and mean absolute deviation

ABSTRACTWe have developed purely sequential methodologies for problems associated with both fixed-width confidence interval estimation and minimum risk point estimation for the normal mean μ when the variance σ2 is assumed unknown. New stopping rules are constructed by replacing the sample variance with appropriate multiples of Gini’s mean difference (GMD) and mean absolute deviation (MAD) in defining the conditions for boundary crossing. A number of asymptotic first-order consistency, efficiency, and risk efficiency properties associated with these new estimation strategies have been investigated. These are followed by summaries obtained from extensive sets of simulations by drawing samples from (i) normal universes or (ii) mixture-normal universes where samples may be reasonably treated as observations from a normal universe in a large majority of simulations. We also include illustrations using sales data and horticulture data. Overall, we empirically feel confident that our newly developed GMD-based or MAD-based methodologies are more robust for practical purposes when we compare them with the sample variance–based methodologies respectively, especially when up to 20% suspect outliers may be expected.

FixedWidth Confidence Interval of P(X < Y) under a Data Dependent Adaptive Allocation Design

The present article is related to a nonparametric fixed-width confidence interval estimation of the parameter µ = P(X < Y ) = R F(y)dG(y), where F and G are two unknown continuous distribution functions. The estimation procedure is based on a sample obtained under some non-iid adaptive situation. We provide various asymptotic results related to the proposed procedure and compare it with a non-adaptive procedure.

Sequential and Two-Stage Fixed-Width Confidence Interval Estimation in Response-Adaptive Designs

Let be a sequence of consistent estimators of a parameter Δ in some response-adaptive design, where the data are not independent and identically distributed. A fixed-width confidence interval estimation involves determining a positive integer ν such that, for a given d (> 0) and α ∈ (0, 1), the coverage probability of the random interval for Δ is at least (1 − α). In this article, we first investigate such confidence intervals for a general structure of response-adaptive allocation design, for two-treatment allocation in clinical trials, in a complete sequential setup. We consider some examples in this context. We then carry out this exercise for a Stein-type two-stage procedure, where the parameters are estimated based on the first-stage data, which are then used for the allocation in the second stage. Fixed-width confidence intervals are obtained in this context. The procedures are evaluated by simulations. We then illustrate the proposed methodology using some real data set.

Two-Stage Procedure Having Exact Third-Order Properties for a Normal Mean

We consider the problem of fixed-width confidence interval estimation of mean μ of a normal distribution with unknown variance σ2 under an additional assumption that the unknown variance σ2 has a known lower bound . We deal with the two-stage procedure proposed by Mukhopadhyay and Duggan (1997) and provide “exact” third-order approximations to the average sample size and the coverage probability with guaranteeing exact consistency. It turns out that the exact third-order approximation gives explicit relations between the coverage probability and the lower bound σ L and that it is superior to the second-order approximation in accuracy via some simulation results.

Higher order approximations by a two-stage procedure for a negative exponential distribution

This paper deals with the problem of fixed-width confidence interval estimation of the location μ of a negative exponential distribution with unknown scale σ . Suppose we have information on the scale parameter σ such that σ > σ L where σ L ( > 0 ) is known to the experimenter from past experiences. We propose a two-stage procedure and provide higher order asymptotic expansions of the expected sample size and the coverage probability.

A two-stage procedure for estimation of the common mean of several normal populations having unequal and unknown variances

The problem of fixed-width confidence interval estimation of the common mean of several normal populations having unknown and unequal variances is considered. A two-stage procedure is developed to tackle the problem and its nice asymptotic properties are studied. Zacks (9) considered sequential procedures to construct fixed-width confidence interval for the common mean of two normal populations assuming one population's variance to be known. Mukhopadhyay and Narayan (6) generalized the results of Zacks (9) when both the variances were unknown. They also considered the problem of constructing fixed-width confidence interval for the common location parameter of two exponential populations having unequal and unknown scale parameters. Costanza, Hamdy and Son (4) dealt with the problem of constructing a confidence interval of pre-assigned width and coverage probability for the common location parameter of several exponential populations, assuming their scale parameters to be unknown and unequal. They established the non- existence of the fixed sample size procedure to deal with this estimation problem. In order to tackle with the problem, along the lines of Stein (8), they developed a 'routine-type' two-stage procedure. For the same estimation problem Chaturvedi, Tiwari and Pandey (2) developed three-stage, accelerated sequential and purely sequential procedures and obtained the associated second-order approximations. In a later communication, Chaturvedi, Tiwari and Pandey (3) considered the problem of bounded risk point estimation of the common location parameter of several exponential distributions and proposed some multistage procedures and derived the second-order approximations. In the present note, we consider the problem of fixed-width confidence interval estimation of the common mean of several normal populations having unknown and possibly unequal variances. In Section 2, we consider the set-up of the estimation problem and establish the failure of the fixed sample size procedure to deal with it. In Section 3, a two-stage procedure is developed for the fixed-width confidence interval estimation problem and its asymptotic properties are obtained.

Fixed Width Confidence Interval of P ( X < Y ) in Partial Sequential Sampling Scheme

The article is related to nonparametric fixed-width confidence interval estimation of the parameter , where F and G are two unknown univariate continuous distribution functions, by adopting a partial sequential sampling scheme. Different asymptotic results associated with the proposed procedures are formulated and examined.

Sequential Estimation for a Functional of the Spectral Density of a Gaussian Stationary Process

Integral functional of the spectral density of stationary process is an important index in time series analysis. In this paper we consider the problem of sequential point and fixed-width confidence interval estimation of an integral functional of the spectral density for Gaussian stationary process. The proposed sequential point estimator is based on the integral functional replaced by the periodogram in place of the spectral density. Then it is shown to be asymptotically risk efficient as the cost per observation tends to zero. Next we provide a sequential interval estimator, which is asymptotically efficient as the width of the interval tends to zero. Finally some numerical studies will be given.