Problem Of Confidence Interval Estimation Research Articles

Confidence interval estimation of the common mean of several gamma populations.

Gamma distributions are widely used in applied fields due to its flexibility of accommodating right-skewed data. Although inference methods for a single gamma mean have been well studied, research on the common mean of several gamma populations are sparse. This paper addresses the problem of confidence interval estimation of the common mean of several gamma populations using the concept of generalized inference and the method of variance estimates recovery (MOVER). Simulation studies demonstrate that several proposed approaches can provide confidence intervals with satisfying coverage probabilities even at small sample sizes. The proposed methods are illustrated using two examples.

Influence function-based empirical likelihood and generalized confidence intervals for the Lorenz curve

This paper aims to solve confidence interval estimation problems for the Lorenz curve. First, we propose new nonparametric confidence intervals using the influence function-based empirical likelihood method. We show that the limiting distributions of the empirical log-likelihood ratio statistics for the Lorenz ordinates are standard chi-square distributions. We also develop “exact” parametric intervals for the Lorenz ordinate based on generalized pivotal quantities when the underlying income distribution is a Pareto distribution or a Lognormal distribution. Extensive simulation studies are conducted to evaluate the finite sample performances of the proposed methods. Finally, we apply our methods to a real income dataset.

INFERNO: Inference-Aware Neural Optimisation

Complex computer simulations are commonly required for accurate data modelling in many scientific disciplines, making statistical inference challenging due to the intractability of the likelihood evaluation for the observed data. Furthermore, sometimes one is interested on inference drawn over a subset of the generative model parameters while taking into account model uncertainty or misspecification on the remaining nuisance parameters. In this work, we show how non-linear summary statistics can be constructed by minimising inference-motivated losses via stochastic gradient descent such that they provide the smallest uncertainty for the parameters of interest. As a use case, the problem of confidence interval estimation for the mixture coefficient in a multi-dimensional two-component mixture model (i.e. signal vs background) is considered, where the proposed technique clearly outperforms summary statistics based on probabilistic classification, a commonly used alternative which does not account for the presence of nuisance parameters.

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.

Purely Sequential and Two-Stage Bounded-Length Confidence Interval Estimation Problems in Fisher’s “Nile” Example

Purely Sequential and Two-Stage Bounded-Length Confidence Interval Estimation Problems in Fisher’s “Nile” Example

On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher’s “Nile” example

ABSTRACTFisher’s “Nile” example is a classic that involves a bivariate random variable (X,Y) having a joint probability density function given by f(x,y;𝜃) = exp(−𝜃x−𝜃−1y), 0<x,y<∞, where 𝜃>0 is a single unknown parameter. We develop (i) fixed-width and (ii) fixed-accuracy confidence intervals for 𝜃 with a preassigned confidence coefficient. In problem (i), we develop a purely sequential estimation strategy along with its asymptotic properties. In problem (ii), we determine that a fixed-sample-size estimation strategy will suffice and yet the requisite sample size would have to be found. We have done that both exactly as well as approximately and we report that for all practical purposes the approximations nearly provide the exact sample size whether it is small, moderate, or large. The last problem we address is bounded-accuracy fixed-sample-size estimation of P𝜃{X>Y}. All theoretical properties are adequately validated by large-scale simulations.

On SPRT and RSPRT for the Unknown Mean in a Normal Distribution with Equal Mean and Variance

We consider a sequence of independent observations from a N(θ, θ) distribution with 0 < θ, c < ∞. Here, θ remains unknown. Given independent observations X 1,…, X n from N(θ, θ), the statistic is complete and sufficient for θ and T has a noncentral with λ =nθ when θ is the true parameter value. The problem is one of testing H 0: θ = θ0 vs. H 1: θ = θ1 where θ0, θ1 (θ0 ≠ θ1) are specified positive numbers with target type-I and type-II error probabilities 0 < α <1 and 0 < β <1, respectively, α + β <1. We first adapt Wald's sequential probability ratio test (SPRT) and describe some crucial characteristics. However, observations may instead arrive with random group sizes sequentially rather than one at a time. Thus, as a follow-up, next we implement Mukhopadhyay and de Silva's (2008) notion of a random SPRT (RSPRT). In the case of both SPRT and RSPRT, we address the issue of truncation. The proposed methodologies are illustrated with appropriate statistical data analysis. In the end, we briefly explore the problem of confidence interval estimation for the parameter θ after termination of the SPRT.

Bootstrapping spectra: Methods, comparisons and application to knock data

The problem of confidence interval estimation for spectra is addressed. Unlike asymptotic techniques, bootstrap techniques provide accurate measures of confidence in that they maintain the preset level, in particular for small sample sizes and non-Gaussian data. We investigate some recently proposed methods, a time domain bootstrap approach, the so-called tapered block bootstrap and a combined time domain and frequency domain approach, the so-called autoregressive-aided periodogram bootstrap. We compare the methods with a well-established frequency domain residual based bootstrap technique in view of confidence accuracy. Confidence interval estimates for spectra of knock data are shown.

Confidence interval estimation of partial area under curve based on combined biomarkers

In diagnostic studies, we often need to combine several markers to increase the diagnostic accuracy. This paper addresses the problem of confidence interval estimation of partial area under receiver-operating characteristic (ROC) curve for the combined marker based on the optimal linear combination proposed by Su and Liu (1993). The proposed approach is developed using the concepts of generalized inference. Numerical study demonstrates that the proposed approach generally can provide reasonable confidence intervals via a few straightforward simulation steps.

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.

Geometric approach for finding HPD-credible sets with applications

An algorithm, based on computational geometry, for computing optimum Bayes credible intervals is presented. A credible interval is the Bayesian version of a classical confidence interval estimate, except it does not require the existence of a pivotal quantity, and it is easier to interpret than the classical confidence interval. For the problem of confidence interval estimation, the standard technique in applied statistics is the use of equal-tailed confidence intervals. The algorithm has been implemented in Java and its performance is tested using Monte Carlo simulation on environmental data from Superfund sites. The results show that the credible interval sets generated by the proposed algorithm is significantly shorter than the interval computed by the standard equal-tailed method.

On a Two-Stage Procedure Having Second-Order Properties with Applications

Under a fairly general setup, we first modify the Stein-type two-stage methodology in order to incorporate some partial information in the form of a known and positive lower bound for the otherwise unknown nuisance parameter, 0(> 0). This revised methodology is then shown to enjoy various customary second-order properties and expansions for functions of the associated stopping variable, under appropriate conditions. Such general machineries are later applied in different types of estimation as well as selection and ranking problems, giving a sense of a very broad spectrum of possibilities. This constitutes natural extensions of these authors' earlier paper (Mukhopadhyay and Duggan (1997a, Sankhya Ser. A, 59, 435 448)) on the fixed-width confidence interval estimation problem exclusively for the mean of a normal distribution having an unknown variance.

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.

Interval estimation of the marginal probability of success for the beta-binomial distribution

Correlated binomial outcomes occur in many areas such as teratology and marketing. We study the problem of confidence interval estimation of the common marginal probability of success for correlated binary observations, focusing on the beta-binomial model. New conservative small sample confidence intervals are proposed for this problem based on extensions of the Tail and Sterne–Crow intervals for independent binomial responses. Some properties of these intervals are derived, comparisons with previously proposed methods are given, and recommendations regarding their use are made.

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.

Importance Sampling for Bootstrap Confidence Intervals

Abstract The use of importance-sampling methods to substantially reduce the amount of resampling necessary for the construction of nonparametric bootstrap confidence intervals is investigated. The classical importance-sampling method of Hammersley and Handscomb (1964) is modified to apply to the estimation of quantiles. Based on this method, a resampling procedure is introduced that recenters the bootstrap distribution of a robust estimator of location to produce estimated confidence limits that are much more accurate (for a given amount of resampling) than those obtained by the usual bootstrap method (Efron 1982). The required recentering is accomplished with a suitable “exponential tilting” similar to that used in another connection by Field and Hampel (1982). This importance-sampling procedure is used to produce bootstrap confidence intervals for location based on a class of estimators that includes symmetric M estimators. These interval estimates are asymptotically optimal in a certain sense. Simulation results for 95% confidence intervals based on a particular robust one-step (from the median) M estimator and its studentized version are presented for sample size 20 for the normal and the normal/uniform (slash) distributions. Studentization is accomplished by dividing the location estimator by the sample analog of its asymptotic standard deviation. The importance-sampling results obtained for bootstrap replication sizes 10, 20, and 40 are compared with the standard bootstrap confidence-interval method based on 400 replications. The simulation size for each case is 10,000. These results suggest that the bootstrap replication size may be reduced by as much as a factor of 10 or more when importance sampling is used. The proposed confidence-interval procedures provide alternatives to those given by Gross (1976), Davison and Hinkley (1986), and DiCiccio and Tibshirani (1987). It seems likely that the general approach proposed in this article could be useful in a variety of other confidence-interval estimation problems.

Three-stage estimation procedures for the negative exponential distributions

Fixed-width confidence interval estimation problems for location parameters of negative exponential populations have been studied. Three-stage sampling procedures have been developed for both the one- and two-sample situations. Our discussions are primarily concerned with second-order expansions of various characteristics of the proposed procedures including those for the achieved coverage probability in either problem. Some simulated results are also presented to indicate the usefulness of our procedures for moderate sample sizes.

Jackknife Estimates of Component and System Availability

The jackknife method is applied to the problem of confidence interval estimation of the long run availability of an individual component or system, and of redundant systems of several components. The method is also applied to some actual data concerning nuclear plants. It is further validated by simulation for selected examples.

Confidence intervals for ranked means

Suppose that observations from populations π1, …, πk (k ≥ 1) are normally distributed with unknown means μ1., μk, respectively, and a common known variance σ2. Let μ[1] μ … ≤ μ[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π1 by Xi (i = 1, …, k), and define the ranked sample means X[1] ≤ … ≤ X[k]. The problem of confidence interval estimation of μ(1), …,μ[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ[i] with minimal probability of coverage γ is (− ∞, X[i]+ h) with h = (σ/n1/2) Φ−1(γ1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ[i] with minimal probability of coverage γ is (Xi[i] – g, + ∞) with g = (σ/n1/2) Φ−1(γ1/i). For the upper confidence interval on μ[i] the maximal probability of coverage is 1– [1 – γ1/k-i+1]i, while for the lower confidence interval on μ[i] the maximal probability of coverage is 1–[1– γ1/i] k-i+1. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X1, …, Xk) for μ[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ[1], …,μ[k]), does not exist.

Interval Estimation in Linear Regression When Both Variables are Subject to Error

Abstract Define the linear functional relationship with parameters, α, β, by μ = α + βv. This paper considers the problem of confidence interval estimation of α and β separately, based on a sample of independent pairs, (x i , y i ), i = 1, 2, ···, n, with Ex i = vi, Ey i = μ i . The pairs, (x i , y i ), are assumed to be independent (i ≠ i′), to have a common (unknown) covariance matrix Σ, and to follow the bivariate normal distribution. It is known that under these circumstances (providing instrumental variates are available) one can define a variance ratio with 2 and (n − 2) degrees of freedom, F 2,n–2 say, as a function of α, β, sample statistics and instrumental variates to yield a confidence region on (α, β) based on In this paper we show, for interval estimation of α only, that taking limits as (min α, max α) satisfying yields confidence at least (1–γ). A similar result holds for interval estimation of β only. In the latter case, the limits obtained differ from those derived by the well-known proced...