Comparing approximations of confidence intervals for the product-moment correlation coefficient

Five approximations in computing confidence intervals for the product-moment correlation coefficient are compared. The α-levels of the approximations are compared with three fixed nominal levels; for varying sample sizes and sample correlation values. It is recommended to use a method of R UBFN (or K RAEMER ) [section 3d (3e)] for accurate approximations and a method of S AMIUDDIN [section 3b] for somewhat less accurate but quick results.     

Distribution–free confidence intervals for the difference between quantiles

An asymptotically distribution–free confidence interval for the difference of the p –th quantiles of two distributions was presented by ALBERS and LOHNBERG (1984). A modification of their procedure is presented for use when the sample sizes are specified.     

Generalized confidence intervals for the process capability index C pm

Process capability indices have been proposed to the manufacturing industry for measuring process reproduction capability. The C _pm index takes into account the degree of process targeting (centering), which essentially measures process performance based on average process loss. To properly and accurately estimate the capability index, numerous conventional approaches have been proposed to obtain lower limits of the classical confidence intervals (CLCLs) for providing process capability information. In particular, lower confidence limits (LCLs) not only provide critical information regarding process performance but are used to determine if an improvement was made in reducing the nonconforming percent and the process expected loss. However, the conventional approach lacks for exact confidence intervals for C _pm involving unknown parameters which is a notable shortcoming. To remedy this, the method of generalized confidence intervals (GCIs) is proposed as an extension of classical confidence intervals (CCIs). For evaluating practical applications, two lower limits of generalized confidence intervals (GLCLs) for C _pm using generalized pivotal quantities (GPQs) are considered, (i) to assess the minimum performance of one manufacturing process/one supplier, and (ii) to assess the smallest performance of several manufacturing processes/several suppliers for equal as well as unequal process variances.     

Asymptotic confidence intervals for the length of the shortt under random censoring

A short t of a one dimensional probability distribution is defined to be an interval which has at least probability t and minimal length. The length of a show and its obvious estimator are significant measures of scale of a distribution and the corresponding random sample, respectively. In this note a non-parametric asymptotic confidence interval for the length of the (uniqueness is assumed) short t is established in the random censorship from the right model. The estimator of the length of the short t is based on the product-limit (PL) estimator of the unknown distribution function. The proof of the result mainly follows from an appropriate combination of the Glivenko-Cantelli theorem and the functional central limit theorem for the PL estimator.     

Minimax estimators of a normal variance

In the estimation problem of unknown variance of a multivariate normal distribution, a new class of minimax estimators is obtained. It is noted that a sequence of estimators in our class converges to the Stein's truncated estimator. Received: March 1998     

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.     

Generalized confidence intervals for process capability indices in the one-way random model

The method of generalized confidence intervals is proposed as an alternative method for constructing confidence intervals for process capability indices under the one-way random model for balanced as well as unbalanced data. The generalized lower confidence limits and the coverage probabilities for three commonly used capability indices were studied via simulation, separately for balanced and unbalanced cases. Simulation results showed that the generalized confidence interval procedure is quite satisfactory both in the balanced and unbalanced cases. Examples are provided to illustrate the results.     

Joint confidence region for the parameters of pareto distribution

The Pareto distribution is commonly used by economists as a model for the distribution of incomes. Separate confidence intervals or approximate confidence intervals for the parameters of Pareto distribution were discussed by some authors. This paper discusses exact joint confidence region for the parameters of Pareto distribution. The method can be used for both complete samples and type II censored samples.     

An approximation method to derive confidence intervals for quantiles with some applications

Standard jackknife confidence intervals for a quantile Q _y (β) are usually preferred to confidence intervals based on analytical variance estimators due to their operational simplicity. However, the standard jackknife confidence intervals can give undesirable coverage probabilities for small samples sizes and large or small values of β. In this paper confidence intervals for a population quantile based on several existing estimators of a quantile are derived. These intervals are based on an approximation for the cumulative distribution function of a studentized quantile estimator. Confidence intervals are empirically evaluated by using real data and some applications are illustrated. Results derived from simulation studies show that proposed confidence intervals are narrower than confidence intervals based on the standard jackknife technique, which assumes normal approximation. Proposed confidence intervals also achieve coverage probabilities above to their nominal level. This study indicates that the proposed method can be an alternative to the asymptotic confidence intervals, which can be unknown in practice, and the standard jackknife confidence intervals, which can have poor coverage probabilities and give wider intervals.     

The asymmetry of judgemental confidence intervals in time series forecasting

In forecasting a time series, one may be asked to communicate the likely distribution of the future actual value, often expressed as a confidence interval. Whilst the accuracy (calibration) of these intervals has dominated most studies to date, this paper is concerned with other possible characteristics of the intervals. It reports a study in which the prevalence and determinants of the symmetry of judgemental confidence intervals in time series forecasting was examined. Most prior work has assumed that this interval is symmetrically placed around the forecast. However, this study shows that people generally estimate asymmetric confidence intervals where the forecast is not the midpoint of the estimated interval. Many of these intervals are grossly asymmetric. Results indicate that the placement of the forecast in relation to the last actual value of a time series is a major determinant of the direction and size of the asymmetry.     

Asymptotic confidence intervals for the effect of closing an area upon the number of birds' nests

Abstract  The methodological discussion in D e R oos -S chaafsma (1981) Section 6 is continued by proposing asymptotic methods as a substitute for laborious confidence interval computations.     

Shortest confidence intervals and UMVU estimators for families of distributions involving truncation parameters

Summary In this paper, using the pivotal quantity method, new shortest-length confidence intervals and uniformly minimum variance unbiased (UMVU) estimators are constructed, where two independent random samples are available from families of distributions involving truncation parameters. Also, in the case of one sample, we give, for some uniform distributions, confidence intervals which are the shortest among all known confidence intervals.     

A note on three-stage confidence intervals for the difference of locations: The exponential case

Fixed-width confidence intervals for the difference of location parameters of two independent negative exponential distributions are constructed via triple sampling when the scale parameters are unknown and unequal. The present three-stage estimation methodology is put forth because (i) it is operationally more convenient than the existing purely sequential counterpart, and (ii) the three-stage and the purely sequential estimation techniques have fairly similar asymptotic second-order characteristics.     

Testing for equality of variance of correlated normal variables

A standard test exists for whether bivariate normal data of arbitrary correlation have equal variances. An extension of this model is useful to test whether two measuring instruments, with which repeated measurements have been made on each of n units, have equal error variance. It is shown that one or two simple F -distributed statistics yield performance comparable with that of the generalized likelihood ratio statistic.     

A consistent and asymptotically efficient two-stage procedure to construct fixed width confidence intervals for the mean

For a normal distribution, a two-stage procedure has been proposed for constructing a fixed width confidence interval for the mean when the variance is unknown. It has all the properties ofStein's two-stage procedure, and at the same time it is asymptotically efficient. We have also discussed the non-normal case as inChow/Robbins [1965].     

On the accuracy of bootstrap confidence intervals for efficiency levels in stochastic frontier models with panel data

We study the construction of confidence intervals for efficiency levels of individual firms in stochastic frontier models with panel data. The focus is on bootstrapping and related methods. We start with a survey of various versions of the bootstrap. We also propose a simple parametric alternative in which one acts as if the␣identity of the best firm is known. Monte Carlo simulations indicate that the parametric method works better than the␣percentile bootstrap, but not as well as bootstrap methods that make bias corrections. All of these methods are valid␣only for large time-series sample size (T), and correspondingly none of the methods yields very accurate confidence intervals except when T is large enough that the identity of the best firm is clear. We also present empirical results for two well-known data sets.      

Minimax estimation for the bounded mean of a bivariate normal distribution

A bivariate normal distribution is considered whose mean lies in an equilateral triangle. We show by a convexity argument that the three point prior having mass 1/3 at each of the edges is least favourable if the length of a side of the equilateral triangle is less than or equal to . Thus the corresponding Bayes estimator is minimax in that case. Numerical studies are given as well.