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.     

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.     

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.     

Conditional assessment of the impact of a Hausman pretest on confidence intervals

In the analysis of clustered and longitudinal data, which includes a covariate that varies both between and within clusters, a Hausman pretest is commonly used to decide whether subsequent inference is made using the linear random intercept model or the fixed effects model. We assess the effect of this pretest on the coverage probability and expected length of a confidence interval for the slope, conditional on the observed values of the covariate. This assessment has the advantages that it (i) relates to the values of this covariate at hand, (ii) is valid irrespective of how this covariate is generated, (iii) uses exact finite sample results, and (iv) results in an assessment that is determined by the values of this covariate and only two unknown parameters. For two real data sets, our conditional analysis shows that the confidence interval constructed after a Hausman pretest should not be used.     

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     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

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.     

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.     

Confidence intervals for batch properties based on both sampling and analytical variance

Summary If different random samples are taken from a heterogeneous batch and each sample is analysed once or in replicate, the average result is a point estimation of the true but unknown mean of the batch property concerned. In this paper, formulae are given for the numbers of degrees of freedom, which are necessary for interval estimations based on both the sampling and the analytical variance. It is essential to distinguish between situations in which estimates of the variances are available and in which they are not.     

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.     

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.     

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.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.