Showing papers on "Coverage probability published in 1985"

On the Number of Bootstrap Simulations Required to Construct a Confidence Interval

Abstract: : The purpose of this document is to make two points about the effect of the number of bootstrap simulations, B, on percentile-t bootstrap confidence intervals. The first point concerns coverage probability; the second, distance of the simulated critical point form the true critical point derived with B=infinity. In both cases the author has in mind applications to smooth statistics, such as the Studentized mean of a sample drawn from a continuous distribution. He indicates the change that have to be made if the distribution of the statistic is not smooth. Additional keywords: Exponentive functions.

Repeated confidence intervals for the median survival time

Abstract: SUMMARY We describe methods for the construction of a confidence interval for median survival time based on right-censored data. These methods are extended to the construction of repeated confidence intervals for the median, based on accumulating data; here, the overall probability that all intervals contain the true median is guaranteed at a fixed level. The use of repeated confidence intervals for median survival time in post- marketing surveillance is discussed. A confidence interval for median survival time provides a useful summary of the survival experience of a group of patients. If confidence intervals are calculated repeatedly, as data accumulates, the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval, and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly. Jennison & Turnbull (1984) have proposed methods for calculating repeated confidence intervals appropriate to such situations. Similar ideas have also been discussed by Lai (1984). In ? 2 we propose a new form of single-sample nonparametric confidence interval; this interval has asymptotically correct coverage probability and Monte Carlo simulations suggest it is superior to its competitors for small sample sizes. Repeated confidence intervals for the median are presented in ? 3 and their stnall sample size performance is assessed by Monte Carlo simulation; an example of their use is given in ? 4. All the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time.

Exact Simultaneous Confidence Intervals for Pairwise Comparisons of Three Normal Means

Abstract: The problem of simultaneously estimating the pairwise differences of means of three independent normal populations with equal variances is considered A computational method involving a bivariate t density is used to form confidence intervals with simultaneous coverage probability equal to 1 — a For equal sample sizes, the method is the Tukey studentized range procedure With unequal sample sizes, the method is superior to the various generalized Tukey methods A table of probability points is presented for small sample sizes A large-sample approximation based on the bivariate normal and studentized range distributions is given

Asymptotic results for stopping times based on u-statistics

Abstract: This paper considers a general stopping time based on U-statistics. The purpose is todevelop a general setting that will apply to many situations where the family ofdistributions from which we sample, is known apart from some unknown parameters. Our interest is in studying asymptotic properties of the stopping time N. First, some convergence results are given where the limiting normality of standardized N is stated in the main theorem in that section (Theorem 2.3). Next, we give the rate of convergence for these results under some additional moment conditions (Theorem 2.9). These results are then utilized for the problems of estimating the mean of a U-statistic by a fixed-width confidence interval of length 2d. The convergence rate of the coverage probability to a pre-determined confidence coefficient is shown (Theorem 3.1) to be of order Some Examples are given where sampling is carried out from a Bernoulli(p),Poisson(δ) or Gamma(δ, β) population.

Sequential Nonparametric Fixed-Width Confidence Intervals for $U$-Statistics

Abstract: A sequential fixed-width confidence interval for the mean of a $U$-statistic, having coverage probability approximately equal to preassigned $\alpha$, is presented. The main result, Theorem 2, shows that the sequential procedure is asymptotically efficient in the sense of Chow and Robbins (1965) and assumes only finiteness of the second moment of the kernel, the weakest possible condition. The paper follows naturally from Sproule (1974) and Sproule (1969), the primary reference.

Modified Confidence Intervals for the Mean of an Autoregressive Process.

Abstract: : The author's motivation is to find asymptotically more accurate confidence intervals for the steady state mean of a simulated process. By this he means that the coverage probability error for the confidence intervals we derive should be of lower order than that of standard confidence intervals. There are several standard methods of setting confidence intervals in simulations, including the regenerative method, batch means, and time series methods. He focuses on improved confidence intervals for the mean of an autoregressive process, and as such our results are useful outside of a simulation setting. Additional keywords: time series analysis; Cornish-Fisher expansions; Edgeworth expansions.

On the corporate of the coverage probability of a sequential confidence interval based on the sign test

Abstract: The convergence rate of the remainder term of the coverage probability of a sequential confidence interval based on the sign test is studied as the prescribed length of the interval tends to zero. Since the coverage probability can be expressed in terms of partial sums of symmetric Bernoulli random variables, it is possible to use and adapt a Berry–Esseen type theorem for a stochastic number of summands in order to obtain an explicit asymptotic bound for the remainder term.

A computational algorithm for the coverage probability of a first order autogressive process

Abstract: In this paper, an algorithm is found to compute for an AR(1) time series Xt and k given intervals . The computational time is approximately linear to k and the error bound is of the order k 2. This algorithm can be used to find the extreme value distribution, the first passage time distribution, and the coverage probability for the prediction intervals.