On sequential fixed-width confidence intervals for the mean and second-order expansions of the associated coverage probabilities (original) (raw)
Related papers
On Sequential Fixed-Size Confidence Regions for the Mean Vector
Journal of Multivariate Analysis, 1997
In order to construct a fixed-size confidence region for the mean vector of an unknown distribution function F, a new purely sequential sampling strategy is proposed first. For this new procedure, under some regularity conditions on F, the coverage probability is shown (Theorem 2.1) to be at least (1&:)&B: 2 d 2 +o(d 2) as d Ä 0, where (1&:) is the preassigned level of confidence, B is an appropriate functional of F, and 2d is the preassigned diameter of the proposed spherical confidence region for the mean vector of F. An accelerated version of the stopping rule is also provided with the analogous second-order characteristics (Theorem 3.1). In the special case of a p-dimensional normal random variable, analogous purely sequential and accelerated sequential procedures as well as a three-stage procedure are briefly introduced together with their asymptotic second-order characteristics.
A survey of confidence interval formulae for coverage analysis
1998
Confidence interval estimators for proportions using normal approximation have been commonly used for coverage analysis of simulation output even though alternative approximate estimators of confidence intervals for proportions were proposed. This is because the normal approximation was easier to use in practice than the other approximate estimators. Computing technology has no problem with dealing these alternative estimators. Recently, one of the approximation methods for coverage analysis which is based on ...
Upper bounds for coverage probabilities of confidence intervals for nonmonotone parametric functions
Journal of Statistical Planning and Inference, 2000
Consider a statistical model F ∈ F and let  = Â(F) be a structural parameter which admits a (1 − )-level two-sided conÿdence interval based on a random sample taken from F. Let g(Â) be some parametric function of interest. The problem of deriving a conÿdence interval for g(Â) directly from that given on  is considered. If g is one-to-one then a (1 − )-level two-sided conÿdence interval is immediately available. If however, g is not one-to-one the problem becomes more complex. In this paper the situation where g is a nonmonotone function is considered. Under the assumption that g has a unique minimum at x = and that g(x) is strictly decreasing (increasing) for x ¡ (x ¿ ) a two-sided conÿdence interval for g(Â) can be obtained from the (1 − )-level conÿdence interval on  whose conÿdence level, while being at least 1 − , is not greater than 1 − =2. Moreover, if in addition g is symmetric then an improved upper bound, smaller than 1 − =2, can be achieved when F is a location or location and scale distribution. (S.K. Bar-Lev).
2007 IEEE International Workshop on Advanced Methods for Uncertainty Estimation in Measurement, 2007
At the application level, it is important to be able to Using this approach, the possibility distribution expression define around the measurement result an interval which will deduced form the probability distribution involves the contain an important part of the distribution of the measured cumulative distribution function , which is more clumsy values, that is, a coverage interval This practice acknowledged by than the density for the usual probability distributions.
A Coverage Probability Approach to Finding an Optimal Binomial Confidence Procedure
The American Statistician, 2014
The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sterne's procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.
Guaranteed Conservative Fixed Width Confidence Intervals via Monte Carlo Sampling
Springer Proceedings in Mathematics & Statistics, 2013
Monte Carlo methods are used to approximate the means, µ, of random variables Y , whose distributions are not known explicitly. The key idea is that the average of a random sample, Y 1 ,. .. ,Y n , tends to µ as n tends to infinity. This article explores how one can reliably construct a confidence interval for µ with a prescribed half-width (or error tolerance) ε. Our proposed two-stage algorithm assumes that the kurtosis of Y does not exceed some user-specified bound. An initial independent and identically distributed (IID) sample is used to confidently estimate the variance of Y. A Berry-Esseen inequality then makes it possible to determine the size of the IID sample required to construct the desired confidence interval for µ. We discuss the important case where Y = f (X X X) and X X X is a random d-vector with probability density function ρ. In this case µ can be interpreted as the integral R d f (x x x)ρ(x x x) dx x x, and the Monte Carlo method becomes a method for multidimensional cubature.
Sequential fixed-width confidence bands for distribution functions under random censoring
Metrika, 1989
Sequential fixed-width confidence bands for a distribution function are derived in the case, when the data are censored from the right. The Breslow-Crowley invariance principle for the Kaplan Meier estimate is extended to the random sample size situation. Also some simulation resuits are reported, which illustrate the behavior of the stopping times.
Statistics & Probability Letters, 2009
In this paper we continue our investigation connected with the new approach developed in Ahmed et al. [Ahmed, S.E., Saleh, A.K.Md.E., Volodin, A., Volodin, I., 2006. Asymptotic expansion of the coverage probability of James–Stein estimators. Theory Probab. Appl. 51 (4) 1–14] for asymptotic expansion construction of coverage probabilities, for confidence sets centered at James–Stein and positive-part James–Stein estimators. The coverage probabilities for these confidence sets depend on the noncentrality parameter τ2τ2, the same as the risks of these estimators. In this paper we consider only the confidence set centered at the positive-part James–Stein estimator. As is shown in the above-mentioned reference, the new approach provides a method to obtain for the given confidence set, an asymptotic expansion of the coverage probability as one formula for both cases τ→0τ→0 and τ→∞τ→∞. We obtain the third terms of the asymptotic expansion for both mentioned cases, that is, the coefficients at τ2τ2 and τ−2τ−2. Numerical illustrations show that the third term has only a small influence on the accuracy of the asymptotic estimation of coverage probability.
Robustness of Normal-Based Multi-Stage Sequential Sampling Procedures
Multistage sequential procedures have been developed to resolve the drawbacks of fixed sample size procedures,to tackle a variety of inference problems,accomplish predetermined optimal criteria and to make operational savings possible by bulk sampling. Although the theory was developed under the normality assumption, several continuous and discrete distribution parameters were investigated within this optimal statistical decision framework. The current study is designed to answer two main questions. First, how sensitive are these sampling procedures to changes in the underlying distributions from normality ?Second, how do shifts in parameters affect the measures of the quality of inference? With first question, our investigation focuses on some departure from normality.The second question concerns the sensitivity of both coverage probability and the Type II error probability of the fixed width confidence intervals to detect possible shifts in the true parameters