Approximate and generalized pivotal quantities for deriving confidence intervals for the offset between two clocks

Development of algorithms that estimate the offset between two clocks has received a lot of attention, with the motivating force being data networking applications that require synchronous communication protocols. Recently, statistical modeling techniques have been used to develop improved estimation algorithms with the focus being obtaining robust estimators in terms of mean squared error. In this paper, we extend the use of statistical modeling techniques to address the construction of confidence intervals for the offset parameter. We consider the case where the distributions of network delays are members of a scale family. Our results include an asymptotic confidence interval and a generalized confidence interval in the sense of [S. Weerahandi, Generalized confidence intervals, Journal of the American Statistical Association 88 (1993) 899–905. Correction in vol. 89, p. 726, 1994]. We compare and contrast the two approaches for obtaining a confidence interval, and illustrate specific applications using exponential, Rayleigh and heavy-tailed Weibull network delays as concrete examples.     

The simultaneous confidence intervals for all distances from the extreme populations for two-parameter exponential populations based on the multiply type II censored samples

Among k independent two-parameter exponential distributions which have the common scale parameter, the lower extreme population (LEP) is the one with the smallest location parameter and the upper extreme population (UEP) is the one with the largest location parameter. Given a multiply type II censored sample from each of these k independent two-parameter exponential distributions, 14 estimators for the unknown location parameters and the common unknown scale parameter are considered. Fourteen simultaneous confidence intervals (SCIs) for all distances from the extreme populations (UEP and LEP) and from the UEP from these k independent exponential distributions under the multiply type II censoring are proposed. The critical values are obtained by the Monte Carlo method. The optimal SCIs among 14 methods are identified based on the criteria of minimum confidence length for various censoring schemes. The subset selection procedures of extreme populations are also proposed and two numerical examples are given for illustration.     

Statistical inference about the shape parameter of the exponential power distribution

After being proposed by Smith & Bain (1975), the exponential power distribution has been discussed by many authors. This paper proposes a simple exact statistical test for the shape parameter of an exponential power distribution, as well as an exact confidence interval for the same parameter. Necessary critical values of the test are given. The method provided in this paper can be used for type II censored data. Comparing this method to the existing approaches, this method requires less calculation or less tables, and is easier to use in practice.     

Statistical estimation of internal available bandwidth

This paper presents a method for using end-to-end available bandwidth measurements in order to estimate available bandwidth on individual internal links. The basic approach is to use a power transform on the observed end-to-end measurements, model the result as a mixture of spatially correlated exponential random variables, carryout estimation by moment methods, then transform back to the original variables to get estimates and confidence intervals for the expected available bandwidth on each link. Because spatial dependence leads to certain parameter confounding, only upper bounds can be found reliably. Simulations with ns2 show that the method can work well and that the assumptions are approximately valid in the examples.     

Simultaneous upper confidence bounds for distances from the best two-parameter exponentlal distribution

Consider k independent exponential distributions possibly with different location parameters and a common scale parameter. If the best population is defined to be the one having the largest mean or equivalently having the largest location parameter, we then derive a set of simultaneous upper confidence bounds for all distances of the means from the largest one. These bounds not only can serve as confidence intervals for all distances from the largest parameter but they also can be used to identify the best population. Relationships to ranking and selection procedures are pointed out. Cases in which scale parameters are known or unknown and samples are complete or type II censored are considered. Tables to implement this procedure are given.     

SHRINKAGE PREDICTION IN THE EXPONENTIAL DISTRIBUTION WITH A PRIOR INTERVAL FOR THE SCALE PARAMETER

The author proposes the best shrinkage predictor of a preassigned dominance level for a future order statistic of an exponential distribution, assuming a prior estimate of the scale parameter is distributed over an interval according to an arbitrary distribution with known mean. Based on a Type II censored sample from this distribution, we predict the future order statistic in another independent sample from the same distribution. The predictor is constructed by incorporating a preliminary confidence interval for the scale parameter and a class of shrinkage predictors constructed here. It improves considerably classical predictors for all values of the scale parameter within its dominance interval containing the confidence interval of a preassigned level.     

Longitudinal network models and permutation-uniform Markov chains

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed here.     

A new class of slash-elliptical distributions

In this paper, we consider a generalization of the modified slash distribution. We define the new family through the quotient between an elliptically distributed random variable and the power of an exponential random variable with parameter equals to 2, both independent. We use the same idea to extend the model for the multivariate case and study general important properties from the resultant family. We perform inference by the method of moments and maximum likelihood. We present a simulation study which indicates satisfactory parameter recovery by using the estimation approaches. Illustrations reveals that it has potential for doing well in real problems.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

On estimation of R=P(Y<X) for exponential distribution under progressive type-II censoring

This paper deals with the estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent exponential random variables, and the data obtained from both distributions are progressively type-II censored. The uniformly minimum variance unbiased estimator and the maximum-likelihood estimator (MLE) are obtained for the stress–strength parameter. Based on the exact distribution of the MLE of R, an exact confidence interval of R has been obtained. Bayes estimate of R and the associated credible interval are also obtained under the assumption of independent inverse gamma priors. An extensive computer simulation is used to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose.     

COMPARISONS AND SELECTION OF TWO-PARAMETER EXPONENTIAL POPULATIONS

For two-parameter exponential populations with the same scale parameter (known or unknown) comparisons are made between the location parameters. This is done by constructing confidence intervals, which can then be used for selection procedures. Comparisons are made with a control, and with the (unknown) “best” or “worst” population. Emphasis is laid on finding approximations to the confidence so that calculations are simple and tables are not necessary. (Since we consider unequal sample sizes, tables for exact values would need to be extensive.)     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

On lower confidence bounds for pcs in truncated location parameter models

We are concerned with deriving lower confidence bounds for the probability of a correct selection in truncated location-parameter models. Two cases are considered according to whether the scale parameter is known or unknown. For each case, a lower confidence bound for the difference between the best and the second best is obtained. These lower confidence bounds are used to construct lower confidence bounds for the probability of a correct selection. The results are then applied to the problem of seleting the best exponential populationhaving the largest truncated location-parameter. Useful tables are provided for implementing the proposed methods.     

Confidence and Likelihood*

Confidence intervals for a single parameter are spanned by quantiles of a confidence distribution, and one‐sided p‐values are cumulative confidences. Confidence distributions are thus a unifying format for representing frequentist inference for a single parameter. The confidence distribution, which depends on data, is exact (unbiased) when its cumulative distribution function evaluated at the true parameter is uniformly distributed over the unit interval. A new version of the Neyman–Pearson lemma is given, showing that the confidence distribution based on the natural statistic in exponential models with continuous data is less dispersed than all other confidence distributions, regardless of how dispersion is measured. Approximations are necessary for discrete data, and also in many models with nuisance parameters. Approximate pivots might then be useful. A pivot based on a scalar statistic determines a likelihood in the parameter of interest along with a confidence distribution. This proper likelihood is reduced of all nuisance parameters, and is appropriate for meta‐analysis and updating of information. The reduced likelihood is generally different from the confidence density. Confidence distributions and reduced likelihoods are rooted in Fisher–Neyman statistics. This frequentist methodology has many of the Bayesian attractions, and the two approaches are briefly compared. Concepts, methods and techniques of this brand of Fisher–Neyman statistics are presented. Asymptotics and bootstrapping are used to find pivots and their distributions, and hence reduced likelihoods and confidence distributions. A simple form of inverting bootstrap distributions to approximate pivots of the abc type is proposed. Our material is illustrated in a number of examples and in an application to multiple capture data for bowhead whales.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

Comparison of conditional and unconditional confidence intervals for the double exponential distribution

Conditional confidence intervals for the location parameter of the double exponential distribution based on maximum likelihood estimators conditioned on a set of ancillary statistics and the corresponding unconditional confidence intervals based on the maximum likelihood estimators alone are compared in two ways. Monte Carlo techniques are used and the conditional approach appears to give slightly better results although agreement as n becomes larger is noted     

A simple and accurate inference procedure for the difference of failure intensities from two exponential distributions

A third order asymptotic method is proposed to obtain approximate observed levels of significance and confidence intervals for the difference of failure intensities from two independent exponential distributions. The proposed method is extremely accurate even for very small sample sizes and it is simple to use. Numerical examples are used to illustrate the difference in accuracy between the proposed method and methods discussed in Bain, Shiue & Engelhardt (1993) and Yu & Burdick (1994), Extensions to censored samples and cost considerations are also discussed.     

End-to-end queuing delay assessment in multi-service ip networks

Packet-based networks are more and more used to transport interactive streaming services like telephony and videophony. To guarantee a good quality for these services, the queuing delay and delay jitter introduced in the transport of voice or video flows over the packet-based network should be kept under control. Because data sources tend to increase their sending rate until (a part of) the network is congested, mixing real-time traffic and data traffic in one queue would lead to unacceptable high delays for real-time services. Therefore, voice and video packets need to get preferential treatment ( e.g. head-of-line priority) over data packets in the network nodes. Therefore, the queuing behavior of the voice and video packets can be studied more or less independently from the traffic generated by data services. Simple methods to assess the end-to-end delay are primordial. Since it is well known that an aggregate of voice (and CBR video) sources is accurately modeled by a Poisson arrival process and that delays in consecutive nodes are more or less statistically independent, this boils down to developing methods to calculate quantiles of the total queuing delay through a system of N statistically independent M/G/1 nodes. This paper develops four methods to calculate quantiles of the total queuing delay: a Gaussian method, a method based on the numerical inversion of the moment generating function of the total queuing delay developed by Abate and Whitt and two methods based on the assumption that the tail distribution of the individual queuing delay of one node is approximately exponential. The Gaussian method is the simplest, but only gives crude results. The method of Abate and Whitt is the most complex and breaks down for large quantiles. The methods based on the assumption of an exponential tail produce results that are more or less equally accurate as long as there is a node where the load is high enough.     

Conditional maxima and inferences for the truncated exponential distribution

This paper derives the conditional distribution of the maximum given the sample total for a random sample from the truncated exponential distribution. Based on that result, the paper develops tests or associated confidence intervals for the truncation parameter θ with another parameter θ assumed unknown.