Confidence Limits For Reliability Research Articles

Terrestrial physical and chemical processes for liquid waste treatment.

Experiences gained from full-scale evaluation of advanced treatment processes used for reclaiming wastewaters should help in the evaluation of potential treatment systems for treatment and reuse of water in space. Water Factory 21 is a 0.66 m3 s-1 (15 million gallons per day) water reclamation plant in California that has been in operation since 1976. The plant receives biologically treated wastewater. Lime treatment is effective for removal of heavy metals. Volatile organic constitutes are efficiently removed by air stripping. Non-volatile organic constituents are removed by activated carbon adsorption and reverse osmosis (RO). RO is a highly effective polishing step, and removes most of the remaining materials including inorganic salts, heavy metals, and organics. RO removed 85% of the total organic carbon, down to about 1 mg l-1, which is lower than in many treated drinking waters. The series of treatment processes used insured virus and pathogen removal, with lime treatment and chlorination together proving highly effective. Sufficient data has been collected to provide statistically reliable confidence limits to be set on the performance of each unit process.

Confidence limits for reliability and tolerance limits in the inverse Weibull distribution

The inverse Weibull distribution has been recently proposed as a suitable model to describe mechanical degradation phenomena. In this paper the statistical properties of the maximum likelihood estimator, R̂(t), of the inverse Weibull reliability, R(t), are investigated. Tables of lower confidence limits for R(t), for confidence levels of 0·90, 0·95 and 0·98, are also provided. Tolerance limits based upon maximum likelihood estimators of the parameters are derived, and it is shown that the same tables needed for obtaining confidence intervals for the reliability allow one to estimate tolerance limits, too. A numerical example is given to clarify the use of the tables.

Applications of schur theory to reliability inference for k of n systems

The results of Hoeffding (1956), Pledger and Proschan (1971), Gleser (1975) and Boland and Proschan (1983) are used to obtain Buehler (1957) 1-α lower confidence limits for the reliability of k of n systems of independent components when the subsystem data have equal sample sizes and the observed failures satisfy certain conditions. To the best of our knowledge, for k ≠ 1 or n, this is the first time the exact optimal lower confidence limits for system reliability have been given. The observed failure vectors are a generalization of key test results for k of n systems, k ≠ n (Soms (1984) and Winterbottom (1974)). Two examples applying the above theory are also given.

Estimating lower confidence limits on system reliability using a Monte-Carlo technique on binomial data

It is known that the use of a Bayes method for computing lower confidence limits for system reliability leads to conservative limits when using a uniform prior distribution. A method of using the component data to modify the beta prior distribution is proposed. It is seen that such a method leads to an improvement in the lower confidence limit, that is, its conservatism is decreased.

Confidence limits for the reliability of systems with redundancy

We consider systems of serial-parallel elements with time to failure following an exponential reliability law. Assuming a testing plan with a bounded number of failures, we obtain an analytical representation of the exact lower confidence limit for the probability of failfree operation of the system.

Towards a systematic understanding of the effects of velocity model errors on depth and time migration of seismic data

The analysis and interpretation of seismic data takes place in three fundamental seismic domains. These are unmigrated time, which corresponds to normal incidence or stacked seismic data, migrated depth, which corresponds to the physical structure of the earth, and finally migrated time, a non-physical intermediate domain. Simplified models may be derived from the seismic data and transformed between the domains using ray theoretical techniques, specifically image rays and normal rays. This ray-theoretical modelling is extremely cheap and may be used to define the velocity zones for depth or time wave-theoretical migration of the seismic data itself. Furthermore, techniques can be developed to analyse the effects of velocity errors on time and depth migration. The positional error zones that result from velocity uncertainty are complex and often quite surprising, in that they relate to the local geology in a way that is often difficult to imagine intuitively. These velocity uncertainty analyses should be an invaluable aid to the interpreter in determining migration velocity sensitivity and, in particular, in placing reliable confidence limits on the spatial predictions made from both time- and depth-migrated seismic data.

Confidence limits for fleet equipment reliability

A negative binomial model is applied to the occurrence of defects in discrete intervals of time to similar equipment fitted to a fleet. The data may be left censored, and the number of items at risk may vary from one time interval to the next. A likelihood approach is adopted for the estimation of parameter values, and a method is suggested by which confidence limits for the reliability estimate may be calculated.

The Line Transect Method: A Nonparametric Estimator Based on Shape Restrictions

SUMMARY A new estimator for population density based on line transect surveys is presented. Its sampling behaviour is investigated through simulation studies. Advantages of the proposed estimator include its simple, intuitive base, its stable behaviour over a wide variety of detection curves, the ease with which it can be computed, and the availability of reliable confidence limits. The simulation results point to a general improvement in efficiency over existing estimators.

Tables for Exact Lower Confidence Limits for Reliability and Quantiles, Based on Least-Squares Estimators of Weibull Parameters

For the 2-parameter Weibull distribution, this paper gives tables to obtain exact lower s-confidence limits for reliability on the basis of the least-squares method and median plotting positions. The tables use a method based on the ancillary property of these estimators. They apply to samples of size N = 3(1)13, censored after the first m observations, m = 3(1)N. The same tables enable one to obtain lower s-confidence limits for population quantiles. The use of the tables is illustrated with a numerical example.

On the confidence limits for Weibull reliability and quantiles: The case of maximum likelihood estimation from small size censored samples

For the two-parameter Weibull distribution, this paper reviews the existing methods of obtaining the interval estimate for reliability R(t), and gives tables of exact lower confidence limits for R(t) in the case of maximum likelihood estimation. These tables extend the previous ones, since they cover samples of size n = 3(1)20, censored after the first m observations, m = 3(1)n, and confidence levels γ = 0·90, 0·95 and 0·98 . A numerical example illustrates how the same tables can be used to obtain lower confidence limits for population quantiles. Bias and variance of the maximum likelihood estimator R̂(t), for some n and m, are also given.

Overnight urine collections to estimate sodium intake.

This study was undertaken to formulate reliable confidence limits for the relationship between nocturnal and 24-hour urine sodium (Na) excretion for use in population compliance studies. Urine excretions were measured in 12 white and 12 black men at three levels of sodium (Na) intake (10, 200, and 400 mEq/day) for 7 days. Nocturnal Na, chloride (Cl), and Cl determined by titrator stick were all highly correlated (r greater than or equal to 0.86, p less than 0.001) with 24-hour UNaV. No significant difference could be attributed to race. Discriminate analysis revealed that the subjects could be categorized with respect to Na intake with accuracies of 95%, 90%, and 85% (low, middle, and high Na intake respectively) by means of two nocturnal urine Cl titrator stick measurements. In addition, two such measurements showing nocturnal UClV less than 10 mEq indicated with 95% accuracy that 24-hour UNaV was less than 60 mEq. According to these data, measurement of nocturnal Na or Cl excretion is a useful means of assessing compliance to a low sodium intake, and the titrator sticks are a convenient and inexpensive tool.

Mechanical Reliability Confidence Limits

Charts are presented relating the lower one-sided confidence limit on the reliability, RL1, to the effective sample size, ne, calculated from the sample sizes used to estimate the failure governing stress and strength distributions, or f(s) and f(S) respectively, and a factor K which is a function of the estimated means and standard deviations of f(s) and f(S). These graphs cover an ne range of 5 to 1000, confidence levels of 0.80, 0.90, 0.95, and 0.99, and lower one-sided limits on the reliability of 0.85 to 0.9145. The equations used to develop these charts are derived and two examples of their applications are given.

Lower Confidence Limits for Series System Reliability from Binomial Subsystem Data

Abstract An examination is made of some methods for determining lower confidence limits for the reliability of series systems using binomial subsystem test data. Key test results form the basis of a theoretical and numerical study which shows that several approximate methods compare favorably with exact methods.

Lower Confidence Limits for Series System Reliability from Binomial Subsystem Data

Abstract An examination is made of some methods for determining lower confidence limits for the reliability of series systems using binomial subsystem test data. Key test results form the basis of a theoretical and numerical study which shows that several approximate methods compare favorably with exact methods.

Bayesian Confidence Limits for the Reliability of Mixed Cascade and Parallel Independent Exponential Subsystems

This paper deals with the theoretical problem of derving Bayesian confidence intervals for the reliability of a system consisting of both cascade and parallel subsystems where each subsystem is independent and has an exponential failure probability density function (pdf). This approach is applicable when test data are available for each individual subsystem and not for the enfire system. The Mellin integral transform is used to analyze the system in a step-by-step procedure until the posterior pdf of the system reliability is obtained. The posterior cumulative distribution function is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the accuracy of the derived pdf. A computer program has been written in FORTRAN IV to evaluate the confidence limits. An example is presented which uses the computer program.

Reparameterization of Semiempirical Techniques to Remove Anomalous Energy Minima

Abstract Semiempirical molecular orbital calculations have been shown to predict double minima potential surfaces which are not confirmed by further experimental or theoretical evidence.1 These same techniques predict barriers to internal rotation which are not only in large error but may not even follow the experimental trends.2 These discrepancies offer a large decrease in self confidence for the application of these techniques to molecular systems where direct experimental evidence is not available, and these are often the circumstances where one wants a theory to be used. Therefore, our research group is in the process of reparameterizing the procedure INDO3 to yield barriers to internal rotation, bond angles, and bond lengths within reliable confidence limits. During this process of determining new parameters we have observed that some sets of parameters will also remove the anomalous energy minima predicted by the original INDO parameters. To illustrate this we consider in this paper the molecule CO2.

Approximate Confidence Limits for System Reliability

Approximate Confidence Limits for System Reliability

Approximate Confidence Limits for System Reliability

Abstract A method is presented for obtaining system confidence limits based on component test results. The technique consists of estimating the asymptotic variance of the maximum likelihood estimate of system reliability, equating this to the estimate of the variance in binomial sampling, and solving for and , the pseudo-numbers of system tests and successes. These are then substituted into the incomplete beta function and confidence limits obtained in the usual way for binomial sampling.

Confidence Limits for System Reliability: A Sequential Method

A sequential method is given for obtaining confidence limits for system reliability when ``pass-fail'' test data have been obtained on components. Costs of testing are examined and a rule is derived for determining the order in which to test the components so that the average cost of testing is minimized. The method is presented first for series systems, but in Section IV the extension to parallel systems is considered and in Section V the procedure is applied to a series-parallel configuration. Comparisons are given with fixed sample-size testing and it is shown that the sequential method is statistically efficient and that many of the difficulties encountered when using fixed sample sizes are eliminated by testing sequentially.

Bayesian Confidence Limits for the Reliability of Mixed Exponential and Distribution-Free Cascade Subsystems

The problem treated here is the theoretical one of deriving exact Bayesian confidence intervals for the reliability of a system consisting of some independent cascade subsystems with exponential failure probability density functions (pdf) mixed with other independent cascade subsystems whose failure pdf's are unknown. The Mellin integral transform is used to derive the posterior pdf of the system reliability. The posterior cumulative distribution function (cdf) is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the derived pdf. A computer program written in Fortran IV is operational. It utilizes multiprecision to obtain the posterior pdf to any desired degree of accuracy in both functional and tabular form. The posterior cdf is tabulated at any desired increments to any required degree of accuracy.