Point estimation

Point Estimation. The estimator for the mean life parameter 0 is given by... 

Reliability Estimation. Both a point estimate and a confidence interval estimate of product rehabUity can be obtained. Point Estimate. The point estimate of the component rehabUity is given by... 

There are a variety of ways to express absolute QRA results. Absolute frequency results are estimates of the statistical likelihood of an accident occurring. Table 3 contains examples of typical statements of absolute frequency estimates. These estimates for complex system failures are usually synthesized using basic equipment failure and operator error data. Depending upon the availability, specificity, and quality of failure data, the estimates may have considerable statistical uncertainty (e.g., factors of 10 or more because of uncertainties in the input data alone). When reporting single-point estimates or best estimates of the expected frequency of rare events (i.e., events not expected to occur within the operating life of a plant), analysts sometimes provide a measure of the sensitivity of the results arising from data uncertainties. 

If there is a lack of specific, appropriate data for a process facility, there can be considerable uncertainty in a frequency estimate like the one above. When study objectives require absolute risk estimates, it is customary for engineers to want to express their lack of confidence in an estimate by reporting a range estimate (e.g., 90% confidence limits of 8 X 10 per year to 1 X 10 per year) rather than a single-point estimate (e.g., 2 X 10per year). For this reason alone it may be necessary for you to require that an uncertainty analysis be performed. 

Frequency Phase 3 Use Branch Point Estimates to Develop a Ere-quency Estimate for the Accident Scenarios. The analysis team may choose to assign frequency values for initiating events and probability values for the branch points of the event trees without drawing fault tree models. These estimates are based on discussions with operating personnel, review of industrial equipment failure databases, and review of human reliability studies. This allows the team to provide initial estimates of scenario frequency and avoids the effort of the detailed analysis (Frequency Phase 4). In many cases, characterizing a few dominant accident scenarios in a layer of protection analysis will provide adequate frequency information. 

In frequentist statistics, by contrast, nuisance parameters are usually treated with point estimates, and inference on the parameter of interest is based on calculations with the nuisance parameter as a constant. This can result in large errors, because there may be considerable uncertainty in the value of the nuisance parameter. 

This equation has been experimentally verified in liquids, and Figure 2 shows that it applies equally well for fluidized solids, provided that G is taken as the flow rate in excess of minimum fluidization requirements. In most practical fluidized beds, bubbles coalesce or break up after formation, but this equation nevertheless gives a useful starting point estimate of bubble size. 

VIEW is the quantification module. All minimal cutsets are stored in the speciiic libraries for the fault trees, supercomponents and sequences. VIEW recalculates the point estimates. It computes and displays the Fussel-Vesely importance, risk increase and risk reduction measures. 

The Systems Module constructs and displays fault trees using EASYFLOW which aic read automatically to generate minimal cutsets that can be transferred, for solution, to SETS. CAFT A. or IRRAS and then transferred to RISKMAN for point estimates and uncertainty analysi,s using Monte Carlo simulations or Latin hypercube. Uncertainty analysis is performed on the systems lev el using a probability quantification model and using Monte Carlo simulations from unavailability distributions. 

Performs a batch solution of PSA functional equations, sequence equations and point estimates using the Big Red Button providing QA by recording analysis steps,... 

Loss of offsite power at nuclear power plants is addressed in EPRI NP-2301, 1982 giving data on the frequency of offsite power loss and subsequent recoveiy at nuclear power plants. Data analysis includes point estimate frequency with confidence limits, assuming a constant rate of occurrence. Recovery time is analyzed with a lognormal distribution for the time to recover. 

The point estimate core damage frequencies for the K-Reactor for both internal and external initiators are given in Table 11.3-6. 

Internal accidents, alone, have 267 sequences with a mean value (2.3E-4) slightly higher than the point estimate. The 5% and 95% confidence values are 1.7E-5 and l.OE-3/ reactor-year, respectively. 

Its unique design suggests several accident scenarios that could not occur at other reactors. For example, failure to supply ECC to 1/16 of the core due to the failure of an ECC inlet valve. On the other hand, some phenomena of concern to other types of reactors seem impossible (e.g., core-concrete interactions). The list of phenomena for consideration came from previous studies, comments of an external review group and from literature review. From this, came the issues selected for the accident progression event tree (APET) according to uncertainty and point estimates. 

A single-number index value representation A point estimate of fatalities/10 exposure hours... 

Our concepts of petroleum reserves and resources and their measurements are changing to reflect the uncertainty associated with these terms. Petroleum reseiwes have been largely calculated deterministically (i.e. single point estimates with the assumption of certainty). In the past decade, reseiwe and resource calculations have incorporated uncertainty into their estimates using probabilistic methodologies. One of the questions now being addressed are such as how certain arc you that the rcsciwcs you estimate arc the actual reseiwes and what is the range of uncertainty associated with that estimate New techniques arc required to address the critical question of how much petroleum we have and under what conditions it can be developed. 

At end, XjYj, if steps do not end at exact point, estimate fraction of vertical step required and report as fractional transfer unit. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Figure 1. Joint 95% posterior probability region— diad fractions. Shimmer bands shown at 95% probability. X, true value , point estimate.

This implies that the diad fraction measurements n and n, are made independently with constant standard deviation 0.05. Figure 3 shows the resulting joint 95% posterior probability region with 95% shimmer bands and point estimates. A second estimate of used here is... 

In both cases the point estimates were close to each other and to those reported by Yamashita et al.. Note that the point... 

Therefore to make meaningful inferences from experiments such as those reported by Yamashita et al. either the error structure must be known or sufficient data must be provided, preferably in the form of optimally designed replicates. This analysis confirms that it is generally insufficient to evaluate only point estimates. In fact these are secondary to evaluating and reporting joint probability regions. 

Applications of the method to the estimation of reactivity ratios from diad sequence data obtained by NMR indicates that sequence distribution is more informative than composition data. The analysis of the data reported by Yamashita et al. shows that the joint 95% probability region is dependent upon the error structure. Hence this information should be reported and integrated into the analysis of the data. Furthermore reporting only point estimates is generally insufficient and joint probability regions are required. 

Once the model functional form has been decided upon and the experimental data have been collected, a value for the model parameters (point estimation) and a confidence region for this value (interval estimation) must be estimated... 

Our first goal is to retrieve a good approximation of the true value 9 by means of some operation on the sample of observations, the point estimate of 0. ... 

Once we have obtained our point estimate, we can ask ourselves what confidence we place in this estimate, how likely it would be, in real life, that actual parameter values differ from the values we have estimated. 

Parameter Point Estimates, Standard Errors, and Coefficients of Variation ... 

As probabilistic exposure and risk assessment methods are developed and become more frequently used for environmental fate and effects assessment, OPP increasingly needs distributions of environmental fate values rather than single point estimates, and quantitation of error and uncertainty in measurements. Probabilistic models currently being developed by the OPP require distributions of environmental fate and effects parameters either by measurement, extrapolation or a combination of the two. The models predictions will allow regulators to base decisions on the likelihood and magnitude of exposure and effects for a range of conditions which vary both spatially and temporally, rather than in a specific environment under static conditions. This increased need for basic data on environmental fate may increase data collection and drive development of less costly and more precise analytical methods. 

The binding of three concentrations of 125I-labeled iodohydroxybenzylpindolol (IHYP) to membranes from turkey erythrocytes was studied in the absence and presence of a range of sotalol concentrations. Table 5.5 presents the results. Plot the total amount of IHYP bound against log [sotalol] and draw smooth curves by eye through each set of points. Estimate the IC50 for each... 

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