Weibull confidence intervals

Alternative methods and algorithms may be used, such as the model-independent approach to compare similarity limits derived from multi-variate statistical differences (MSD) combined with a 90% confidence interval approach for test and reference batches (21). Model-dependent approaches such as the Weibull function use the comparison of parameters obtained after curve fitting of dissolution profiles. See Chapters 8 and 9 for further discussion of these methods. 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

Fig. 3 Characteristic strength versus (effective) volume in a double logarithmic plot. Shown are test results on specimens of different size. The straight line shows the Weibull extrapolation based on the four point bending test results. The dashed lines are the 90 % confidence intervals of the prediction.

Figure 15. Weibull plot of biaxial flexural strength data, Op tor LIST dental porcelain with and without ion exchange. Dotted lines are 95% confidence interval m is Weibull modulus, Uq is characteristic strength, and 05% is fracture stress at 5% fracture probability. Data from [71]...

Figure 9 Weibull plot of breakdown voltage (showing upper and lower 95% confidence intervals).

One exemplary simple DCD algorithm is shown in Fig. 5. The cumulative probabilities of known coolant damages in the field are looped by the DCD algorithm. The form parameter b of the Weibull-Distribution is the search criterion to detect the crossover between two damage causes. To detect the crossover, the data volume is dissected into useful subsets and fitted by separate Weibull distributions. The search criterion to detect the crossover between two potential damage causes is the comparison of the confidence intervals of form parameters b and bi of two subsequent intervals. The result is the maximum of the possible number of selectable data lots, which implies a potential maximum of damage causes. 

For the sake of brevity, only one plot is provided as an example Fig. 3 presents the superimposition of the nonparametric reliability curve with 95% confidence interval for the TTC subsystem with the Weibull fit determined with the MLE method. 

It is dear that the difference between sample and population may become larger, the smaller a sample is. The minimum sample size of N = 30 as specified in standards represents a compromise between the large cost of specimen preparation and accuracy, although it should be noted that for N = 30 the uncertainty remains relatively large. Figure 12.10 shows the confidence intervals which result from the sampling procedure, for the Weibull modulus and the function (the scatter of the... 

Figure 12.10 Confidence intervals in dependence of sample size (a) fora and (b) forthe Weibull modulus. The confidence intervals forthe characteristic strength depend on the modulus.

With information such as that of Figure 1, an estimate of m can be corrected for bias. If the above example of ten strength measurements yielded a maximum likelihood estimate for m of 10.0, then the unbiased estimate of the median m would be equivalent to 10.0/1.11 = 9.0. Similar methods could be used to calculate the unbiased estimate of m at any probability of occurrence of interest. Unbiased confidence intervals w ould then involve a similar correction at two probabilities. The unbiased 90 percent confidence interval are estimated by calculating the bias correction to the 5 and 95 percent contours as follows 10.0/0.74 = 13.5 10.0/1.83 = 5.46. Therefore the unbiased 90 percent confidence interval on the Weibull modulus would be 5.46 to 13.5. 

Zussman et al. [108] measured the mechanical properties of single carbon nanofibers introduced in subsection 10.4.1. The average bending modulus of nanofibers were 63 7 GPa and the failure stress was between 0.32 and 0.9 GPa, where failure occurring at the gauge region. Carbon fibers were prepared from PAN homopolymers, hence it is not easy to compare with carbon microfibers based on PAN copolymers. They also carried out Weibull-analysis which gave a failure stress of 640 MPa where 63% of nanofibers would break with a confidence interval of 95%. 

Idem, What Are Confidence Bounds 1996-2006, available online at http //www. weibull.com/Life-DataWeb/what are confidence intervals or bounds. htm. 

In many applications, the Weibull analysis is applied to predict the part reliability or unreliability based on limited data with the help of modern computer technology. The limited data will inevitably introduce some statistical uncertainty to the results. Figure 6.11 shows the unreliability as a function of time in a Weibull plot. The six data points reasonably fit on the straight line and verify that the data are a Weibull distribution. The hourglass curves plotted on each side of the Weibull line represent the bounds of 90% confidence intervals for fhis analysis. The width of the intervals depends on the sample size it narrows when more samples are analyzed. 

The most important statistical subjects relevant to reverse engineering are statistical average and statistical reliability. Most statistical averages of material properties such as tensile strength or hardness can be calculated based on their respective normal distributions. However, the Weibull analysis is the most suitable statistical theory for reliability analyses such as fatigue lifing calculation and part life prediction. This chapter will introduce the basic concepts of statistics based on normal distribution, such as probability, confidence level, and interval. It will also discuss the Weibull analysis and reliability prediction. 

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