Weibull determination

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

The above process above could also be performed for the 3-parameter Weibull distribution to compare the correlation coefficients and determine the better fitting distributional model. Computer-based techniques have been devised as part of the approach to support businesses attempting to determine the characterizing distributions... 

Gn L) is often difficult to determine for a given load distribution, but when is large, an approximation is given by the Maximum Extreme Value Type I distribution of the maximum extremes with a scale parameter, 0, and location parameter, v. When the initial loading stress distribution,/(L), is modelled by a Normal, Lognormal, 2-par-ameter Weibull or 3-parameter Weibull distribution, the extremal model parameters can be determined by the equations in Table 4.11. These equations include terms for the number of load applications, n. The extremal model for the loading stress can then be used in the SSI analysis to determine the reliability. 

We ean use a Monte Carlo simulation of the random variables in equation 4.83 to determine the likely mean and standard deviation of the loading stress, assuming that this will be a Normal distribution too. Exeept for the load, F, whieh is modelled by a 2-parameter Weibull distribution, the remaining variables are eharaeterized by the Normal distribution. The 3-parameter Weibull distribution ean be used to model... 

Many distributions can be represented in closed form except for the Normal and Lognormal types. The CDF for these distributions can only be determined numerically. For example, the 3-parameter Weibull distribution s CDF is in closed form, where ... 

Because the Normal distribution is difficult to work with, we can use a 3-parameter Weibull distribution as an approximating model. Given the mean, /i, and standard deviation, cr, for a Normal distribution (assuming [3 = 3.44), the parameters xo and 6 can be determined from ... 

Suppose again that both the stress and strength distributions of interest are of the Normal type, where the loading stress is given as L A (350,40) MPa and the strength distribution is S A (500, 50) MPa. The Normal distribution eannot be used with the integral transform method, but ean be approximated by the 3-parameter Weibull distribution where the CDF is in elosed form. It was determined above that the loading stress parameters for the 3-parameter Weibull distribution were ... 

Repair and maintenance records were analyzed to determine failure rates and distribution of failure modes. Preliminary findings are reported which include the Weibull distribution characteristics. Failure mode distributions are approximate. Overall mean-time-between-failure is given for the kiln, leach tank, screwfeeder, tank pump, tank gearbox, and kiln gearbox. The study was confined to an analysis of unscheduled repairs and failures. 

DuyvesteynW S, ShimoniE and LabuzaTP(2001), Determination of the end of shelf life for milk using Weibull Hazard Method , Food Science and Technology, 34, 143-148. 

As an example of investigations of mechanical effects, may be mentioned that of Weibull(Ref). He detonated charges of expls in air and underwater in order to determine mechanical effects on surrounding media. He found that when the chge was exploded in air, the distribution of impulse around the charge depended on its form, whereas in underwater expins the impulse was distributed in the form of a circle and did not depend on the shape of the chge... 

Fig. 8.6 Estimated risk of liver cancer, P(d), in relation to dose of aflatoxin, d, as determined with different dose-incidence models. The models for the different curves. are as follows OH. one-hit model MS, multi-stage model W, Weibull model MH, multihit model MB, Mantel-Bryan (log-probit model) (from Krewski and Van Ryzin, 1981).

The analysis of covariance between a continuous variable (P is the curve shape parameter from the Weibull function) and a discrete variable (process) was determined using the general linear model (GLM) procedure from the Statistical Analysis System (SAS). The technique of the heterogeneity of slopes showed that there was no significant difference (Tables 5 and 6). 

ISO (2005). Milk Products and Milk-Based Foods—Determination of Fat Content by the Weibull-Berntrop Gravimetric Method (Standard ISO 8262-3/IDF 124-3). International Organization for Standardization, Geneva. 

Statistical models. A number of statistical dose-response extrapolation models have been discussed in the literature (Krewski et al., 1989 Moolgavkar et al., 1999). Most of these models are based on the notion that each individual has his or her own tolerance (absorbed dose that produces no response in an individual), while any dose that exceeds the tolerance will result in a positive response. These tolerances are presumed to vary among individuals in the population, and the assumed absence of a threshold in the dose-response relationship is represented by allowing the minimum tolerance to be zero. Specification of a functional form of the distribution of tolerances in a population determines the shape of the dose-response relationship and, thus, defines a particular statistical model. Several mathematical models have been developed to estimate low-dose responses from data observed at high doses (e.g., Weibull, multi-stage, one-hit). The accuracy of the response estimated by extrapolation at the dose of interest is a function of how accurately the mathematical model describes the true, but unmeasurable, relationship between dose and response at low doses. 

Equation (9.32) is a linear Fredholm integral equation of the first kind. It is also known as an unfolding or deconvolution equation. One can preanalyze the data and try to solve this first-kind integral equation. Besides the complexity of this equation, there is a paucity of numerical methods for determining the unknown function / (h) [208,379] with special emphasis on methods based on the principle of maximum entropy [207,380]. The so-obtained density function may be approximated by several models, gamma, Weibull, Erlang, etc., or by phase-type distributions. 

A statistical analysis of the liber tensile strength values determined on a series of fiber samples can be easily made by using the two-parameter Weibull distribution described above. Using the form of the Weibull expression given in Eq. (10.2), we can write the probability of failure F(o) of the liber at a stress a, as... 

With experimental data for V as a function of 1/N, they were able to determine by inverse Laplace transform, the function fik) and the values of m for the Weibul distribution for Kibushi clay im = 1), TiOg im = 1), and Si02 im = 0.5). Such studies require that each powder have a reproducible initial packing, which can be achieved by fluidization with air. If the tap frequency is too high, consolidation will not take place so frequencies of 1 Hz are often used with an amplitude of tap of 1 cm. For these conditions, it is often found that p is achieved in 1000 to 2000 sec. 

In the future, the BMCqs and MLEqi for lethality will be determined, presented, and discussed. Results from the above models will be compared with the log probit EPA (2000) benchmark dose software (http //www.epa.gov/ncea/ bmds.htm). In all cases, the MLE and BMC at specific response levels will be considered. Other statistical models such as the Weibull may also be considered. Since goodness-of fit-tests consider an average fit, they may not be valid predictors of the fit in the low-exposure region of interest. In this case, the output of the different models will be plotted and compared visually with the experimental data to determine the most appropriate model. The method that results in values consistent with the experimental data and the shape of the exposure-response curve will be selected for AEGL derivations. 

Coppard et al (1989) measured the breakdown field of polyethylene plaques loaded with metallic particles in very small quantities. Their results concern the range of low p. One of the purposes of this study was to determine the distribution probability F E y) to distinguish between the Gumbel and the Weibull distributions. But it appears to be very difficult a task experimentally, because a large number of samples are needed for the study. However, they were able to verify that the average of the breakdown fields varies with p in accordance with (2.68). 

Fig. I a, b. Plots for the determination of uniformity coefficients (reliability function) of phenolic foam samples during tensile tests (a) normal (Gaussian) distribution (b) Weibull s distribution )...

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