Weibull zeros

Data that is not evenly distributed is better represented by a skewed distribution such as the Lognormal or Weibull distribution. The empirically based Weibull distribution is frequently used to model engineering distributions because it is flexible (Rice, 1997). For example, the Weibull distribution can be used to replace the Normal distribution. Like the Lognormal, the 2-parameter Weibull distribution also has a zero threshold. But with increasing numbers of parameters, statistical models are more flexible as to the distributions that they may represent, and so the 3-parameter Weibull, which includes a minimum expected value, is very adaptable in modelling many types of data. A 3-parameter Lognormal is also available as discussed in Bury (1999). 

A, the fitted Weibull parameters listed in Table 2 and the remaining model parameters as listed in Figure 4 (panel a) or the assumed zero order release rates of 4%, 5%, and 6.7% per hour (panel b). 

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. 

Considering all the above data, the U.S. EPA (1991) selected the unit risk of 8.5 x 10 per pg/m, derived from the Weibull time-to-tumor model, as the recommended upper bound estimate of the carcinogenic potency of sulfur mustard for a lifetime exposure to HD vapors. However, U.S. EPA (1991) stated that "depending on the unknown true shape of the dose-response curve at low doses, actual risks may be anywhere from this upper bound down to zero". The Weibull model was considered to be the most suitable because the exposures used were long-term, the effect of killing the test animals before a full lifetime was adjusted for, and the sample size was the largest obtainable from the McNamara et al. (1975) data. 

The typical absorption profiles are represented by the first-order absorption and the zero-order absorption. Atypical absorption profiles can be described by parallel first-order absorption, mixed first-order and zero order absorption, or Weibull-type absorption. 

Any model can be applied to in vitro dissolution data and fitted by linear or non-linear regression, as appropriate. Sometimes a first-order model [A(t) = A - Ae kt where A(t) is the amount dissolved after time t, A is the initial amount and k is the first-order dissolution rate constant] or even a zero-order model (A(t) = A-Akt) is sufficiently sophisticated to determine a dissolution rate that is representative for the whole process. However, a more general equation that is commonly applied to dissolution data is the Weibull equation (Langenbucher 1976) ... 

Various mathematical models have evolved that attempt to incorporate some of the biological concepts and hypotheses. Some of the most commonly used models are the Probit, the Multi-Hit, the One-Hit, the Multi-Stage, and the Weibull. All of these models have the defined property that for zero dose, the risk is zero. However, since the spontaneous background incidence is not zero for most tumors, the models incorporate background, utilizing the concept of independence as ascribed by the correction of Abbott (14) i.e.,... 

Based on the estimated density functions, the mean and the 95%-quantile are superimposed by the dashed lines in Figure 4-13a and Figure 4.13b whereby the latter coincides with the re-order level ensuring a 95% a-service level. For both sites the density functions of total consumption are bimodal and skewed to the right. The first mode is at zero consumption and is inherited from the Markov chain of the production models. It corresponds to situations when a pipeline inspection coincides with a cracker. shut-down. The second mode is inherited from the Weibull distribution determining the pipeline inspection time which also causes the skewness. 

For the failure times of the controller and sensor, we associate the exponential laws. For failure of the hydraulic motor, a Weibull distribution is used. The repair time of the component is taken to zero. 

A statistical definition of brittleness can be formulated in terms of the Weibull distribution of fracture probability for a material (Derby et al., 1992). The Weibull modulus m (see Eq. 2) can range from zero (totally random fracture behaviour, where the failure probability is the same at all stresses, equivalent to an ideally brittle material) to infinity (representing a precisely unique, reproducible fracture stress, equivalent to an ideally non-brittle material). 

In the above relations, is the stress level below which the probability of failure is zero, in other words, the probability of survival is 1.0. The Weibull modulus, m, principally has values in the range O-oo. In metals, the value of m is 100 and for ceramics m < 3, but this value depends on the soundness of the ceramics. Well-controlled engineering ceramics with fewer flaws may even have an m value in the range of 5-10. 

The percent chamfer failures were calculated for Weibull moduli between 2 and 30, and are plotted in Figures 1 and 2 the effects of corner stress intensity and average flaw size on percent expected chamfer failures are included. The corner stress concentration was assumed to be zero percent for Figure 1, and 5 percent for Figure 2. As expected, the 5 percent corner stress concentration causes the percent of expected chamfer failures to increase with increasing Weibull modulus. This is because, as Weibull modulus increases, the material becomes more sensitive to peak stresses due to the increased stress intensity located at the chamfer. 

This parameter allows for a lower limit stress a below which the probabihty of failure is zero. The underlying distribution is called three-parameter Weibull distribution. In the following, we will usually assume <7i = 0, however. 

Taking 100% overtopping for zero freeboard (the actual data are only a little lower), a Weibull curve can be fitted through the data. Equation (15.5) can be used to predict the number or percentage of overtopping waves or to establish the armor crest level for an allowable percentage of overtopping waves. 

Fig. 22.12. Shape parameter k2 of the Weibull volume distribution versus overtopping probability Pot, emergent and zero-freeboard structures. Prom Caceres et al. ...

In this respect, it has been shown that the reduction of the uncertainty in the estimates depends on the number of records collected in the field inspections, and that, in some particular cases, the proposed procedure conservatively leads to estimations of the scale factor of the Weibull distribution heavily shifted towards zero. Future work will be focused on solutions to avoid such anomalous situations. 

The three parameters m, a, and (Tq are constants for the material m is the Weibull modulus and (T is the maximum value of stress under which the specimen will not fail, which is usually set to zero. Rearranging, setting <7equal to the largest value of stress in the... 

However, the problem is only apparently solved because a third parameter Xmin is introduced whose value is rather unknown. Note that the need to introduce a third parameter may be mandatory if the Weibull exponent is low, as in the example before, or the quantity to be measured very low as in the case of toughness of brittle material that in ferritic steels may be as low as 50 MPa,ym, below the transition temperature NTD. When m is high things may change. For instance, going back to the previous example, if a good quality steel is considered with m — 40, a maximum fatigue limit of 400 MPa and So = 350 MPa, the probability to get a failure at let s say 250 MPa stress amplitude is P(250) = 0.000001 almost zero. In this last case to fix a minimum value at 250 would not have any practical effect. 

Equations 7.3 and 7.4 are called the three-parameter Weibull functions. The three parameters are o, Oq, and m. For ceramics, a two-parameter form is used. In this, the a is considered to be zero. Then the function can be written as ... 

Figure 8.27. Weibull probability plot with selected set of parameters. For zero location parameter, a straight line results.

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