Weibull shape parameters

The difference between Eqs. 4 and 5 becomes significant for large ul0 values. Similar considerations for other Weibull-shape parameters are given in Livingstone and Imboden (1993). 

In association with each invariant, the a values correspond to the Weibull shape parameters and the /3 values correspond to Weibull scale parameters. A two-parameter Weibull probability density function has the following form... 

A higher strength variability is foimd in ceramic fibers (such as alumina), which have a relatively low Weibull shape parameter. Typical Weibull parameters for commercial fibers are shown in Table 1. The statistical variability of bundles and composites is a function of the fiber variability but is generally lower than... 

The Weibull shape parameter a depends upon the average volume per overtopping wave War, where... 

Influence on life time The failure probability at the life time specified with the parameter MTTF is 63.2% for a constant failure rate (i.e. Weibull shape parameter = 1) or approximately 50% for an increasing failure rate (e.g. Weibull shape parameter b = 3.5). That means, at this time half (or even more than the half) of the component s population are expected to be failed. Because of this, the MTTF as a specification parameter is often criticized. 

As an example of the type of functions possible with just one of the functions, Figure 8.12 shows various Weibull probability density plots for a range of Weibull shape parameters. The ability of this rather simple function to take on many different functional forms is one of the reasons it is frequently used for modeling of reliability data. The shape can range from an exponential like decrease with x to approximate a Gaussian for a shape parameter around 4 and an even more sharply... 

Effect of Weibull Shape Parameter on Probability Plot... 

When n < 0.7, the ln[—ln(l — a)] against In t plots show curvature and linearity is improved if the latter parameter is replaced by t. This reduces the Weibull distribution to a log-normal distribution. Since both exponential and normal distributions are special cases of the more general gamma distribution, Kolar-Anic and Veljkovic [441] compared the applicability of the Weibull and the gamma distributions. The shape parameter of the latter (e) was shown to depend exclusively on the shape parameter of the former (n). 

According to Table 1, semi-invariants of higher order characterize the shape of the profile in terms of variance, skewness, and kurtosis. The outstanding merit of the Weibull distribution is that its shape parameter a provides a summarizing measure for this property. For other distributions, the characterization of the shape is less obvious. 

Three cases have been constructed by Weibull functions according to Eqs. (la) and (lb), as these best reflect systematic differences in the sequence. In all cases, a reference profile is defined by extent Eo l.O, scale parameter / = 2.0, and shape parameter a = 1.5. In each case, one parameter is altered to illustrate its influence. 

Table 11 contains the pertinent parameter estimates and the residual error for each release model. From these results and from Figs 4, 5 and 6 it can be concluded that the release of diclofenac sodium is fitted by the Weibull distribution. P>1 (P being the shape parameter) is characteristic for a slower initial rate (diclofenac sodium is insoluble at pH 1.2) followed by an acceleration to the final plateau (sigmoid). In the direct compression optimization, after infinite time, the fraction released (F.rJ is estimated to be only 90% [13]. 

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). 

One has simply to assume a particular probability distribution for A with the survival function available in a closed form, namely the exponential, Erlang, Rayleigh, and Weibull. Table 9.1 summarizes the probability density functions, survival functions, and hazard rates for the above-mentioned distributions. In these expressions, A is the scale parameter and p and v are shape parameters with k, A, p > 0 and v = 1, 2,.... ... 

The Weibull distribution allows noninteger shape parameter values, and the kinetic profile is similar to that obtained by the Erlang distribution for p, > 1. When 0 < p < 1, the kinetic profile presents a log-convex form and the hazard rate decreases monotonically. This may be the consequence of some saturated clearance mechanisms that have limited capacity to eliminate the molecules from the compartment. Whatever the value of p, all profiles have common ordinates, p(l/X) = exp(-l). 

The Weibull distribution is completely described by the shape parameter / and the characteristic time T. We should mention that T is related to but not identical to the mean time to failure (MTTF). T is actually the time by which 63% of the original population fails. Figure 5.9.8 shows how the time dependence of the failure rate changes with the shape parameter at constant T. 

With shape parameter / < 1 a decreasing failure rate can be simulated, insuch as on the left side of the bathtub curve (Fig. 5.9.6). For / =1 the Weibull distribution reduces to a pure exponential with a constant failure rate that is given by the reciprocal of the characteristic time T ... 

Fig. 5.9.8 Weibull distribution for three different shape parameters / ...

P = shape parameter or the Weibull slope, p > 0 6 = scale parameter or the characteristic life 8 = location parameter or the minimum life... 

The Weibull function has two parameters. The first is 8 or a shape parameter and the second is j a scale parameter. The scale parameter determines when, in time, a given portion of the population will fail (say 75%) at a given time /(t). The shape parameter enables the Weibull distribution to be applied to any portion of the bathtub cmve as follows ... 

We generally find statistically significant effects, of the expected sign, in the duration models. While we do not report the estimated shape parameters of the Weibull distribution in Tables 3.4 or 3.5, they indicate negative duration dependence as claim duration increases, the rate of exit from claimant status falls. Hence, the longer a claimant stays on a workers compensation claim, the less likely he is to leave it. 

A systematic analysis has been made for the statistical approach to describe secondary drop size distributions. Two groups were identified. An empirical one based on the Weibull distribution where the scale and shape parameters can change according to the degree of control desired over the size and frequency range. The second group is semiempirical and is associated with a log-normal distribution function. The statistical meaning of the log-normal expresses the multiplicative nature of the secondary atomization process. 

The shape parameter m is usually called as the Weibull coefficient. A bigger m makes the variance of the material strength smaller. 

Components with increasing failure rate have Weibull distributed lifetimes with scale parameter equal to 0.903/MTTF and shape parameter equal to 1.5. This distribution has a mean which approximately equals the MTTF. 

Weibull life distribution with parameters that fit historical MTTF. Shape parameter 1.5. Reflected in the 4 repair scenarios. Each scenario is assigned a probability of occurrence. Long term availability is not reflected. 

To be more specific, consider as an example the second uncertainty factor degree of ageing. Ageing is for all parameters described/reflected in the traditional analysis using a Weibull distribution with shape parameter 1.5. However, the uncertainty assessment in Table 1 shows that the results from the traditional analysis are highly sensitive to the selection of... 

Using a software tool (RehaSoft Corporation2007), and taking into accoimt the collected data, it was selected the bi-parametric Weibull distribution as the most adequate one, with the following parameters e = 2.3439 (shape parameter)... 

Weibull rfP, m ), where t]P (i = 0,1,..., k) is the scale parameter of the dth failure mode at stress 5), and is its shape parameter, which is changeless at any stress level. Then the observed lifetime... 

Pascual, F. 2006. Accelerated Life Test Planning with independent Weibull competing risks with known shape paramete. IEEE Transactions on reliability, 56(1) 85-93. 

However, therein discussed test has been always generated by the sample from the Weibull, gamma or generalized gamma, i.e. all shape parameters have been equal. In this section we derive the test for sample based on t>j(y), which can have a different shape parameters coj, respectively. The latter is caused by the missing time-to-failure mechanism. 

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