Weibull parameter estimates

Figure 5. Predicted failure probability of disk versus stress, or, using the Weibull parameters estimated from the 3 point flexure bar data. Solid line is for specimen 3 and dotted line is specimen 9. Experi mental rupture data also shown.

Considering the familiar form of a regression equation, Y = aX + b, the left side of Equation 4 corresponds to 7, Incr eorresponds to X, m corresponds to a, and -mlncTd corresponds to b. Using aa) and F aa)) pairs in Equation 4, a and b are obtained by the least squares procedure. Then the Weibull parameter estimates are calculated as m = a and... 

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. 

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]. 

Table 1. Weibull subsystems reliability parameter estimates. 

Assuming, without loss of generaUty, that the times between failures are modeled by a Weibull distribution, the GRP parameter estimation problem consists on estimating the scale (a) and shape (fi) parameters of the WeibiiU distribution, and the parameter q, as shown in Equation 4. 

The CARES code can estimate Weibull parameters by either maximum likelihood (ML) or le st squares (LS). Since insufficient volume flaw data existed at RT and 1000°C, parameters obtained at these temperatures were neither reasonable nor consistent with each other. The volume flaw... 

Another interesting item worth commenting on is the significant difference between the Weibull modulus my for the three-point flexure bar rupture data (my = 11.96, oey = 612.7 MPa) and that of the thermal shocked disk (my = 6.91, (j0y = 345.9 MPa) experimental rupture data (shown in Figure 5) as determined by CAKES/Life maximum likelihood parameter estimation. Least squares regression (using an Excel spreadsheet) on the CARES/Life disk predictions... 

Figure 2. Estimation of Weibull parameters T, b, tg = 0 and arithmetic mean value x, using non-linear regression methods for prognosis of the comprehensive failure mode, product fleet field observation time 24 months in service exemplified by the case study electric actuator.

The Weibull parameters m and do are estimated from a sample of strength measurements di, d2,..., cr . The Maximum Likelihood (ML) and the General Linear Regression (GLR)... 

The methods for estimating Weibull parameters are also used in the estimation of statistically-based design values, namely A-basis and B-basis material properties (or simply A-basis and B-basis values), which are the 95% confidence lower bounds on the first and tenth percentiles of a Weibull distribution, respectively. They are of great interest to the engineer in the design of structural and mechanical components however, recent research on their estimation has remained limited [1, 4, 5, 19, 20]. 

Several methods are available for the estimation of Weibull parameters. Among them, the most commonly used are the ML and the GLR methods [1, 5, 19]. 

An important reason for the estimation of Weibull parameters is the need to determine the (lOOp)th percentile, ap, for a predefined failure probability p ... 

The previous comparison based on sample sizes suggests that an experimenter should use the ML method for estimating the confidence lower bounds. Therefore, only this method will be considered for a detailed study of the effects of Weibull parameters and sample size. 

Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Parameters for Advaneed Ceramics , ASTM C1239-94a (American Society for Testing and Materials, Philadelphia, PA, 1994). 

D. I. Gibbons and L. C. Vance, M Simulation Study of Estimators for the Parameters and Percentiles in the Two-Parameter Weibull Distribution, General Motors Research Publication No. GMR-3041, General Motors, Detroit, Mich., 1979. 

Note the parameters for the 3-parameter Weibull distribution, xo and 6, ean be estimated given the mean, /i, and standard deviation, cr, for a Normal distribution (assuming [3 = 3.44) by ... 

For each of the 36 bus sections tliat had not already failed, the Weibull distribution was used to detennine tlie probability of failure before tlie next outage. Under assumption (a), tliis probability is P(T < 3301T > 209) i.e., tlie conditional probability of failure before 330 days, given tliat tlie bus section lias survived 209 days. Under assumption (b), tlie corresponding probability is P(T < 330 T > 230). For part (b), tlie estimates of the Weibull distribution parameters used in part (a) were modified to take into consideration tlie absence of failures for 3 additional weeks. 

Using die procedure described in Section 20.4, obtain grapliical estimates of the parameters a and (3 for die Weibull distribudon of time to failure on die basis of the data in Table 21.6.1. 

Given next are the different methods for estimating distribution parameters on exponential, Weibull, normal, log normal, and extreme-value hazard papers. The methods are explained with the aid of simulated data from known distributions. Thus, we can judge from the hazard plots how well the hazard-plotting method does. 

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

The mean represents the overall rate of the relevant process and corresponds to the abscissa of the center of gravity of the PDF and the mean value of the CDF. It is exactly reflected by the rate parameter of the Weibull distribution t63 2% is exact for mono-exponential and may be used as a shorthand estimate for any CDF of similar shape. 

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