Weibull general

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. 

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). 

Although the Noyes-Whitney equation has been used widely, it has been shown to be inadequate in modeling either S-shape experimental data or data with a steeper initial slope. Therefore, a more general function, based on the Weibull distribution [8], was proposed [9] and applied empirically and successfully to all types of dissolution curves [10] ... 

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). 

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

When carbon (S) spins are locked following a CP 90° pulse, the HB term becomes nonsecular and, after these oscillations vanish, the S spin-lock magnetization is fractionally reduced. Thus, the observed decay is generally a sum of two Weibull functions, usually exponentials (Equation 20 p = 1) the initial slope reflects a Tle dominated by a spin-lattice process and the final slope yields a Tt dominated... 

The general problem that we will focus on in this section is the escape of drug molecules1 from a cylindrical vessel. Initially, theoretical aspects are presented demonstrating that the Weibull function can describe drug release kinetics from cylinders, assuming that the drug molecules move inside the matrix by a Fickian... 

The above reasoning shows that the stretched exponential function (4.14), or Weibull function as it is known, may be considered as an approximate solution of the release problem. The advantage of this choice is that it is general enough for the description of drug release from vessels of various shapes, in the presence or absence of different interactions, by adjusting the values of the parameters a and b. Monte Carlo simulation methods were used to calculate the values of the parameters a and (mainly) the exponent b [87]. 

In 1951, Weibull [116] described a more general function that can be applied to all common types of dissolution curves. This function was introduced in the pharmaceutical field by Langenbucher in 1972 [117] to describe the accumulated fraction of the drug in solution at time t, and it has the following form 1... 

This measure of heterogeneity generalizes the notion of heterogeneity as a departure from the classical first-order model initially introduced [121] for the specific case of the Weibull function. In addition, the above equation can also be used for comparison between two experimentally obtained dissolution profiles [131]. 

This technique was employed in calculating the reliability contours depicted in Fig. 11.4. The reliability contours represent a homogeneously stressed material element, and for dimensionless , the Weibull parameter /3 has units of stress (volume)1/a. Here a, = 5, j3, = 0.2, ac = 35, /3C = 2, abc — 35, and f bc = 2.32. The three surfaces correspond to 0tj = 0.95, 0.5, and 0.05. Note that the reliability contours retain the general behavior of the deterministic failure surface from which they were generated. In general, as the a values increase, the spacing between contours diminishes. Eventually the contours would not be distinct and they would effectively map out a... 

The investigation of failures of manufactured components and systems, especially in the electronics and aerospace industries, has generated a variety of statistical models on which data analysis may be based. Each model uses a specific distibution of failure probabilities, and it is important to select a model that matches the actual distribution inherent in the product concerned. In the case of dielectric breakdown, where a large number of quite different modes of failure are known to occur, sometimes even together, the application of a particular statistical failure model must be approached with great caution. Nevertheless, one treatment, based on a Weibull distribution of failure probability, has taken root, and is most generally used in practice. For a dielectric, the Weibull failure probability function has the form... 

We can regard a fiber as consisting of a chain of links. We assume that fiber failure occurs when the weakest link fails. This is called the weakest-link assumption. It turns out that such a weak-link material is well described by the statistical distribution known as the Weibull distribution (Weibull, 1939,1951). We first describe the general Weibull treatment for brittle materials and then describe its application for fibers. 

The BMCs were generally lower than the NOAELs when analyzed with either statistical estimate. The mean NOAEL/BMC ratios for the 1%, 5%, and 10% response were 1.60, 1.16, and 0.99, respectively, when using a probit analysis, and 3.59, 1.59, and 1.17, respectively, when using the Weibull analysis. It is interesting to note that comparable means from a Weibull analysis of developmental toxicity data were considerably greater, the developmental toxicity means of the NOAEL/BMC ratios were 29, 5.9, and 2.9 (Allen et al. 1994). 

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) ... 

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. 

Bertholon model can be generalized into a model to 7 parameters characterizing the three phases of the bathtub curve a first Weibull law with P < I for the phase of youth failure, an exponential law for the phase of occasional failures and a second Weibull with P > for the wear phase. It corresponds to three blocks in series, the first is a Weibull, initiated at t = 0 (y = 0) and limited to duration T, and the other two corresponding to the Bertholon model. The occurrence of failure can be simulated by the formula 4 under Excel. 

The generalized gamma distribution is one of the most studied probabihty density functions of statistics since many of the important nondiscrete density functions can be derived from it. For example, /(y (2,0, V2A)) is the one-sided normal distribution, and/(y (l, /2— 1,2)) is the / -distribution. In the special case of = a — 1 the gamma distribution is called a Weibull distribution and in case of a = 1 we obtain the Gamma distribution. 

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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