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Analyzing Toxicity Data Using Statistical Models for Time-To-Death: An Introduction
Published in Michael C. Newman, Alan W. McIntosh, Metal Ecotoxicology, 2020
Philip M. Dixon, Michael C. Newman
The Weibull distribution, a generalization of the exponential distribution, has a hazard function that can take a variety of shapes, not just a flat line like that of the exponential distribution. Weibull distributions are described by two positive parameters: a, the scale parameter that determines the spread and location of the values, and β, the shape parameter that determines the shape of the hazard or survivor functions. If β = 1, the Weibull reduces to the exponential distribution. If β < 1, the hazard is initially high and declines with time. If β > 1, the hazard increases with time, and the survivor function is S-shaped (Figure 5). An intuitive interpretation of the Weibull distribution and the role of β is that the Weibull describes the “weakest link” mode of failure.3 Consider an individual to be composed of β parts, each of which has a constant hazard of failing. If the individual dies when any one of the β parts fails, then the time to-death will fit a Weibull distribution. The Weibull distribution (and its special case, the exponential distribution) is the only distribution for which the accelerated time and the proportional hazards models are identical4 because of the mathematical form of the hazard and survivor functions.
Asset and Liability Management: Recent Advances
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
where r1 and r2 are independent copies of r, and d__ denotes equality in distribution. The distribution is described by the following parameters: α ∈ (0,2] (index of stability), β ∈ [–1,1] (skewness parameter), μ ∈ R (location parameter), and σ ∈ [0,∞) (scale parameter). The variable is then represented as r~Sα,β (μ,σ). Gaussian distribution is actually a special case of stable distribution when α = 2, β = 0. The smaller the stability index is, the stronger the leptokurtic nature of the distribution becomes, i.e., with higher peak and fatter tails. If the skewness parameter is equal to zero, as in the case of Gaussian distribution, the distribution is symmetric. When β > 0 (β < 0), the distribution is skewed to the right (left). If β = 0 and μ = 0, then the stable random variable is called symmetric α-stable (SαS). The scale parameter generalizes the definition of standard deviation. The stable analog of variance is variation, υα, which is given by σα.
Monte Carlo Simulation
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
Kevin Cronin, James P. Gleeson
Any probability distribution can be characterized by its parameters, which in turn can be estimated from the corresponding sample statistics. Generally, a probability distribution function has three parameters that can be geometrically interpreted as defining the location, scale, and shape of the distribution. A location parameter represents the position of the distribution on the x-axis by specifying an abscissa such as the minimum value or the average of the distribution. Changing the location parameter shifts the distribution left or right along the x-axis. The scale parameter represents the width or dispersion of the distribution (such as the standard deviation in the normal distribution). Changing this parameter compresses or expands the distribution without changing its basic shape. The shape parameter represents the shape of distribution usually characterized by the skewness of the distribution. Note that not all distributions have a shape parameter and some have more than one (such as the beta distribution).
Simultaneous optimization of quality and censored reliability characteristics with constrained randomization experiment
Published in Quality Technology & Quantitative Management, 2022
Shanshan Lv, Zhen He, Guodong Wang, Geoff Vining
After obtaining the expression for the predicted responses, the next step is to determine the factor levels that optimize both the quality and reliability characteristics. The scale parameter is the characteristic lifetime of the Weibull distribution. Maximizing product reliability was obtained by the modeling the logarithm of the scale parameter. According to the motivating example, the product reliability should be at least 90% at 2000 h, which means that the logarithm of the scale parameter should be larger than 8.1476. Translating the goals to a desirability function, a larger-the-better function with lower bound Lr = 8.1476 and target value Tr = 8.5 is considered for the reliability characteristic. The quality characteristic goal can also be translated to a nominal-the-best desirability function with the lower bound Lq = 75, target value Tq = 80, and upper bound Uq = 85. Thus, the overall desirability function for the quality and reliability characteristics is
Weibull distribution analysis of roselle and coconut-shell reinforced vinylester composites
Published in Australian Journal of Mechanical Engineering, 2021
S. Navaneethakrishnan, V. Sivabharathi, S. Ashokraj
The slope of the line is 27.229, which is the value of the shape parameter m. A m < 1:0 indicates that the material has a decreasing failure rate. Similarly, an m = 0 indicates constant failure rate and a m > 1:0 indicates an increasing failure rate. The value is computed as = 42.34 using the point the line intersects the Y-axis (−101.93) in = exp (-(y/m)). Therefore, m = 27.229 indicates that the composite material tends to fracture with a higher probability for every unit increase in applied tension load. The scale parameter measures the spread in the distribution of data. Then, the distribution function was:
Stochastic model-based assessment of power systems subject to extreme wind power fluctuation
Published in SICE Journal of Control, Measurement, and System Integration, 2021
Kaito Ito, Kenji Kashima, Masakazu Kato, Yoshito Ohta
The parameter α represents the degree of non-Gaussianity. In particular, coincides with the Gaussian distribution with zero-mean and variance . The scale parameter σ acts like the standard deviation of the Gaussian distribution. The parameter is omitted when such as . For , we have by definition. We write when such that . The property of the tail of stable distributions is characterized as follows [12, Property 1.2.15].