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Education and Training for Radiation Protection in Nuclear Power Plants
Published in Kenneth L. Miller, Handbook of Management of Radiation Protection Programs, 2020
A solid background in applied mathematics is an important element of a health physicist’s basic education. Basic calculus and differential equations courses seem essential. Further study would be of maximum value if concentrated on applied and engineering mathematics, including numerical analysis. Understanding statistical analysis is particularly important to health physicists. Education should go beyond a first course which introduces the basic concepts and Gaussian and Poisson distributions. It is becoming increasingly important to be familiar with the lognormal and the Weibull distributions. Many of the phenomena of interest to health physicists, such as the distribution of occupational radiation doses,17 are best described by lognormal distributions. The Weibull distribution also is becoming more popular, not only in the reliability assessment but in other areas such as radiobiological models.27,28 Health physicists have a special need to understand the relatively new field of “judgment under uncertainty”, which is identifying the problems in human logic that cause people to grossly misjudge problems that are amenable to statistical analysis.29
Pairwise Survival Analysis of Infectious Disease Transmission Data
Published in Leonhard Held, Niel Hens, Philip O’Neill, Jacco Wallinga, Handbook of Infectious Disease Data Analysis, 2019
The contact interval distribution in these models can be any parametric failure time distribution. Many standard stochastic S(E)IR models assume an exponential distribution, which can be parametrized by the rate parameter λ. The Weibull distribution is a generalization of the exponential distribution that is parameterized by θ = (λ, γ) where λ is the rate parameter and γ is the shape parameter. It has the cumulative hazard function H(τ, θ) = λtγ. The log-logistic distribution is a two-parameter distribution that allows non-monotonic hazards. It has the cumulative hazard H(τ, θ) = ln[1 + (λt)γ] where λ is the rate parameter and γ is the shape parameter.
Survival Analysis
Published in Jianrong Wu, Statistical Methods for Survival Trial Design, 2018
The hazard function of the Weibull distribution depends on the scale parameter and the shape parameter . When the shape parameter , the Weibull distribution reduces to the exponential distribution, and and correspond to the decreasing and increasing hazard functions, respectively (Figure 2.4).
Improved maximum likelihood estimation of the shape-scale family based on the generalized progressive hybrid censoring scheme
Published in Journal of Applied Statistics, 2022
Several authors have discussed the estimation of the Weibull model parameters, including [31,37] who derived confidence intervals using some pivotal quantities based on progressively censored samples [2]. Derived the estimates of the parameters of the Weibull model based on the classical and Bayesian approaches [20] presented reliability and quantile analyzes of the Weibull distribution [5] derived the maximum likelihood estimates for the Weibull model parameters based on complete and censored data [22] presented some methods for estimating the parameters of the Weibull model [31,38] derived the MLEs for the Weibull model parameters based on progressive type-II censored samples, and [26] derived the empirical Bayes estimates of the Weibull model parameters. For further discussion of the Weibull distribution, see Zhang et al. [39,40].
A mixed control chart for monitoring failure times under accelerated hybrid censoring
Published in Journal of Applied Statistics, 2021
Muhammad Aslam, Muhammad Ali Raza, Rehan Ahmad Khan Sherwani, Muhammad Farooq, Jun Yong Jeong, Chi-Hyuck Jun
This paper proposed a mixed control chart using the accelerated hybrid censoring scheme that monitors the attribute and variable characteristics sequentially. The comparison based on simulation study shows that the proposed control chart has the ability to detect the small to moderate process shift quickly as compared to the existing control chart. This implies that the use of the proposed control chart in the industry is helpful to reduce the number of defective products. The application of the proposed control chart has been illustrated through an example. The idea of the proposed control chart can be extended for future research by considering the cost of the model. In this paper, we assumed that the scale parameter of Weibull distribution is known. However, if it is unknown, it can be estimated by using past failure data or engineering knowledge before applying the proposed control chart.
Optimal design of repetitive group sampling plans for Weibull and gamma distributions with applications and comparison to the Birnbaum–Saunders distribution
Published in Journal of Applied Statistics, 2018
S. Balamurali, P. Jeyadurga, M. Usha
Suppose that the lifetime ‘t’ of a product follows a Weibull distribution. Then the cumulative distribution function (cdf) of the Weibull distribution is given by δ is the shape parameter and λ is the scale parameter. In this study, we assume that the shape parameter is known. It is important to note that the cdf depends only on t/λ because the shape parameter is known. The mean life of a product under Weibull distribution is given by t0 under Weibull distribution is given as follows: t0in terms of the specified mean life µ0. That is, t0=aµ0, where ‘a’ is a constant and it is also called as an experiment termination ratio. Therefore, the failure probability of a product before the experiment time t0 under Weibull distribution is written as follows: p1 and LQL is denoted by p2. A number of authors studied the acceptance sampling plans for ensuring the product lifetime where the lifetime follows Weibull distribution (see, for example, Aslam and Jun [2], Aslam et al. [10], Aslam et al. [11], Jun et al. [31] and Balamurali et al. [17,18]).