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The EWMA Control Chart for Lifetime Monitoring with Failure censoring Reliability Tests and Replacement
Published in Ioannis S. Triantafyllou, Mangey Ram, Statistical Modeling of Reliability Structures and Industrial Processes, 2023
Let us now define by Vs the elapsed time until the sth failure in a life test. It is well known that the time between events in a Poisson process follows an exponential distribution. It is also known that the exponential distribution with parameter 1/r is equal to the gamma distribution with parameters (1, 1/r). Therefore, the time until the observation of the sth failure is gamma distributed with shape parameter s and scale parameter n/r0 where 1/r0 is the target value of the failure rate. Keeping in mind the relationship between the gamma and the chi-square distribution it is obvious that 2n1r0Vs is chi-square distributed with 2s degrees of freedom.
Estimation and inference
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
Populations are modeled by probability distributions. A probability distribution is generally defined as a formula involving the variable x, say, and a few symbols which take specific values in an application. These symbols are known as parameters and in conjunction with the general formula, the parameters determine numerical characteristics of a population such as its mean, standard deviation, and quantiles, and in the case of binary data a proportion with a particular attribute. For example, a normal distribution has two parameters which are its mean μ and standard deviation σ. The exponential distribution for the time between events in a Poisson process is often defined by the rate parameter λ, events per unit time, in which case its mean is 1/λ time units. The uniform distribution is defined by two parameters a and b, which specify the range of the variable, and its mean is (a + b)/2.
Statistical Estimation of Warranty Costs
Published in Wallace R. Blischke, D. N. Prabhakar Murthy, Warranty Cost Analysis, 2019
Wallace R. Blischke, D. N. Prabhakar Murthy
Since for the exponential distribution M(t) = λt, this is particularly easily estimated, namely. () E^[Cs(W)]=cs[1+λ^W]
A new methodology for estimating seismic resilience of buildings under successive damage-retrofit processes during the recovery time
Published in Structure and Infrastructure Engineering, 2022
Rayehe Khaghanpour-Shahrezaee, Mohammad Khanmohammadi
This model is implemented to calculate the performance of a damaged building affected by aftershock sequences when the building is partially recovered. To estimate the damageability of this building, the recovery level at the time of an aftershock should be determined. The recovery level, a time-based random variable, is calculated by the recovery model. As usual, the transition matrix only has diagonal and upper-triangular elements. Under an aftershock scenario, the building can transfer from a lower damage state to a higher damage state. However, the inverse condition could happen when repairs are performed, that the building can transfer from a higher damage state to a lower one. In this research, the Poisson process is used to simulate the transition matrix of a damaged building from a damage state to the intact building state under recovery conditions (Zhang & Burton, 2021). Based on the Poisson process, the time interval between any two events is assumed as an exponential distribution.
Adaptive Reweighted Variance Estimation for Monte Carlo Eigenvalue Simulations
Published in Nuclear Science and Engineering, 2020
The exponential distribution is a continuous distribution that is often used to model the time elapsed between events occurring at a constant rate . We proposed a spectral estimate to estimate the variance of the sample mean, where and is a positive integer.11 Our numerical results11 show that still underestimates the true variance in almost all locations for the test problems. This paper presents new data-adaptive estimators that yield better performance for the same test problems used in Ref. 11.
Purely Sequential Fixed Accuracy Confidence Intervals for P(X < Y) under Bivariate Exponential Models
Published in American Journal of Mathematical and Management Sciences, 2018
As it is very well known, an exponential distribution is widely used to model many reliability experiments and failure time data. One may refer to Balakrishnan and Basu (1995) and Ahsanullah and Hamedani (2010) for a good overview on many such practical applications. A more realistic approach that fits in today's world is to consider a model where (X, Y) follows a bivariate exponential model. A few simple scenarios could be that of failure times of two devices which are goverened by a common external source, or survival times of two groups of patients under a common dose factor. There have been several other forms of bivariate exponential models introduced in the literature, including those by Gumbel (1960), Downton (1970) and Arnold and Strauss (1988).