The construction of the gamma degradation model with covariates

In this paper, we consider the problem of constructing the gamma degradation model as the most frequently used one for description of a degradation process and prediction of non-failure operation time. A stochastic process Z (t) characterizing degradation process is referred to as gamma degradation process with a shape parameter v(t) and scale parameter о, if 1. Z (0) = 0 . 2. Z (t) is a stochastic process with independent increments. 3. the increments AZ (t) = Z (t + At) _ Z (t) follows the gamma distribution with the probability density function ( _t ^ (t)_ _e_t/о ^ Av(t)) о j оТА^О) , where Av(t) = v(t + At) _ v(t) is the shape parameter and о > 0 is the scale parameter. The gamma distribution is a repeatable distribution (if random variables ^ and follow the gamma distribution with a scale parameter о and shape parameters v ₁ and v ₂ , then ^ + follows the gamma distribution with the scale parameter о and the shape parameter v ₁ + v ₂ ) that explains the fact of using the gamma distribution as a distribution of increments. Non-failure operation time, which depends on covariate x , is equal to Tx = sup{t: Zx (t) < z}, where z is the critical value of degradation index, when the failure is fixed. Then, the reliability function for gamma degradation model is equal to Sx(t) = P{Tx > t} = P{Zx(t) < z} = FGamma [£ CT,^j , where m _x (t) = ctv ^ ( | is a positive monotone increasing trend function, r(x; в) is a positive covariate function. The main problem of using the gamma degradation model is the absence of mathematical methods for testing the statistical hypothesis of goodness-of-fit for the model. We propose an approach to testing goodness-of-fit of the gamma degradation model with covari-ates, which implies the investigation of test statistic distributions with computer simulation methods in interactive mode of testing hypothesis. The non-parametric goodness-of-fit tests of Kolmogorov, Cramer-von Mises-Smirnov and Anderson-Darling are recommended for testing this hypothesis. In this paper, we have carried out the research of statistics distributions and the power of the considered goodness-of-fit tests for the gamma degradation model by means of computer simulation methods. It has been shown that the distributions of the test statistics in the case of composite hypotheses depend on the type of the parametric trend function, the type of the parametric covariate function, and the design of experiment (the moments of time, in which degradation index was measured, and the values of covariates, for which objects were observed). We have carried out an empirical analysis of the power of considered tests for various pairs of competing hypotheses. It has been shown, that the proposed method of testing the goodness-of-fit hypothesis enables to test the assumption of a distribution of degradation increments, as well as the assumptions about the trend function and the covariate function. We have also considered an example of construction of the gamma degradation model for Carbon Film resistors data.

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