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Optimal Cusum Control Chart for Censored Reliability Data with Log-logistic Distribution

Sadeghpour Gildeh B., Taghizadeh Ashkavaey M.

Computational Methods in Science and Technology

2015

Vol. 21, No. 4

221--227

The goal of this work is to detect any potentially harmful change in a process. The reliability tests are assumed to generate type-I right-censored data following a log-logistic distribution with scale parameter (η) and shape parameter (β). For this purpose, we have constructed a likelihood ratio based simultaneous cumulative sum (CUSUM) control chart that targets changes in both the failure mechanism and the characteristic life (the simultaneous CUSUM chart for detecting shifts in the shape and the scale parameters). This control chart displays best performance for combinations with larger positive or negative shifts in the shape parameter, signaling on average in 5 samples in an out-of-control situation, while targeting an in-control average run length of 370. The simultaneous CUSUM chart’s performance is highly dependent on the values of ,β and on the interaction between them and the censoring rates and shift sizes.

Model error in flood frequency estimation

Strupczewski W., Węglarczyk S., Singh V.

Acta Geophysica Polonica

2002

Vol. 50, nr 2

279-319

Asymptotic bias in large quantiles and moments for three parameter estimation methods, including the maximum likelihood method (MLM), moments method (MOM) and linear moments method (LMM), is derived when a probability distribution function (PDF) is falsely assumed. It is illustrated using an alternative set of PDFs consisting of five two-parameter PDFs that are lower-bounded at zero, i.e., Log-Gumbel (LG), Log-logistic (LL), Log-normal (LN), Linear Diffusion (LD) and Gamma (Ga) distribution functions. The stress is put on applicability of LG and LL in the real conditions, where the hypothetical distribution (H) differs from the true one (T). Therefore, the following cases are considered: H=LG; T=LL, LN, LD and Ga, and H=LL, LN, LD and Ga, T=LG. It is shown that for every pair (H; T) and for every method, the relative bias (RB) of moments and quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (cv), except that RB of moments for MOM is zero. The value of RB is smallest for MOM and the largest for MLM. The bias of LMM occupies an intermediate position. Since MLM used as the approximation method is irreversible, the asymptotic bias of the MLM-estimate of any statistical characteristic is not asymmetric as is for the MOM and LMM. MLM turns out to be the worst method if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e., acceptance of LN, LD or Ga as a hypothetical distribution while LG or LL as the true one, gives the MLM-bias of reasonable magnitude in upper quantiles. Therefore, one should be highly reluctant in choosing the LG and LL in flood frequency analysis, especially if MLM is to be applied.