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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 25 Jul 2024 10:34:50 GMT2024-07-25T10:34:50ZEstimating the parameters of a generalized lambda distribution
http://hdl.handle.net/10985/10868
Estimating the parameters of a generalized lambda distribution
FOURNIER, Benjamin; RUPIN, Nicolas; BIGERELLE, Maxence; NAJJAR, Denis; IOST, Alain; WILCOX, R
The method of moments is a popular technique for estimating the parameters of a generalized lambda distribution (GLD), but published results suggest that the percentile method gives superior results. However, the percentile method cannot be implemented in an automatic fashion, and automatic methods, like the starship method, can lead to prohibitive execution time with large sample sizes. A new estimation method is proposed that is automatic (it does not require the use of special tables or graphs), and it reduces the computational time. Based partly on the usual percentile method, this new method also requires choosing which quantile u to use when fitting a GLD to data. The choice for u is studied and it is found that the best choice depends on the final goal of the modeling process. The sampling distribution of the new estimator is studied and compared to the sampling distribution of estimators that have been proposed. Naturally, all estimators are biased and here it is found that the bias becomes negligible with sample sizes n⩾2×103. The .025 and .975 quantiles of the sampling distribution are investigated, and the difference between these quantiles is found to decrease proportionally to View the MathML source. The same results hold for the moment and percentile estimates. Finally, the influence of the sample size is studied when a normal distribution is modeled by a GLD. Both bounded and unbounded GLDs are used and the bounded GLD turns out to be the most accurate. Indeed it is shown that, up to n=106, bounded GLD modeling cannot be rejected by usual goodness-of-fit tests.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10985/108682007-01-01T00:00:00ZFOURNIER, BenjaminRUPIN, NicolasBIGERELLE, MaxenceNAJJAR, DenisIOST, AlainWILCOX, RThe method of moments is a popular technique for estimating the parameters of a generalized lambda distribution (GLD), but published results suggest that the percentile method gives superior results. However, the percentile method cannot be implemented in an automatic fashion, and automatic methods, like the starship method, can lead to prohibitive execution time with large sample sizes. A new estimation method is proposed that is automatic (it does not require the use of special tables or graphs), and it reduces the computational time. Based partly on the usual percentile method, this new method also requires choosing which quantile u to use when fitting a GLD to data. The choice for u is studied and it is found that the best choice depends on the final goal of the modeling process. The sampling distribution of the new estimator is studied and compared to the sampling distribution of estimators that have been proposed. Naturally, all estimators are biased and here it is found that the bias becomes negligible with sample sizes n⩾2×103. The .025 and .975 quantiles of the sampling distribution are investigated, and the difference between these quantiles is found to decrease proportionally to View the MathML source. The same results hold for the moment and percentile estimates. Finally, the influence of the sample size is studied when a normal distribution is modeled by a GLD. Both bounded and unbounded GLDs are used and the bounded GLD turns out to be the most accurate. Indeed it is shown that, up to n=106, bounded GLD modeling cannot be rejected by usual goodness-of-fit tests.Application of the generalized lambda distributions in a statistical process control methodology
http://hdl.handle.net/10985/10792
Application of the generalized lambda distributions in a statistical process control methodology
FOURNIER, Benjamin; RUPIN, Nicolas; BIGERELLE, Maxence; NAJJAR, Denis; IOST, Alain
In statistical process control (SPC) methodology, quantitative standard control charts are often based on the assumption that the observations are normally distributed. In practice, normality can fail and consequently the determination of assignable causes may result in error. After pointing out the limitations of hypothesis testing methodology commonly used for discriminating between Gaussian and non-Gaussian populations, a very flexible family of statistical distributions is presented in this paper and proposed to be introduced in SPC methodology: the generalized lambda distributions (GLD). It is shown that the control limits usually considered in SPC are accurately predicted when modelling usual statistical laws by means of these distributions. Besides, simulation results reveal that an acceptable accuracy is obtained even for a rather reduced number of initial observations (approximately a hundred). Finally, a specific user-friendly software have been used to process, using the SPC Western Electric rules, experimental data originating from an industrial production line. This example and the fact that it enables us to avoid choosing an a priori statistical law emphasize the relevance of using the GLD in SPC.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10985/107922006-01-01T00:00:00ZFOURNIER, BenjaminRUPIN, NicolasBIGERELLE, MaxenceNAJJAR, DenisIOST, AlainIn statistical process control (SPC) methodology, quantitative standard control charts are often based on the assumption that the observations are normally distributed. In practice, normality can fail and consequently the determination of assignable causes may result in error. After pointing out the limitations of hypothesis testing methodology commonly used for discriminating between Gaussian and non-Gaussian populations, a very flexible family of statistical distributions is presented in this paper and proposed to be introduced in SPC methodology: the generalized lambda distributions (GLD). It is shown that the control limits usually considered in SPC are accurately predicted when modelling usual statistical laws by means of these distributions. Besides, simulation results reveal that an acceptable accuracy is obtained even for a rather reduced number of initial observations (approximately a hundred). Finally, a specific user-friendly software have been used to process, using the SPC Western Electric rules, experimental data originating from an industrial production line. This example and the fact that it enables us to avoid choosing an a priori statistical law emphasize the relevance of using the GLD in SPC.Comments on the mixture detection rule used in SPC control charts
http://hdl.handle.net/10985/10859
Comments on the mixture detection rule used in SPC control charts
FOURNIER, Benjamin; RUPIN, Nicolas; BIGERELLE, Maxence; NAJJAR, Denis; IOST, Alain
When calculating independently the false alarm rate of the eight usual runs rules used in SPC control chart, it appears that the proposed rule designed to detect mixture patterns corresponds to a Type-I error strongly lower than the seven other rules. This discrepancy is underlined and the mixture rule is showed to be useless both for in-control and out-of-control processes. Thus a modification of the mixture detection rule is proposed and the impact of this new mixture rule is then illustrated and discussed using Monte Carlo calculations.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10985/108592007-01-01T00:00:00ZFOURNIER, BenjaminRUPIN, NicolasBIGERELLE, MaxenceNAJJAR, DenisIOST, AlainWhen calculating independently the false alarm rate of the eight usual runs rules used in SPC control chart, it appears that the proposed rule designed to detect mixture patterns corresponds to a Type-I error strongly lower than the seven other rules. This discrepancy is underlined and the mixture rule is showed to be useless both for in-control and out-of-control processes. Thus a modification of the mixture detection rule is proposed and the impact of this new mixture rule is then illustrated and discussed using Monte Carlo calculations.Application of Lambda Distributions and Bootstrap analysis to the prediction of fatigue lifetime and confidence intervals
http://hdl.handle.net/10985/10819
Application of Lambda Distributions and Bootstrap analysis to the prediction of fatigue lifetime and confidence intervals
FOURNIER, Benjamin; RUPIN, Nicolas; BIGERELLE, Maxence; NAJJAR, Denis; IOST, Alain
Dealing with fatigue lifetime prediction, this paper aims to report on a new statistical method combining the Lambda Distributions and the Bootstrap technique. This method is first applied for determining the Probability Density Function (PDF) of the C and n coefficients in the Paris relationship of a fatigue crack propagation curve. Then, introducing the initial crack's length distribution, the fatigue lifetime prediction is obtained and discussed considering various standard deviations of the initial crack's length. It is shown that the scattering of the initial crack's length needs to be taken into account in predicting lifetime, and that the stochastic nature of the crack's propagation is not self-sufficient to explain completely the experimental asymmetry of the PDF lifetime. This paper shows that the Lambda Distributions are a powerful tool for modelling the PDF lifetime, compared with traditional Gaussian or lognormal PDF
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10985/108192006-01-01T00:00:00ZFOURNIER, BenjaminRUPIN, NicolasBIGERELLE, MaxenceNAJJAR, DenisIOST, AlainDealing with fatigue lifetime prediction, this paper aims to report on a new statistical method combining the Lambda Distributions and the Bootstrap technique. This method is first applied for determining the Probability Density Function (PDF) of the C and n coefficients in the Paris relationship of a fatigue crack propagation curve. Then, introducing the initial crack's length distribution, the fatigue lifetime prediction is obtained and discussed considering various standard deviations of the initial crack's length. It is shown that the scattering of the initial crack's length needs to be taken into account in predicting lifetime, and that the stochastic nature of the crack's propagation is not self-sufficient to explain completely the experimental asymmetry of the PDF lifetime. This paper shows that the Lambda Distributions are a powerful tool for modelling the PDF lifetime, compared with traditional Gaussian or lognormal PDF