Analysis of Reliability and Quality Control V1

<p>Preface ix</p>
<p>Chapter 1. Elements of Analysis of Reliability and Quality Control&nbsp; 1</p>
<p>1.1. Introduction 1</p>
<p>1.1.1. The importance of true physical acceleration life models (accelerated tests = true acceleration or acceleration)&nbsp; 3</p>
<p>1.1.2. Expression for linear acceleration relationships 4</p>
<p>1.2. Fundamental expression of the calculation of reliability 5</p>
<p>1.3. Continuous uniform distribution 9</p>
<p>1.3.1. Distribution function of probabilities (density of probability)&nbsp; 10</p>
<p>1.3.2. Distribution function 10</p>
<p>1.4. Discrete uniform distribution (discrete U)&nbsp; 12</p>
<p>1.5. Triangular distribution 13</p>
<p>1.5.1. Discrete triangular distribution version&nbsp; 13</p>
<p>1.5.2. Continuous triangular law version&nbsp; 14</p>
<p>1.5.3. Links with uniform distribution&nbsp; 14</p>
<p>1.6. Beta distribution 15</p>
<p>1.6.1. Function of probability density&nbsp; 16</p>
<p>1.6.2. Distribution function of cumulative probability 18</p>
<p>1.6.3. Estimation of the parameters (p, q) of the beta distribution&nbsp; 19</p>
<p>1.6.4. Distribution associated with beta distribution 20</p>
<p>1.7. Normal distribution 20</p>
<p>1.7.1. Arithmetic mean 20</p>
<p>1.7.2. Reliability 22</p>
<p>1.7.3. Stabilization and normalization of variance error&nbsp; 23</p>
<p>1.8. Log–normal distribution (Galton)&nbsp; 28</p>
<p>1.9. The Gumbel distribution 28</p>
<p>1.9.1. Random variable according to the Gumbel distribution (CRV, E1 Maximum)&nbsp; 29</p>
<p>1.9.2. Random variable according to the Gumbel distribution (CRV E1 Minimum) 30</p>
<p>1.10. The Frechet distribution (E2 Max)&nbsp; 31</p>
<p>1.11. The Weibull distribution (with three parameters) 32</p>
<p>1.12. The Weibull distribution (with two parameters)&nbsp; 35</p>
<p>1.12.1. Description and common formulae for the Weibull distribution and its derivatives 37</p>
<p>1.12.2. Areas where the extreme value distribution model can be used 39</p>
<p>1.12.3. Risk model 40</p>
<p>1.12.4. Products of damage 41</p>
<p>1.13. The Birnbaum Saunders distribution&nbsp; 42</p>
<p>1.13.1. Derivation and use of the Birnbaum Saunders model 43</p>
<p>1.14. The Cauchy distribution 45</p>
<p>1.14.1. Probability density function&nbsp; 45</p>
<p>1.14.2. Risk function&nbsp; 48</p>
<p>1.14.3. Cumulative risk function&nbsp; 48</p>
<p>1.14.4. Survival function (reliability) 49</p>
<p>1.14.5. Inverse survival function&nbsp; 49</p>
<p>1.15. Rayleigh distribution 50</p>
<p>1.16. The Rice distribution (from the Rayleigh distribution) 52</p>
<p>1.17. The Tukey–lambda distribution 53</p>
<p>1.18. Student s (t) distribution 55</p>
<p>1.18.1. t–Student s inverse cumulative function law (T)&nbsp; 57</p>
<p>1.19. Chi–square distribution law ( 2)&nbsp; 57</p>
<p>1.19.1. Probability distribution function of chi–square law ( 2) 57</p>
<p>1.19.2. Probability distribution function of chi–square law ( 2) 58</p>
<p>1.20. Exponential distribution 59</p>
<p>1.20.1. Example of applying mechanics to component lifespan 63</p>
<p>1.21. Double exponential distribution (Laplace)&nbsp; 66</p>
<p>1.21.1. Estimation of the parameters&nbsp; 66</p>
<p>1.21.2. Probability density function&nbsp; 66</p>
<p>1.21.3. Cumulated distribution probability function 67</p>
<p>1.22. Bernoulli distribution 68</p>
<p>1.23. Binomial distribution 71</p>
<p>1.24. Polynomial distribution 75</p>
<p>1.25. Geometrical distribution 75</p>
<p>1.25.1. Hypergeometric distribution (the Pascal distribution) versus binomial distribution&nbsp; 76</p>
<p>1.26. Hypergeometric distribution (the Pascal distribution) 78</p>
<p>1.27. Poisson distribution 80</p>
<p>1.28. Gamma distribution 81</p>
<p>1.29. Inverse gamma distribution 85</p>
<p>1.30. Distribution function (inverse gamma distribution probability density) 85</p>
<p>1.31. Erlang distribution (characteristic of gamma distribution, )&nbsp; 85</p>
<p>1.32. Logistic distribution 89</p>
<p>1.33. Log–logistic distribution 91</p>
<p>1.33.1. Mathematical statistical characteristics of log–logistic distribution 91</p>
<p>1.33.2. Moment properties 92</p>
<p>1.34. Fisher distribution (F–distribution or Fisher Snedecor)&nbsp; 92</p>
<p>1.35. Analysis of component lifespan (or survival) 95</p>
<p>1.36. Partial conclusion of Chapter 1 96</p>
<p>1.37. Bibliography 97</p>
<p>Chapter 2. Estimates, Testing Adjustments and Testing the Adequacy of Statistical Distributions 99</p>
<p>2.1. Introduction to assessment and statistical tests 99</p>
<p>2.1.1. Estimation of parameters of a distribution 100</p>
<p>2.1.2. Estimation by confidence interval 102</p>
<p>2.1.3. Properties of an estimator with and without bias&nbsp; 103</p>
<p>2.2. Method of moments 106</p>
<p>2.3. Method of maximum likelihood 106</p>
<p>2.3.1. Estimation of maximum likelihood&nbsp; 107</p>
<p>2.3.2. Probability equation of reliability–censored data 108</p>
<p>2.3.3. Punctual estimation of exponential law&nbsp; 109</p>
<p>2.3.4. Estimation of the Weibull distribution&nbsp; 110</p>
<p>2.3.5. Punctual estimation of normal distribution 111</p>
<p>2.4. Moving least–squares method&nbsp; 113</p>
<p>2.4.1. General criterion: the LSC&nbsp; 114</p>
<p>2.4.2. Examples of nonlinear models 118</p>
<p>2.4.3. Example of a more complex process&nbsp; 122</p>
<p>2.5. Conformity tests: adjustment and adequacy tests&nbsp; 123</p>
<p>2.5.1. Model of the hypothesis test for adequacy and adjustment&nbsp; 125</p>
<p>2.5.2. Kolmogorov Smirnov Test (KS 1930 and 1936)&nbsp; 126</p>
<p>2.5.3. Simulated test (1st application)&nbsp; 131</p>
<p>2.5.4. Simulated test (2nd application)&nbsp; 131</p>
<p>2.5.5. Example 1 132</p>
<p>2.5.6. Example 2 (Weibull or not?)&nbsp; 135</p>
<p>2.5.7. Cramer Von Mises (CVM) test&nbsp; 139</p>
<p>2.5.8. The Anderson Darling test 140</p>
<p>2.5.9. Shapiro Wilk test of normality&nbsp; 145</p>
<p>2.5.10. Adequacy test of chi–square ( 2) 145</p>
<p>2.6. Accelerated testing method 151</p>
<p>2.6.1. Multi–censored tests 152</p>
<p>2.6.2. Example of the exponential model&nbsp; 152</p>
<p>2.6.3. Example of the Weibull model&nbsp; 152</p>
<p>2.6.4. Example for the log normal model&nbsp; 153</p>
<p>2.6.5. Example of the extreme value distribution model (E–MIN) 153</p>
<p>2.6.6. Example of the study on the Weibull distribution&nbsp; 154</p>
<p>2.6.7. Example of the BOX COX model&nbsp; 156</p>
<p>2.7. Trend tests 157</p>
<p>2.7.1. A unilateral test 158</p>
<p>2.7.2. The military handbook test (from the US Army)&nbsp; 160</p>
<p>2.7.3. The Laplace test&nbsp; 160</p>
<p>2.7.4. Homogenous Poisson Process (HPP)&nbsp; 160</p>
<p>2.8. Duane model power law 164</p>
<p>2.9. Chi–Square test for the correlation quantity&nbsp; 166</p>
<p>2.9.1. Estimations and 2 test to determine the confidence interval 167</p>
<p>2.9.2. t—test of normal mean&nbsp; 170</p>
<p>2.9.3. Standard error of the estimated difference, s&nbsp; 171</p>
<p>2.10. Chebyshev s inequality 171</p>
<p>2.11. Estimation of parameters 173</p>
<p>2.12. Gaussian distribution: estimation and confidence interval 174</p>
<p>2.12.1. Confidence interval estimation for a Gauss distribution 175</p>
<p>2.12.2. Reading to help the statistical values tabulated 175</p>
<p>2.12.3. Calculations to help the statistical formulae appropriate to normal distribution 175</p>
<p>2.12.4. Estimation of the Gaussian mean of unknown variance 175</p>
<p>2.13. Kaplan Meier estimator 178</p>
<p>2.13.1. Empirical model using the Kaplan Meier approach 179</p>
<p>2.13.2. General expression of the KM estimator 180</p>
<p>2.13.3. Application of the ordinary and modified Kaplan Meier estimator 181</p>
<p>2.14. Case study of an interpolation using the bi–dimensional spline function 181</p>
<p>2.15. Conclusion 183</p>
<p>2.16. Bibliography 184</p>
<p>Chapter 3. Modeling Uncertainty 187</p>
<p>3.1. Introduction to errors and uncertainty 187</p>
<p>3.2. Definition of uncertainties and errors as in the ISO norm 189</p>
<p>3.3. Definition of errors and uncertainty in metrology 191</p>
<p>3.3.1. Difference between error and uncertainty 192</p>
<p>3.4. Global error and its uncertainty 202</p>
<p>3.5. Definitions of simplified equations of measurement uncertainty 204</p>
<p>3.5.1. Expansion factor k and range of relative uncertainty 206</p>
<p>3.5.2. Determination of type A and B uncertainties according to GUM 208</p>
<p>3.6. Principal of uncertainty calculations of type A and type B 229</p>
<p>3.6.1. Standard and expanded uncertainties&nbsp; 231</p>
<p>3.6.2. Components of type A and type B uncertainties 232</p>
<p>3.6.3. Error on repeated measurements: composed uncertainty 232</p>
<p>3.7. Study of the basics with the help of the GUMic software package: quasi–linear model 239</p>
<p>3.8. Conclusion 245</p>
<p>3.9. Bibliography 245</p>
<p>Glossary&nbsp; 249</p>
<p>Index 257</p>