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《纺织复合材料》课程参考文献（Composite Materials Handbook，Volume 1）CHAPTER 8 STATISTICAL METHODS

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MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-22 C BETA = (BETAL + BETAH) / 2.0 ALPHA = FNALPH(X,NOBS,BETA,XGM) RETURN END C------------------------------------------------------------------------------------------------- FUNCTION FNALPH(X,NOBS,BETA,XGM) C C COMPUTE MLE FOR TWO-PARAMETER WEIBULL SCALE PARAMETER (ALPHA) C XGM IS THE GEOMETRIC MEAN OF THE X'S C DIMENSION X(NOBS) RN = FLOAT(NOBS) C SUMZ = 0.0 DO 20 I = 1, NOBS SUMZ = SUMZ I (X(l)/XGM)**BETA 20 CONTINUE C FNALPH = XGM*(SUMZ/RN)**(1./BETA) C RETURN END C------------------------------------------------------------------------------------------------- C------------------------------------------------------------------------------------------------- Function GFUNCT(X,NOBS,BETA,XGM) C C COMPUTE G FUNCTION USED IN ESTIMATING THE TWO-PARAMETER WEIBULL C SHAPE PARAMETER (BETA). C XGC IS THE GEOMETRIC MEAN OF THE X'S USED IN ESTIMATING ALPHA. C DIMENSION X(NOBS) RN = FLOAT(NOBS) C ALPHA = FNALPH(X,NOBS,BETA,XGM) SUMYZ = 0.0 DO 10 I = 1, NOBS SUMYZ = SUMYZ + ALOG(X(I))*((X(I)/ALPHA)**BETA - 1.) 10 CONTINUE C GFUNCT = (SUMYZ/RN) - 1.0/BETA C RETURN END C------------------------------------------------------------------------------------------------- FIGURE 8.3.4.2.1 FORTRAN routines for calculating two-parameter Weibull shape and shale parameter, estimates, concluded. Equation 8.3.4.2.1(d) is a monotonically decreasing continuous function of β . Designate the lefthand side of Equation 8.3.4.2.1(d) divided by n as G( β) and obtain a solution for β by the following iterative procedure. Let Sy denote the standard deviation of 1 n y ,..., y where y1 i = ln(x ) for i = 1,...,n. Calculate I = 1.28/Sy as an initial guess at the solution and calculate G(I). If G(I) > 0, then find the smallest positive integer k such that G(1/ 2 )<0 k and let L = I/ 22 and H = I/ 2k-1 . If G(I) < 0, then find the smallest positive integer k such that G(2 I) > 0 k and let L = 2 k-1 I and H = 2 kI. In either case, the interval (L,H) contains the solution to G( β) = 0 . Now calculate G(M) where M = (L + H)/2. If G(M) = 0, then the solution is β = M. If G(M) > 0, then let H = M. If G(M) < 0 then let L = M. The new interval (L,H) still contains the solution to G( β) = 0 but is only half as long as the old interval. Calculate a new M-value and begin the process of interval halving again. The process is repeated until H-L < 2I/106 . The solution to G( β) = 0 is then taken to be M = (L + H)/2. The solution is in error by at most I/106 .

MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-23 8.3.4.2.2 Goodness-of-fit test for the two-parameter Weibull distribution The two-parameter Weibull distribution is considered by comparing the cumulative Weibull distribution function (Section 8.1.4) that best fits the data with the cumulative distribution function of the data. Using the shape and scale parameter estimates from Section 8.3.4.2.1, let z(i) (i) = x / , for i = 1,...,n β α 8.3.4.2.2(a) The Anderson-Darling test statistic is n (i) (n+1-i) i=1 1- 2i AD = n 1- exp( ) - - n z z n ∑     −     A 8.3.4.2.2(b) and the observed significance level is { } OSL = 1/ 1+ exp[-0.10 +1.24ln( ) + 4.48 ] * * AD AD 8.3.4.2.2(c) where * AD = 1+ 0.2 n AD F H G I K J 8.3.4.2.2(d) This OSL measures the probability of observing an Anderson-Darling statistic at least as extreme as the value calculated if in fact the data are a sample from a two-parameter Weibull distribution. If OSL ≤ 0.05, one may conclude (at a five percent risk of being in error) that the population does not have a twoparameter Weibull distribution. Otherwise, the hypothesis that the population has a two-parameter Weibull distribution is not rejected. For further information on this procedure, see Reference 8.3.4.2. 8.3.4.2.3 Basis values for the two-parameter Weibull distribution If the unstructured data set is from a population with a two-parameter Weibull distribution, the B-basis value is B = qexp -V β n R S | T | U V | W | 8.3.4.2.3(a) where q = (0.10536) 1/ α β 8.3.4.2.3(b) and V is the value in Table 8.5.8 corresponding to a sample of size n. A numerical approximation to the V values is given in Equation 8.5.8(h). To calculate the A-basis value, use the appropriate V value from Table 8.5.9 substituting 8.3.4.2.3(c) for 8.3.4.2.3(b). 1/ q (0.01005) ˆ ˆ β =α 8.3.4.2.3(c) 8.3.4.3 Normal distribution In order to compute a basis value for a normally distributed population, it is necessary to obtain estimates of the population mean and standard deviation. Section 8.3.4.3.1 gives the equations for calculating these parameters. Section 8.3.4.3.2 provides the procedure for goodness-of-fit for the normal distri-

MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-24 bution, and Section 8.3.4.3.3 gives the procedure for calculating basis values. The example problem in Section 8.3.7.2 demonstrates these procedures. 8.3.4.3.1 Estimating the mean and standard deviation parameters for the normal distribution The population mean and standard deviation are estimated using the sample mean x and sample standard deviation s. x = 1 n x i=1 n ∑ i s = 1 n-1 ( i=1 n ∑ x -x) i 2 8.3.4.3.2 Goodness-of-fit test for the normal distribution The normal distribution is considered by comparing the cumulative normal distribution function (Section 8.1.4) that best fits the data with the cumulative distribution function of the data. Let z(i) = s , for i = 1,...,n x x (i) − 8.3.4.3.2(a) where x(i) is the ith smallest sample observation, x is the sample average, and s is the sample standard deviation. The Anderson-Darling test statistic is AD = 1-2i n ln F (z ) + ln 1- f (z ) -n i=1 n ∑ o t 0 (i) 0 (n+1-i) 8.3.4.3.2(b) where F0 is the standard normal distribution function (Equation 8.1.4(e)). The observed significance level is OSL = 1/ 1+ exp[-0.48 + 0.78ln(AD*) + 4.58AD*] l q 8.3.4.3.2(c) where * AD = 1+ 0.2 n AD F H G I K J 8.3.4.3.2(d) This OSL measures the probability of observing an Anderson-Darling statistic at least as extreme as the value calculated if in fact the data are a sample from a normal distribution. If OSL ≤ 0.05, one may conclude (at a five percent risk of being in error) that the population is not normally distributed. Otherwise, the hypothesis that the population is normally distributed is not rejected. For further information on this procedure, see Reference 8.3.4.2. 8.3.4.3.3 Basis values for the normal distribution If the unstructured data set is from a population with a normal distribution, the B-basis value is B= - xks B 8.3.4.3.3(a) where kB is the appropriate one-sided tolerance-limit factor from Table 8.5.10. A numerical approximation to the kB values is given in Equation 8.5.10

MIL-HDBK-17-1F Volume 1,Chapter 8 Statistical Methods To calculate the A-basis value,replace ke with the appropriate value of kA from Table 8.5.11 or the numerical approximation in Equation 8.5.11. 8.3.4.4 Lognormal distribution The lognormal distribution is a positively skewed distribution that is simply related to the normal distri- bution.If something is lognormally distributed,then its logarithm is normally distributed.The natural(base e)logarithm is used in MIL-HDBK-17.See Section 8.1.4 for the definition of the lognormal distribution. The example problem in Section 8.3.7.3 demonstrates the application of the procedures in Section 8.3.4.3 for a lognormal distribution. In order to fit test the goodness-of fit of the lognormal distribution,take the logarithm of the data and perform the Anderson-Darling test for normality from Section 8.3.4.3.Using the natural logarithm,let 20= ln(区o)-XL,fori=l,,n 8.3.4.4(a SL where x(is the ith smallest sample observation,xL and sL are the mean and standard deviation of the In(xi)values. The Anderson-Darling statistics is computed using Equation 8.3.4.3(b)and the observed significance level(OSL)is computed using Equation 8.3.4.3(c).This OSL measures the probability of observing an Anderson-Darling statistic at least as extreme as the value calculated if in fact the data are a sample from a lognormal distribution.If OSL<0.05,one may conclude(at a five percent risk of being in error)that the population is not lognormally distributed.Otherwise,the hypothesis that the population is lognormally distributed is not rejected.For further information on this procedure,see Reference 8.3.4.2. The following procedure should be used to calculate basis values for unstructured data that is as- sumed to be a sample from a lognormal population.The equations presented in Section 8.3.4.3 are used to calculate the basis values.However,the calculations are performed using the logarithms of the data rather than the original observations.The computed B-basis value must then be transformed back to the original units by applying the inverse of the log transformation which was used. 8.3.4.5 Nonparametric basis values These procedures should be used to compute basis values for unstructured data when one is unwill- ing to assume a particular population model,usually because the Weibull,normal,and lognormal models all provide inadequate fits to the data.One of two methods should be used,depending on the sample size. 8.3.4.5.1 Nonparametric basis values for large samples To calculate a B-basis value for n>28,determine the value r corresponding to the sample size n from Table 8.5.12.For sample sizes between tabulated values,select the r value associated with the largest tabulated sample size that is smaller than the actual n.The B-basis value is the rth lowest observation in the data set.For example,in a sample of size n=30,the lowest(r 1)observation is the B-basis value. A numerical approximation to the tabulated r values as a function of n is given in Section 8.5.12.The ex- ample problem in Section 8.3.7.4 demonstrates this procedure.Further information on this procedure may be found in Reference 8.3.4.5.1. For n>298,an A-basis value can calculated using the sample procedure,with the r value selected from Table 8.5.13. 8-25

MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-25 To calculate the A-basis value, replace kB with the appropriate value of kA from Table 8.5.11 or the numerical approximation in Equation 8.5.11. 8.3.4.4 Lognormal distribution The lognormal distribution is a positively skewed distribution that is simply related to the normal distribution. If something is lognormally distributed, then its logarithm is normally distributed. The natural (base e) logarithm is used in MIL-HDBK-17. See Section 8.1.4 for the definition of the lognormal distribution. The example problem in Section 8.3.7.3 demonstrates the application of the procedures in Section 8.3.4.3 for a lognormal distribution. In order to fit test the goodness-of fit of the lognormal distribution, take the logarithm of the data and perform the Anderson-Darling test for normality from Section 8.3.4.3. Using the natural logarithm, let (i) L z = ln( ) - s , for i = 1,...,n x x (i) L 8.3.4.4(a) where x(i) is the ith smallest sample observation, xL and sL are the mean and standard deviation of the ln(xi) values. The Anderson-Darling statistics is computed using Equation 8.3.4.3(b) and the observed significance level (OSL) is computed using Equation 8.3.4.3(c). This OSL measures the probability of observing an Anderson-Darling statistic at least as extreme as the value calculated if in fact the data are a sample from a lognormal distribution. If OSL ≤ 0.05, one may conclude (at a five percent risk of being in error) that the population is not lognormally distributed. Otherwise, the hypothesis that the population is lognormally distributed is not rejected. For further information on this procedure, see Reference 8.3.4.2. The following procedure should be used to calculate basis values for unstructured data that is assumed to be a sample from a lognormal population. The equations presented in Section 8.3.4.3 are used to calculate the basis values. However, the calculations are performed using the logarithms of the data rather than the original observations. The computed B-basis value must then be transformed back to the original units by applying the inverse of the log transformation which was used. 8.3.4.5 Nonparametric basis values These procedures should be used to compute basis values for unstructured data when one is unwilling to assume a particular population model, usually because the Weibull, normal, and lognormal models all provide inadequate fits to the data. One of two methods should be used, depending on the sample size. 8.3.4.5.1 Nonparametric basis values for large samples To calculate a B-basis value for n > 28, determine the value r corresponding to the sample size n from Table 8.5.12. For sample sizes between tabulated values, select the r value associated with the largest tabulated sample size that is smaller than the actual n. The B-basis value is the rth lowest observation in the data set. For example, in a sample of size n = 30, the lowest (r = 1) observation is the B-basis value. A numerical approximation to the tabulated r values as a function of n is given in Section 8.5.12. The example problem in Section 8.3.7.4 demonstrates this procedure. Further information on this procedure may be found in Reference 8.3.4.5.1. For n > 298, an A-basis value can calculated using the sample procedure, with the r value selected from Table 8.5.13

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