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A fit of a mixture of bivariate normals to lumber stiffness—strength dataAuthor(s): Steve P. Verrill; Frank C. Owens; David E. Kretschmann; Rubin Shmulsky
Source: Res. Paper FPL-RP-696. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 1-46.
Publication Series: Research Paper (RP)
Station: Forest Products Laboratory
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DescriptionIt has been common practice to assume that a two-parameter Weibull probability distribution is suitable for modeling lumber strength properties. In a series of papers published from 2012 to 2018, Verrill et al. demonstrated theoretically and empirically that the modulus of rupture (MOR) distribution of a visual grade of lumber or of lumber that has been “binned” by modulus of elasticity (MOE) is not a twoparameter Weibull. Instead, the tails of the MOR distribution are thinned via “pseudo-truncation.” The theoretical portion of Verrill et al.’s argument was based on the assumption of a bivariate normal–Weibull MOE–MOR distribution for the full (“mill run”) population of lumber. Verrill et al. felt that it was important to investigate this assumption. In a recent pair of papers, they reported results obtained from a sample of size 200 drawn from a mill run population. They found that normal, lognormal, three-parameter beta, and Weibull distributions did not fit the sample MOR distribution of these data. Instead, it appeared that the MOR data might be fit by a skew normal distribution or a mixture of two univariate normals. In this paper, we investigate whether the joint MOE–MOR data from Verrill et al.’s recent mill run study can be well modeled as a mixture of two bivariate normals.
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CitationVerrill, Steve P.; Owens, Frank C.; Kretschmann, David E.; Shmulsky, Rubin. 2018. A fit of a mixture of bivariate normals to lumber stiffness—strength data. Research Paper FPL-RP-696. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 1-46.
Keywordsnormal distribution, bivariate normal distribution, mixed normal distribution, mixed bivariate normal distribution, skew normal distribution, twoparameter Weibull distribution, three-parameter Weibull distribution, bivariate Gaussian–Weibull distribution, pseudo-truncated Weibull distribution, pseudo-truncated mixed normal distribution, machine stress-rated lumber, MSR lumber, binned MOE lumber, probability density functions, mill run, thin tail, lumber property distribution, chi-squared goodness-of-fit test for a mixture of bivariate normals.
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- Distributions of MOE and MOR in eight mill-run lumber populations (four mills at two times)
- Statistical models for the distribution of modulus of elasticity and modulus of rupture in lumber with implications for reliability calculations
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