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Title: A fit of a mixture of bivariate normals to lumber stiffness—strength data

Source: Res. Paper FPL-RP-696. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 1-46.

Author(s)Verrill, Steve P.; Owens, Frank C.; Kretschmann, David E.; Shmulsky, Rubin

Publication Year: 2018  View PDF »

Category: Research Papers
Associated Research Project(s):   FPL-4714-2B   FPL-4851-2B

Abstract: It 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.

Keywords: normal 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.

Publication Review Process: Informally Refereed (Peer-Reviewed)

File size: 686 kb(s)

Date posted: 09/21/2018

This publication is also viewable on Treesearch:  view
RITS Product ID: 92362
Current FPL Scientists associated with this product (listed alphabetically)
Kretschmann, David E.
Research General Engineer
Verrill, Steve P.
Mathematical Statistician

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