A fit of a mixture of bivariate normals to lumber stiffness–strength data

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.

Citation

Verrill, 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. 46 p.