U.S. flag

An official website of the United States government


Main content area

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

S. P. Verrill, Frank Charles Owens 1971-, David E. Kretschmann, Rubin Shmulsky, and Forest Products Laboratory (U.S.)
Distribution (Probability theory), Lumber, Mechanical properties, Elastic properties, Fracture, Statistics, Simulation methods, Weibull distribution, and Gaussian distribution
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 two parameter 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.
1 online resource (23, [21] pages) : illustrations (some color).
United States Department of Agriculture, Forest Service, Forest Products Laboratory
Research paper
USDA publications
Permanent URL: