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    Author(s): Steve P. Verrill; David E. Kretschmann; James W. Evans
    Date: 2016
    Source: USDA Forest Service, Forest Products Laboratory, Research Paper, FPL-RP-685
    Publication Series: Research Paper (RP)
    Station: Forest Products Laboratory
    PDF: Download Publication  (1.0 MB)


    Two important wood properties are stiffness (modulus of elasticity, MOE) and bending strength (modulus of rupture, MOR). In the past, MOE has often been modeled as a Gaussian and MOR as a lognormal or a two- or threeparameter Weibull. It is well known that MOE and MOR are positively correlated. To model the simultaneous behavior of MOE and MOR for the purposes of wood system reliability calculations, a 2012 paper by Verrill, Evans, Kretschmann, and Hatfield introduced a bivariate Gaussian–Weibull distribution and the associated pseudo-truncated Weibull. In that paper, they obtained an asymptotically efficient estimator of the parameter vector of the bivariate Gaussian– Weibull. This estimator requires data from the full bivariate MOE,MOR distribution. In practice, such data are often not available. Instead, in some cases “Machine Stress-Rated” (MSR) data are available. An MSR data set consists of MOE,MOR pairs, where a pair is accepted into the data set (a piece of lumber is accepted) if and only if the MOE value lies between predetermined lower and upper bounds. For such a data set, the asymptotically efficient methods appropriate for a full data set cannot be used. In this paper we present an approach that is effective for MSR data.

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    • This article was written and prepared by U.S. Government employees on official time, and is therefore in the public domain.


    Verrill, Steve P.; Kretschmann, David E.; Evans, James W. 2016. Maximum likelihood estimation of the parameters of a bivariate Gaussian-Weibull distribution from machine stress-rated data. Research Paper, FPL-RP-685. Madison, WI. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 72 p.


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    reliability, modulus of rupture, modulus of elasticity, normal distribution, Weibull distribution, bivariate Gaussian-Weibull distribution, bivariate normal-Weibull distribution, pseudo-truncated Weibull distribution, likelihood methods, machine stress-rated data, MSR data

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