Asymptotically Efficient Estimation of a Bivariate Gaussian–Weibull Distribution and an Introduction to the Associated Pseudo-truncated Weibull

TLDR: In this article, a bivariate Gaussian-Weibull distribution and the associated pseudo-truncated Weibull was proposed to model the simultaneous behavior of stiffness and bending strength of wood.

Abstract: Two important wood properties are stiffness (modulus of elasticity or MOE) and bending strength (modulus of rupture or MOR). In the past, MOE has often been modeled as a Gaussian and MOR as a lognormal or a two or three parameter 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, we introduce a bivariate Gaussian–Weibull distribution and the associated pseudo-truncated Weibull. We use asymptotically efficient likelihood methods to obtain an estimator of the parameter vector of the bivariate Gaussian–Weibull, and then obtain the asymptotic distribution of this estimator.

Frequently asked questions

Q1. What have the authors contributed in "Asymptotically efficient estimation of a bivariate gaussian-weibull distribution and an introduction to the associated pseudo-truncated weibull" ?

To model the simultaneous behavior of MOE and MOR for the purposes of wood system reliability calculations, the authors introduce a bivariate Gaussian–Weibull distribution and the associated pseudotruncated Weibull. 

Q2. What are the main issues associated with the division of a percentile?

Of course there are stochastic issues associated with variable loads, uncertainty in estimation, and the division of a percentile with no consideration of population variability. 

Q3. Why are the authors comparing bins with higher MOE boundaries?

Because MOE and MOR are correlated, bins with higher MOE boundaries also tend to contain board populations with higher MOR values. 

Q4. What is the distribution of the X,W?

Then W is distributed as a Weibull with shape parameter β and scale parameter 1/γ , and the pair X,W have their joint “bivariate Gaussian–Weibull” distribution. 

Q5. what is the density of a pseudo-truncated Weibull?

In Appendix K of Verrill et al. (2012a), the authors show that as ρ → 1, the density of a pseudo-truncated Weibull density converges to the density of a truncated Weibull. 

Q6. What is the definition of an allowable strength property?

In essence, an allowable strength property is calculated by estimating a fifth percentile of a population (actually a 95% content, one-sided lower 75% tolerance bound) and then dividing that value byReceived October 19, 2012; Accepted May 8, 2013. 

Q7. How do the authors obtain the distribution of a PTW?

In this article, the authors obtain the distribution of a PTW and show how to obtain estimates of its parameters and its quantiles by fitting a bivariate Gaussian–Weibull to the full MOE–MOR distribution. 

Q8. What is the definition of a PTW?

if the full joint MOE–MOR population were distributed as a bivariate Gaussian–Weibull, the subpopulation would be distributed as a “pseudo-truncated Weibull” (PTW).