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United States Department of Agriculture Procedures for Estimation of Weibull Parameters James W. Evans David E. Kretschmann David W. Green Forest Forest Products General Technical Report February Service Laboratory FPL–GTR–264 2019 Abstract Contents The primary purpose of this publication is to provide an 1 Introduction .................................................................. 1 overview of the information in the statistical literature on 2 Background .................................................................. 1 the different methods developed for fitting a Weibull distribution to an uncensored set of data and on any 3 Estimation Procedures .................................................. 1 comparisons between methods that have been studied in the 4 Historical Comparisons of Individual statistics literature. This should help the person using a Estimator Types ........................................................ 8 Weibull distribution to represent a data set realize some advantages and disadvantages of some basic methods. It 5 Other Methods of Estimating Parameters of should also help both in evaluating other studies using the Weibull Distribution .......................................... 11 different methods of Weibull parameter estimation and in 6 Discussion .................................................................. 12 discussions on American Society for Testing and Materials Standard D5457, which appears to allow a choice for the 7 Conclusion ................................................................. 13 method to estimate the parameters of a Weibull distribution 8 References .................................................................. 14 from a data set. Because in D5457 the method to estimate parameters is to some extent optional, the resulting fitted distribution used to derive the reference resistance properties of wood-based materials and structural connections might result in different values developed under the standard. The maximum-likelihood method appears to be the method that should be used as the default, with other methods requiring some type of justification for their use in wood utilization research. Keywords: Weibull distribution, estimation of parameters, maximum likelihood estimation, regression estimators, simple percentile estimators, method of moments February 2019 Evans, James W.; Kretschmann, David E.; Green, David W. 2019. Procedures for estimation of Weibull parameters. General Technical Report FPL–GTR–264. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 17 p. A limited number of free copies of this publication are available to the public from the Forest Products Laboratory, One Gifford Pinchot Drive, Madison, WI 53726-2398. This publication is also available online at www.fpl.fs.fed.us. Laboratory publications are sent to hundreds of libraries in the United States and elsewhere. The Forest Products Laboratory is maintained in cooperation with the University of Wisconsin. The use of trade or firm names in this publication is for reader information and does not imply endorsement by the United States Department of Agriculture (USDA) of any product or service. In accordance with Federal civil rights law and U.S. Department of Agriculture (USDA) civil rights regulations and policies, the USDA, its Agencies, offices, and employees, and institutions participating in or administering USDA programs are prohibited from discriminating based on race, color, national origin, religion, sex, gender identity (including gender expression), sexual orientation, disability, age, marital status, family/parental status, income derived from a public assistance program, political beliefs, or reprisal or retaliation for prior civil rights activity, in any program or activity conducted or funded by USDA (not all bases apply to all programs). Remedies and complaint filing deadlines vary by program or incident. Persons with disabilities who require alternative means of communication for program information (e.g., Braille, large print, audiotape, American Sign Language, etc.) should contact the responsible Agency or USDA’s TARGET Center at (202) 720–2600 (voice and TTY) or contact USDA through the Federal Relay Service at (800) 877–8339. Additionally, program information may be made available in languages other than English. To file a program discrimination complaint, complete the USDA Program Discrimination Complaint Form, AD-3027, found online at http://www.ascr.usda/gov/complaint_filing_cust.html and at any USDA office or write a letter addressed to USDA and provide in the letter all of the information requested in the form. To request a copy of the complaint form, call (866) 632–9992. Submit your completed form or letter to USDA by: (1) mail: U.S. Department of Agriculture, Office of the Assistant Secretary for Civil Rights, 1400 Independence Avenue, SW, Washington, D.C. 20250–9410; (2) fax: (202) 690–7442; or (3) email: [email protected]. USDA is an equal opportunity provider, employer, and lender. Procedures for Estimation of Weibull Parameters James W. Evans, Mathematical Statistician David E. Kretschmann, Research General Engineer (retired1) David W. Green, Supervisory Research General Engineer Emeritus USDA Forest Service, Forest Products Laboratory, Madison, Wisconsin, USA 1 Introduction literature available on the Weibull distribution. It is intended to provide basic information on the more well-known Two-parameter and three-parameter Weibull distributions methods for estimating Weibull parameters from data. We are widely used to represent the strength distribution of start by giving background on the Weibull distribution. structural lumber and engineering-designed wood subassemblies. Current design practices for wood and 2 Background related material are based on an estimated fifth percentile of The “Weibull” distributional form (Eq. (1)) was first derived the strength distribution. This deterministic method of through an extreme-value approach by Fisher and Tippett design is known as allowable stress design (ASD). Although (1928). As noted by Mann (1968), it became known as the the fifth percentile can be based on a distributional form, Fisher–Tippett Type III distribution of smallest values or as such as the Weibull distribution, the real interest in the the third asymptotic distribution of smallest (extreme) Weibull distribution is occuring as wood construction values. In 1939, a Swedish scientist, Waloddi Weibull practices in the United States and Canada are revised from (1939a), derived the same distribution in an analysis of deterministic procedures to reliability-based design (RBD) breaking strengths using only certain practical requirements. procedures. The two-parameter Weibull distribution is the Several examples of its use were given by Weibull (1951). underlying basis of the calculations in load and resistance Two other papers by Weibull (1939b, 1952) also used the factor design (LRFD), a subset of RBD that is discussed in distribution. Use of the distribution became common in American Society for Testing and Materials (ASTM) reliability analyses after World War II, and the name D5457–04a (ASTM 2009). This design procedure starts Weibull became firmly associated with the distribution. with fitting a two-parameter Weibull distribution to either a complete data set or a set of data representing the lower tail In its three-parameter form, the Weibull family is of the distribution. The fit can be made using either represented by the density function, maximum-likelihood methods or a regression-based − aa−1 estimation procedure. Initially, the standard also considered fx( ) = abxc1 ( −) / b exp −−( xc) / b { } { } a method of moments estimate, but this was dropped before >> the standard was approved. Because of the sensitivity of (x cab ; , 0) (1) reliability calculations to the distributional form and, where a is the shape parameter, b the scale parameter, and potentially, method of fit, the wood engineering community c the location parameter. A more common representation of is interested in looking at (1) evaluating the effect of using a the Weibull distribution in the wood community is to use the two-parameter instead of a three-parameter Weibull cumulative distribution function, distribution, (2) the effect of estimation method, and (3) the a effect of using censored data sets compared with using full Fx( ) =−−−1 exp{( x c) / b} (2) data sets. Simulations such as those performed by Durrans and others (1998) encouraged the wood community to The family of two-parameter Weibull distributions follows address some of these issues. Unfortunately, there is a from Equation (1) when c = 0. general lack of knowledge in the wood community of the extensive work available in the statistics literature on these 3 Estimation Procedures issues, which could lead to needless work in areas in which the information is already available. The primary purpose of Numerous methods of estimating Weibull parameters have this publication is to provide an overview of the basic been suggested by many authors. Estimation procedures are information in the statistical literature on the different generally categorized into one of four major categories: methods developed for fitting a Weibull distribution to an (1) method of moment estimators, (2) linear estimators, such uncensored set of data and on any comparisons between as least-squares-type estimators, (3) estimators based on few methods that have been studied in the statistics literature.
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