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Why ZMPE When You Can MUPE?

Methods Track

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METH-1 Hu Paper Why ZPME


Multiplicative error terms are commonly used in the cost analysis field because experience tells us that the error of an individual observation (e.g., cost) is generally proportional to the magnitude of the observation rather than some fixed amount. In such cases, it is appropriate to hypothesize a multiplicative error term for a cost estimating relationship (CER). The Unmanned Space Vehicle Cost Model, Eighth Edition (USCM8) CERs were developed using the Minimum-Unbiased-Percentage Error (MUPE) method to model multiplicative errors. The MUPE method is also known as an iterative, weighted least squares regression. MUPE provides unbiased percentage error regression results—no transformation or adjustment (to correct the bias in unit space) is needed for fitting a MUPE equation.

A popular alternative for hypothesizing the multiplicative error in CERs is the Minimum- Percentage Error under the Zero-Percentage Bias (ZMPE) method. Although the ZMPE method also delivers zero percentage errors for the sample data points, it does not offer any other statistical properties, such as the significance levels of the coefficients. Also, the ZMPE method can be easily trapped in local minima, especially when regressing complicated CERs.

This paper examines the properties of the MUPE and ZMPE methods, along with the pros and cons of using each for CER development and cost uncertainty analysis. It further evaluates whether the estimated multiplicative CERs derived from these methods are actually unbiased. Since the derived CERs and the corresponding prediction intervals will be used to quantify the cost estimating risk, using a validated and supportable method is essential to cost uncertainty analysis. The conclusions identified in this paper form the basis for the decision to use the MUPE regression technique to develop the USCM9 CERs.


Dr. Shu-Ping Hu
Educated at National Taiwan University (B.S., Mathematics) and the University of California, Santa Barbara (M.S., Mathematics, and Ph. D., Statistics), Dr. Hu is a Technical Expert at Tecolote Research, Inc. She joined Tecolote in 1984 and has served as a company expert in all statistical matters. She has advocated an iterative regression technique (MUPE) to model a multiplicative error term without bias, developed correction factors (PING Factor) to adjust the downward bias in log-error models, and developed various distributions and correlation methods implemented in the ACE RI$K simulation tool.
She has over 12 years of experience in USCM CER development and the related database. She also has 19 years of experience in designing, developing, modifying, and integrating statistical software packages for fitting various types of regression equations, learning curves, cost risk analysis, and other PC-based models.

Alfred Smith
General Manager for the Software Products/Services Group. Over 20 years experience leading, executing or contributing to life cycle cost model development for a wide variety of systems such as Aircraft Carriers, Submarines, Multi Mission Aircraft, Amphibious Assault Vehicles, International Space Station and Customs Modernization. Over 10 years of experience teaching cost estimating, statistical analysis and risk modeling. Tecolote lead for the development, distribution and support of the ACEIT suite of tools. Education: 1986 University College, London, England, Master of Science with Distinction; 1978 Royal Military College, Kingston, Bachelor of Engineering.