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Moments of the truncated normal distribution


  • William Horrace



The truncated (below zero) normal distribution is considered. Some existing results are surveyed, and a recursive moment formula is used to derive the first four central moments in terms of the mean and variance of the underlying normal and in terms of lower moments of the truncated distribution. Bounding and monotonicity of the moments of the truncated distribution are considered and some previously unknown features of the distribution are presented. Moment results are used to derive a test of the distributional form. The distribution is commonly used in economics, particularly in the stochastic frontier literature. Application to the stochastic frontier model is briefly considered. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • William Horrace, 2015. "Moments of the truncated normal distribution," Journal of Productivity Analysis, Springer, vol. 43(2), pages 133-138, April.
  • Handle: RePEc:kap:jproda:v:43:y:2015:i:2:p:133-138
    DOI: 10.1007/s11123-013-0381-8

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    References listed on IDEAS

    1. Anil Bera & Subhash Sharma, 1999. "Estimating Production Uncertainty in Stochastic Frontier Production Function Models," Journal of Productivity Analysis, Springer, vol. 12(3), pages 187-210, November.
    2. Hong, Han & Shum, Matthew, 2003. "Econometric models of asymmetric ascending auctions," Journal of Econometrics, Elsevier, vol. 112(2), pages 327-358, February.
    3. Waldman, Donald M., 1982. "A stationary point for the stochastic frontier likelihood," Journal of Econometrics, Elsevier, vol. 18(2), pages 275-279, February.
    4. Menezes, C & Geiss, C & Tressler, J, 1980. "Increasing Downside Risk," American Economic Review, American Economic Association, vol. 70(5), pages 921-932, December.
    5. Amemiya, Takeshi, 1974. "Multivariate Regression and Simultaneous Equation Models when the Dependent Variables Are Truncated Normal," Econometrica, Econometric Society, vol. 42(6), pages 999-1012, November.
    6. Carree, Martin A., 2002. "Technological inefficiency and the skewness of the error component in stochastic frontier analysis," Economics Letters, Elsevier, vol. 77(1), pages 101-107, September.
    7. Aigner, Dennis & Lovell, C. A. Knox & Schmidt, Peter, 1977. "Formulation and estimation of stochastic frontier production function models," Journal of Econometrics, Elsevier, vol. 6(1), pages 21-37, July.
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    Cited by:

    1. William C. Horrace & Ian A. Wright, 2016. "Stationary Points for Parametric Stochastic Frontier Models," Center for Policy Research Working Papers 196, Center for Policy Research, Maxwell School, Syracuse University.
    2. Brian A'Hearn & John Komlos, 2015. "The Decline in the Nutritional Status of the U.S. Antebellum Population at the Onset of Modern Economic Growth," CESifo Working Paper Series 5691, CESifo Group Munich.

    More about this item


    Moments; Stochastic frontier model; Probability; C12; C16; D24;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • D24 - Microeconomics - - Production and Organizations - - - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity


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