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Moments of the truncated normal distribution

Author

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  • William Horrace
Abstract
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

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    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. Robin C. Sickles & William C. Horrace (ed.), 2014. "Festschrift in Honor of Peter Schmidt," Springer Books, Springer, edition 127, number 978-1-4899-8008-3, February.
    8. 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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    Citations

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    Cited by:

    1. Siqi Chen & Wenhao Gui, 2020. "Estimation of Unknown Parameters of Truncated Normal Distribution under Adaptive Progressive Type II Censoring Scheme," Mathematics, MDPI, vol. 9(1), pages 1-33, December.
    2. William C. Horrace & Ian A. Wright, 2020. "Stationary Points for Parametric Stochastic Frontier Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(3), pages 516-526, July.
    3. Ogasawara, Haruhiko, 2021. "A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    4. Jun Cai & William C. Horrace & Christopher F. Parmeter, 2021. "Density deconvolution with Laplace errors and unknown variance," Journal of Productivity Analysis, Springer, vol. 56(2), pages 103-113, December.
    5. Christopher J. Adcock, 2022. "Properties and Limiting Forms of the Multivariate Extended Skew-Normal and Skew-Student Distributions," Stats, MDPI, vol. 5(1), pages 1-42, March.
    6. Christopher F. Parmeter & Shirong Zhao, 2023. "An alternative corrected ordinary least squares estimator for the stochastic frontier model," Empirical Economics, Springer, vol. 64(6), pages 2831-2857, June.
    7. Moshe Pollak & Michal Shauly-Aharonov, 2019. "A Double Recursion for Calculating Moments of the Truncated Normal Distribution and its Connection to Change Detection," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 889-906, September.
    8. 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.
    9. Kiril Tenekedjiev & Simon Cooley & Boyan Mednikarov & Guixin Fan & Natalia Nikolova, 2021. "Reliability Simulation of Two Component Warm-Standby System with Repair, Switching, and Back-Switching Failures under Three Aging Assumptions," Mathematics, MDPI, vol. 9(20), pages 1-40, October.

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    More about this item

    Keywords

    Moments; Stochastic frontier model; Probability; C12; C16; D24;
    All these keywords.

    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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