IDEAS home Printed from https://ideas.repec.org/a/spr/empeco/v60y2021i6d10.1007_s00181-020-01868-6.html
   My bibliography  Save this article

Evaluating the cdf of the Skew Normal distribution

Author

Listed:
  • Christine Amsler

    (Michigan State University)

  • Alecos Papadopoulos

    (Athens University of Economics and Business)

  • Peter Schmidt

    (Michigan State University)

Abstract

In this paper, we consider various methods for evaluating the cdf of the Skew Normal distribution. This distribution arises in the stochastic frontier model because it is the distribution of the composed error, which is the sum (or difference) of a Normal and a Half-Normal random variable. The cdf must be evaluated in models in which the composed error is linked to other errors using a Copula, in some methods of goodness of fit testing, or in the likelihood of models with sample selection bias. We investigate the accuracy of the evaluation of the cdf using expressions based on the bivariate Normal distribution, and also using simulation methods and some approximations. We find that the expressions based on the bivariate Normal distribution are quite accurate in the central portion of the distribution, and we propose several new approximations that are accurate in the extreme tails. By a simulated example, we show that the use of approximations instead of the theoretical exact expressions may be critical in obtaining meaningful and valid estimation results.

Suggested Citation

  • Christine Amsler & Alecos Papadopoulos & Peter Schmidt, 2021. "Evaluating the cdf of the Skew Normal distribution," Empirical Economics, Springer, vol. 60(6), pages 3171-3202, June.
    • Handle: RePEc:spr:empeco:v:60:y:2021:i:6:d:10.1007_s00181-020-01868-6
    DOI: 10.1007/s00181-020-01868-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00181-020-01868-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00181-020-01868-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Christine Amsler & Peter Schmidt & Wen-Jen Tsay, 2019. "Evaluating the CDF of the distribution of the stochastic frontier composed error," Journal of Productivity Analysis, Springer, vol. 52(1), pages 29-35, December.
    2. Loperfido, Nicola, 2002. "Statistical implications of selectively reported inferential results," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 13-22, January.
    3. Antonella Capitanio, 2010. "On the approximation of the tail probability of the scalar skew-normal distribution," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 299-308.
    4. Wen-Jen Tsay & Cliff Huang & Tsu-Tan Fu & I.-Lin Ho, 2013. "A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models," Journal of Productivity Analysis, Springer, vol. 39(3), pages 259-269, June.
    5. Yalta, A. Talha & Schreiber, Sven, 2012. "Random Number Generation in gretl," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 50(c01).
    6. Alecos Papadopoulos, 2015. "The half-normal specification for the two-tier stochastic frontier model," Journal of Productivity Analysis, Springer, vol. 43(2), pages 225-230, April.
    7. Meeusen, Wim & van den Broeck, Julien, 1977. "Efficiency Estimation from Cobb-Douglas Production Functions with Composed Error," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(2), pages 435-444, June.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zhao, Shirong, 2021. "Quantile estimation of stochastic frontier models with the normal–half normal specification: A cumulative distribution function approach," Economics Letters, Elsevier, vol. 206(C).
    2. Gaygysyz Guljanov & Willi Mutschler & Mark Trede, 2022. "Pruned Skewed Kalman Filter and Smoother: With Application to the Yield Curve," CQE Working Papers 10122, Center for Quantitative Economics (CQE), University of Muenster.
    3. Simos G. Meintanis & Christos K. Papadimitriou, 2022. "Goodness--of--fit tests for stochastic frontier models based on the characteristic function," Journal of Productivity Analysis, Springer, vol. 57(3), pages 285-296, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Tai-Hsin Huang & Nan-Hung Liu & Subal C. Kumbhakar, 2018. "Joint estimation of the Lerner index and cost efficiency using copula methods," Empirical Economics, Springer, vol. 54(2), pages 799-822, March.
    2. Christopher F. Parmeter, 2018. "Estimation of the two-tiered stochastic frontier model with the scaling property," Journal of Productivity Analysis, Springer, vol. 49(1), pages 37-47, February.
    3. Huang, Tai-Hsin & Chen, Kuan-Chen & Lin, Chung-I, 2018. "An extension from network DEA to copula-based network SFA: Evidence from the U.S. commercial banks in 2009," The Quarterly Review of Economics and Finance, Elsevier, vol. 67(C), pages 51-62.
    4. Alecos Papadopoulos, 2023. "The noise error component in stochastic frontier analysis," Empirical Economics, Springer, vol. 64(6), pages 2795-2829, June.
    5. Tai-Hsin Huang & Nan-Hung Liu, 2014. "Bank competition in transition countries: Are those markets really in equilibrium?," Empirical Economics, Springer, vol. 47(4), pages 1283-1316, December.
    6. Martini, Gianmaria & Scotti, Davide & Viola, Domenico & Vittadini, Giorgio, 2020. "Persistent and temporary inefficiency in airport cost function: An application to Italy," Transportation Research Part A: Policy and Practice, Elsevier, vol. 132(C), pages 999-1019.
    7. Huang, Tai-Hsin & Chiang, Dien-Lin & Lin, Chung-I, 2017. "A new approach to estimating a profit frontier using the censored stochastic frontier model," The North American Journal of Economics and Finance, Elsevier, vol. 39(C), pages 68-77.
    8. Huang, Tai-Hsin & Chiang, Dien-Lin & Chao, Shih-Wei, 2017. "A new approach to jointly estimating the Lerner index and cost efficiency for multi-output banks under a stochastic meta-frontier framework," The Quarterly Review of Economics and Finance, Elsevier, vol. 65(C), pages 212-226.
    9. Christine Amsler & Peter Schmidt & Wen-Jen Tsay, 2019. "Evaluating the CDF of the distribution of the stochastic frontier composed error," Journal of Productivity Analysis, Springer, vol. 52(1), pages 29-35, December.
    10. Shih-Tang Hwu & Tsu-Tan Fu & Wen-Jen Tsay, 2021. "Estimation and efficiency evaluation of stochastic frontier models with interval dependent variables," Journal of Productivity Analysis, Springer, vol. 56(1), pages 33-44, August.
    11. Simos G. Meintanis & Christos K. Papadimitriou, 2022. "Goodness--of--fit tests for stochastic frontier models based on the characteristic function," Journal of Productivity Analysis, Springer, vol. 57(3), pages 285-296, June.
    12. Paul Hewson & Keming Yu, 2008. "Quantile regression for binary performance indicators," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(5), pages 401-418, September.
    13. Tom Kompas & Tuong Nhu Che & R. Quentin Grafton, 2004. "Technical efficiency effects of input controls: evidence from Australia's banana prawn fishery," Applied Economics, Taylor & Francis Journals, vol. 36(15), pages 1631-1641.
    14. Lundgren, Tommy & Marklund, Per-Olov & Zhang, Shanshan, 2016. "Industrial energy demand and energy efficiency – Evidence from Sweden," Resource and Energy Economics, Elsevier, vol. 43(C), pages 130-152.
    15. Andriakopoulos, Konstantinos & Ladas, Augoustinos & Andriakopoulos, Panagiotis, 2020. "Bank efficiency and leasing in U.S.A. banking system," MPRA Paper 112645, University Library of Munich, Germany.
    16. Jan Kluge & Sarah Lappöhn & Kerstin Plank, 2023. "Predictors of TFP growth in European countries," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 50(1), pages 109-140, February.
    17. Giovanni Calice & Levent Kutlu & Ming Zeng, 2021. "Understanding US firm efficiency and its asset pricing implications," Empirical Economics, Springer, vol. 60(2), pages 803-827, February.
    18. Khanal, Aditya & Koirala, Krishna & Regmi, Madhav, 2016. "Do Financial Constraints Affect Production Efficiency in Drought Prone Areas? A Case from Indonesian Rice Growers," 2016 Annual Meeting, February 6-9, 2016, San Antonio, Texas 230087, Southern Agricultural Economics Association.
    19. Wu, Yanrui, 1995. "The productive efficiency of Chinese iron and steel firms A stochastic frontier analysis," Resources Policy, Elsevier, vol. 21(3), pages 215-222, September.
    20. Firna Varina & Sri Hartoyo & Nunung Kusnadi & Amzul Rifin, 2020. "The Determinants of Technical Efficiency of Oil Palm Smallholders in Indonesia," International Journal of Economics and Financial Issues, Econjournals, vol. 10(6), pages 89-93.

    More about this item

    Keywords

    Skew Normal distribution; Bivariate Normal distribution; Stochastic frontier; Simulation; Computational software;
    All these keywords.

    JEL classification:

    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C87 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Econometric Software
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:empeco:v:60:y:2021:i:6:d:10.1007_s00181-020-01868-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.