IDEAS home Printed from https://ideas.repec.org/a/spr/jstada/v4y2017i1d10.1186_s40488-017-0059-2.html
   My bibliography  Save this article

Recent developments on the moment problem

Author

Listed:
  • Gwo Dong Lin

    (Academia Sinica)

Abstract

We consider univariate distributions with finite moments of all positive orders. The moment problem is to determine whether or not a given distribution is uniquely determined by the sequence of its moments. There is a huge literature on this classical topic. In this survey, we will focus only on the recent developments on the checkable moment-(in)determinacy criteria including Cramér’s condition, Carleman’s condition, Hardy’s condition, Krein’s condition and the growth rate of moments, which help us solve the problem more easily. Both Hamburger and Stieltjes cases are investigated. The former is concerned with distributions on the whole real line, while the latter deals only with distributions on the right half-line. Some new results or new simple (direct) proofs of previous criteria are provided. Finally, we review the most recent moment problem for products of independent random variables with different distributions, which occur naturally in stochastic modelling of complex random phenomena.

Suggested Citation

  • Gwo Dong Lin, 2017. "Recent developments on the moment problem," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-17, December.
    • Handle: RePEc:spr:jstada:v:4:y:2017:i:1:d:10.1186_s40488-017-0059-2
    DOI: 10.1186/s40488-017-0059-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1186/s40488-017-0059-2
    File Function: Abstract
    Download Restriction: no
    File URL: https://libkey.io/10.1186/s40488-017-0059-2?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
    ---><---

    References listed on IDEAS

    as
    1. Christian Kleiber, 2014. "The Generalized Lognormal Distribution and the Stieltjes Moment Problem," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1167-1177, December.
    2. Gwo Dong Lin, 1997. "On the moment problems," Statistics & Probability Letters, Elsevier, vol. 35(1), pages 85-90, August.
    3. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    4. Ostrovska, Sofiya & Stoyanov, Jordan, 2005. "Stieltjes classes for M-indeterminate powers of inverse Gaussian distributions," Statistics & Probability Letters, Elsevier, vol. 71(2), pages 165-171, February.
    5. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    6. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    7. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    8. Christian Kleiber, 2013. "On moment indeterminacy of the Benini income distribution," Statistical Papers, Springer, vol. 54(4), pages 1121-1130, November.
    9. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    10. Stoyanov, Jordan & Tolmatz, Leonid, 2004. "New Stieltjes classes involving generalized gamma distributions," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 213-219, August.
    11. Martin A. Lariviere, 2006. "A Note on Probability Distributions with Increasing Generalized Failure Rates," Operations Research, INFORMS, vol. 54(3), pages 602-604, June.
    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. Ostrovska, Sofiya & Turan, Mehmet, 2019. "q-Stieltjes classes for some families of q-densities," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 118-123.
    2. Gaillac, Christophe & Gautier, Eric, 2021. "Non Parametric Classes for Identification in Random Coefficients Models when Regressors have Limited Variation," TSE Working Papers 21-1218, Toulouse School of Economics (TSE).
    3. Botosaru, Irene, 2023. "Time-varying unobserved heterogeneity in earnings shocks," Journal of Econometrics, Elsevier, vol. 235(2), pages 1378-1393.
    4. P. Patie & A. Vaidyanathan, 2022. "Non‐classical Tauberian and Abelian type criteria for the moment problem," Mathematische Nachrichten, Wiley Blackwell, vol. 295(5), pages 970-990, May.
    5. Botosaru, Irene, 2020. "Nonparametric analysis of a duration model with stochastic unobserved heterogeneity," Journal of Econometrics, Elsevier, vol. 217(1), pages 112-139.
    6. Tibor KPogany, 2018. "On a statistical approximation model of probabilitydensity function of non-negative random variables," Biostatistics and Biometrics Open Access Journal, Juniper Publishers Inc., vol. 8(4), pages 69-72, November.
    7. Arcia-Garibaldi, Guadalupe & Cruz-Romero, Pedro & Gómez-Expósito, Antonio, 2018. "Future power transmission: Visions, technologies and challenges," Renewable and Sustainable Energy Reviews, Elsevier, vol. 94(C), pages 285-301.
    8. Wei, Yixi & Ma, Jiang-Hong, 2021. "Determinacy of a distribution with finitely many mass points by finitely many moments," Statistics & Probability Letters, Elsevier, vol. 176(C).
    9. Jornet, Marc, 2021. "Beyond the hypothesis of boundedness for the random coefficient of the Legendre differential equation with uncertainties," Applied Mathematics and Computation, Elsevier, vol. 391(C).

    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. López-García, Marcos, 2009. "Characterization of solutions to the Stieltjes-Wigert moment problem," Statistics & Probability Letters, Elsevier, vol. 79(10), pages 1337-1342, May.
    2. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    3. Zhiqiang Zheng & Balaji Padmanabhan & Steven O. Kimbrough, 2003. "On the Existence and Significance of Data Preprocessing Biases in Web-Usage Mining," INFORMS Journal on Computing, INFORMS, vol. 15(2), pages 148-170, May.
    4. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Other publications TiSEM 73e2f01b-ad4d-4447-95ba-a, Tilburg University, School of Economics and Management.
    5. Carlos R. Handy & Daniel Vrinceanu & Carl B. Marth & Harold A. Brooks, 2015. "Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure," Mathematics, MDPI, vol. 3(4), pages 1-24, November.
    6. Allen C. Goodman & Miron Stano, 2000. "Hmos and Health Externalities: A Local Public Good Perspective," Public Finance Review, , vol. 28(3), pages 247-269, May.
    7. Bode, Sven & Michaelowa, Axel, 2003. "Avoiding perverse effects of baseline and investment additionality determination in the case of renewable energy projects," Energy Policy, Elsevier, vol. 31(6), pages 505-517, May.
    8. Ala, Guido & Fasshauer, Gregory E. & Francomano, Elisa & Ganci, Salvatore & McCourt, Michael J., 2017. "An augmented MFS approach for brain activity reconstruction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 3-15.
    9. Bettina Campedelli & Andrea Guerrina & Giulia Romano & Chiara Leardini, 2014. "La performance della rete ospedaliera pubblica della regione Veneto. L?impatto delle variabili ambientali e operative sull?efficienza," MECOSAN, FrancoAngeli Editore, vol. 2014(92), pages 119-142.
    10. Haider A. Khan, 2004. "General Conclusions: From Crisis to a Global Political Economy of Freedom," Palgrave Macmillan Books, in: Global Markets and Financial Crises in Asia, chapter 9, pages 193-211, Palgrave Macmillan.
    11. Penn Loh & Zoë Ackerman & Joceline Fidalgo & Rebecca Tumposky, 2022. "Co-Education/Co-Research Partnership: A Critical Approach to Co-Learning between Dudley Street Neighborhood Initiative and Tufts University," Social Sciences, MDPI, vol. 11(2), pages 1-17, February.
    12. Broekhuis, Manda & Vos, Janita F.J., 2003. "Improving organizational sustainability using a quality perspective," Research Report 03A43, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    13. O'Brien, Raymond & Patacchini, Eleonora, 2003. "Testing the exogeneity assumption in panel data models with "non classical" disturbances," Discussion Paper Series In Economics And Econometrics 0302, Economics Division, School of Social Sciences, University of Southampton.
    14. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2002. "Perfection and Stability of Stationary Points with Applications in Noncooperative Games," Discussion Paper 2002-108, Tilburg University, Center for Economic Research.
    15. Edcarlos D. Silva & J. C. Albuquerque & T. R. Cavalcante, 2021. "Fourth-order nonlocal type elliptic problems with indefinite nonlinearities," Partial Differential Equations and Applications, Springer, vol. 2(2), pages 1-22, April.
    16. YongSeog Kim & W. Nick Street & Gary J. Russell & Filippo Menczer, 2005. "Customer Targeting: A Neural Network Approach Guided by Genetic Algorithms," Management Science, INFORMS, vol. 51(2), pages 264-276, February.
    17. Montijano, J.I. & Rández, L. & Van Daele, M. & Calvo, M., 2020. "On the numerical stability of the exponentially fitted methods for first order IVPs," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    18. Yanling Li & Zita Oravecz & Shuai Zhou & Yosef Bodovski & Ian J. Barnett & Guangqing Chi & Yuan Zhou & Naomi P. Friedman & Scott I. Vrieze & Sy-Miin Chow, 2022. "Bayesian Forecasting with a Regime-Switching Zero-Inflated Multilevel Poisson Regression Model: An Application to Adolescent Alcohol Use with Spatial Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 376-402, June.
    19. Jensen, Nathan M. & Li, Quan & Rahman, Aminur, 2007. "Heard melodies are sweet, but those unheard are sweeter : understanding corruption using cross-national firm-level surveys," Policy Research Working Paper Series 4413, The World Bank.
    20. Oscar J. Cacho & Robyn L. Hean & Russell M. Wise, 2003. "Carbon‐accounting methods and reforestation incentives," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 47(2), pages 153-179, June.

    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:jstada:v:4:y:2017:i:1:d:10.1186_s40488-017-0059-2. 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.