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Maximum Likelihood Estimation of Isotonic Normal Means with Unknown Variances

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  • Shi, Ning-Zhong
  • Jiang, Hua

Abstract

To analyze the isotonic regression problem for normal means, it is usual to assume that all variances are known or unknown but equal. This paper then studies this problem in the case that there are no conditions imposed on the variances. Suppose that we have data drawn fromkindependent normal populations with unknown means[mu]i's and unknown variances[sigma]2i's, in which the means are restricted by a given partial ordering. This paper discusses some properties of the maximum likelihood estimates of[mu]i's and[sigma]2i's under the restriction and proposes an algorithm for obtaining the estimates.

Suggested Citation

  • Shi, Ning-Zhong & Jiang, Hua, 1998. "Maximum Likelihood Estimation of Isotonic Normal Means with Unknown Variances," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 183-195, February.
    • Handle: RePEc:eee:jmvana:v:64:y:1998:i:2:p:183-195
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    References listed on IDEAS

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    1. Shi, N. Z., 1994. "Maximum Likelihood Estimation of Means and Variances from Normal Populations Under Simultaneous Order Restrictions," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 282-293, August.
    2. Dykstra, Robert R., 1988. "Trails South: The Wagon-Road Economy in the Dodge City-Panhandle Region. By C. Robert Haywood. Norman: University of Oklahoma Press, 1986. xv + 312 pp. Maps, illustrations, notes, bibliography, and in," Business History Review, Cambridge University Press, vol. 62(1), pages 157-158, April.
    3. G. W. Cran, 1980. "Amalgamation of Means in the Case of Simple Ordering," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 29(2), pages 209-211, June.
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    Cited by:

    1. Yuling Li & Wei Gao & Ning-Zhong Shi, 2007. "A note on multinomial maximum likelihood estimation under ordered restrictions and the EM algorithm," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(1), pages 105-114, July.
    2. Mondal, Anjana & Sattler, Paavo & Kumar, Somesh, 2023. "Testing against ordered alternatives in a two-way model without interaction under heteroscedasticity," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    3. Hoferkamp, Carol & Das Peddada, Shyamal, 2002. "Parameter Estimation in Linear Models with Heteroscedastic Variances Subject to Order Restrictions," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 65-87, July.
    4. Ma, Tiefeng & Wang, Songgui, 2010. "Estimation of means of multivariate normal populations with order restriction," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 594-602, March.
    5. Tao, Jian & Shi, Ning-Zhong & Guo, Jianhua & Gao, Wei, 2002. "Stepwise procedures for the identification of minimum effective dose with unknown variances," Statistics & Probability Letters, Elsevier, vol. 57(2), pages 121-131, April.
    6. Jamshidian, Mortaza, 2004. "On algorithms for restricted maximum likelihood estimation," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 137-157, March.

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