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Poisson Counts, Square Root Transformation and Small Area Estimation

Author

Listed:
  • Malay Ghosh

    (University of Florida)

  • Tamal Ghosh

    (University of Florida)

  • Masayo Y. Hirose

    (Kyushu University)

Abstract

The paper intends to serve two objectives. First, it revisits the celebrated Fay-Herriot model, but with homoscedastic known error variance. The motivation comes from an analysis of count data, in the present case, COVID-19 fatality for all counties in Florida. The Poisson model seems appropriate here, as is typical for rare events. An empirical Bayes (EB) approach is taken for estimation. However, unlike the conventional conjugate gamma or the log-normal prior for the Poisson mean, here we make a square root transformation of the original Poisson data, along with square root transformation of the corresponding mean. Proper back transformation is used to infer about the original Poisson means. The square root transformation makes the normal approximation of the transformed data more justifiable with added homoscedasticity. We obtain exact analytical formulas for the bias and mean squared error of the proposed EB estimators. In addition to illustrating our method with the COVID-19 example, we also evaluate performance of our procedure with simulated data as well.

Suggested Citation

  • Malay Ghosh & Tamal Ghosh & Masayo Y. Hirose, 2022. "Poisson Counts, Square Root Transformation and Small Area Estimation," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 449-471, November.
    • Handle: RePEc:spr:sankhb:v:84:y:2022:i:2:d:10.1007_s13571-021-00269-8
    DOI: 10.1007/s13571-021-00269-8
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    References listed on IDEAS

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    1. M. Ghosh & T. Kubokawa & Y. Kawakubo, 2015. "Benchmarked empirical Bayes methods in multiplicative area-level models with risk evaluation," Biometrika, Biometrika Trust, vol. 102(3), pages 647-659.
    2. Sugasawa, Shonosuke & Kubokawa, Tatsuya, 2017. "Transforming response values in small area prediction," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 47-60.
    3. Yu, Guan, 2009. "Variance stabilizing transformations of Poisson, binomial and negative binomial distributions," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1621-1629, July.
    4. Eric V. Slud & Tapabrata Maiti, 2006. "Mean‐squared error estimation in transformed Fay–Herriot models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 239-257, April.
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