Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Bayesian Estimation of Shape Parameter of Pareto Income Distribution Using LINEX Loss Function

  • Saxena, Sharad (Operations & Quantitative Methods Area, Institute of Management Technology) ;
  • Singh, Housila P. (School of Studies in Statistics, Vikram University)
  • Published : 2007.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

The economic world is full of patterns, many of which exert a profound influence over society and business. One of the most contentious is the distribution of wealth. Way back in 1897, an Italian engineer-turned-economist named Vilfredo Pareto discovered a pattern in the distribution of wealth that appears to be every bit as universal as the laws of thermodynamics or chemistry. The present paper proposes some Bayes estimators of shape parameter of Pareto income distribution in censored sampling. Asymmetric LINEX loss function has been considered to study the effects of overestimation and underestimation. For the prior distribution of the parameter involved a number of priors including one and two-parameter exponential, truncated Erlang and doubly truncated gamma have been contemplated to express the belief of the experimenter s/he has regarding the parameter. The estimators thus obtained have been compared theoretically and empirically with the corresponding estimators under squared error loss function, some of which were reported by Bhattacharya et al. (1999).

Keywords

References

  1. Aitchison, J. and Dunsmore, I. R. (1975). Statistical Prediction Analysis. Cambridge University Press, London
  2. Arnold, B. C. (1983). Pareto Distributions. International Cooperative Publishing House. Arnold, B. C. and Press, S. J. (1983). Bayesian inference for Pareto population. Journal of Econometrics, 21, 287-306 https://doi.org/10.1016/0304-4076(83)90047-7
  3. Arnold, B. C. and Press, S. J. (1989). Bayesian estimation and prediction for Pareto data. Journal of American Statistical Association, 84, 1079-1084 https://doi.org/10.2307/2290086
  4. Arnold, B. C. and Press, S. J. (1986). Bayesian analysis of censored and grouped data from Pareto populations. Bayesian Inference and Decision Techniques (P. Goel and A. Zellner eds.), North-Holland, Amsterdam
  5. Berger, J. O. (1980). Statistical Decision Theory: Foundations, Concepts and Methods. Springer-Verlag, New York
  6. Bhattacharya, S. K., Chaturvedi, A. and Singh, N. K. (1999). Bayesian estimation for the Pareto income distribution. Statistical Papers, 40, 247-262 https://doi.org/10.1007/BF02929874
  7. Buchanan, M. (2002). Wealth happens. Harvard Business Review, April, 3-7
  8. Chattopadhyay, S., Chaturvedi, A. and Sengupta, R. N. (2000). LINEX loss function and its statistical applications. A review. Decision, 26, 51-66
  9. Dagum, C. (1988). Gini ratio. The New Palgrave: A Dictionary of Economics (Eatwell, John, Murray Milgate and Peter Newman eds.), 1
  10. Davis, H. T. and Feldstein, M. L. (1979). The generalized Pareto law as a model for progressively censored survival data. Biometrika, 66, 299-306 https://doi.org/10.1093/biomet/66.2.299
  11. DeGroot, M. H. (1970). Optimal Statistical Decisions. McGraw-Hill, New York
  12. De Lima, P. J. F. (1997). On the robustness of non-linearity tests to moment condition failure. Journal of Econometrics, 76, 251-280 https://doi.org/10.1016/0304-4076(95)01791-7
  13. De Vany, A. and Walls, W. D. (1999). Uncertainty in movie industry: Does star power reduce the terror of the box office? Journal of Cultural Economics, 23, 285-318 https://doi.org/10.1023/A:1007608125988
  14. De Vany, A. (2003). Hollywood Economics. Routledge, London
  15. Ferguson, T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York
  16. Ganguly, A., Singh, N. K, Choudhuri, H. and Bhattacharya, S. K (1992). Bayesian estimation of the GINI index for PID. Test, 1, 93-104 https://doi.org/10.1007/BF02562665
  17. Geisser, S. (1984). Predicting Pareto and exponential observable. Canadian Journal of Statistics, 12, 143-152 https://doi.org/10.2307/3315178
  18. Geisser, S. (1985). Interval prediction for Pareto and exponential observables. Journal of Econometrics, 29, 173-185 https://doi.org/10.1016/0304-4076(85)90038-7
  19. Gomes, C. P., Selman, B. and Crato, N. (1997). Heavy-tailed distributions in combinatorial search. Principles and Practice of Constraint Programming, CP 97. (Gert Smolka, ed.) Lecture Notes in Computer Science 1330, Springer, 121-135
  20. Hagstroem, K G. (1960). Remarks on Pareto distributions. Scandinavian Actuarial Journal, 59-71
  21. Harris, C. M. (1968). The Pareto distribution as a queue service discipline. Operations Research, 16, 307-313 https://doi.org/10.1287/opre.16.2.307
  22. Huber, P. J. (1981). Robust Statistics. John Wiley & Sons, New York
  23. Launer, R. L. and Wilkinson, G. N. (eds.) (1979). Robustness in Statistics. Academic Press, New York
  24. Liang, T. C. (1993). Convergence rates for empirical Bayes estimation of the scale parameter in a Pareto distribution. Computational Statistics and Data Analysis, 16, 34-45
  25. Lwin, T. (1972). Estimation of the tail of the Paretian law. Scandinavian Actuarial Journal, 55,170-178
  26. Mandelbrot, B. (1960). The Pareto-levy law and the distribution of income. International Economics Review, 1, 79-106 https://doi.org/10.2307/2525289
  27. Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, University of Chicago, 36, 394-419
  28. Mandelbrot, B. (1967). The variation of some other speculative prices. Journal of Business, University of Chicago, 40, 393-413
  29. Nigm, A. M. and Hamdy, H. I. (1987). Bayesian prediction bands for the Pareto lifetime model. Communications in Statistics - Theory and Methods, 16, 1791-1772
  30. Pareto, V. (1897). Cours d' Economie Politique. Rouge et Cie, Paris
  31. Pigou, A. C. (1932). The Economics of Welfare. The McMillan Co., London
  32. Raiffa, H. and Schlaifer, R. (1961). Applied Statistical Decision Theory. Harvard University Press, Boston
  33. Sinha, S. and Pan, R. K. (2005). Blockbusters, Bombs and Sleepers: The income distribution of movies. Econophysics of Wealth Distribution (A. Chatterjee, B. K. Chakrabarti and S. Yarlagadda eds.), Springer, 43-47
  34. Sinha, S. and Raghavendra, S. (2004). Hollywood blockbusters and long-tailed distributions: An empirical study of the popularity of movies. The European Physical Journal-B. 42, 293-296 https://doi.org/10.1140/epjb/e2004-00382-7
  35. Sornette, D. and Zajdenweber, D. (1999). Economic returns of research: The Pareto law and its implications. The European Physical Journal-B, 8, 653-664 https://doi.org/10.1007/s100510050733
  36. Upadhyaya, S. K. and Shastri, V. (1997). Bayes results for classical Pareto distribution via Gibbs sampler with doubly censored observations. IEEE Transactions on Reliability, 46, 56-59 https://doi.org/10.1109/24.589927
  37. Varian, H. R. (1975). A Bayesian approach to real estate assessment. Studies in Bayesian Econometrics and Statistics in honour of Leonard J. Savage (Stephen E. Fienberg and Arnold Zellner eds.), North-Holland, Amsterdam, 195-208
  38. Wetherill, G. B. (1961). Bayesian sequential analysis. Biometrika, 48, 281-292 https://doi.org/10.1093/biomet/48.3-4.281
  39. Zellner, A. and Geisel, M. S. (1968). Sensitivity of control to uncertainty and form of the criterion function. The Future of Statistics (Donald G. Watts ed.), Academic Press, New York,269-289
  40. Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. John Wiley & Sons
  41. Zellner, A. (1986). Bayesian estimation and prediction using asymmetric loss functions. Journal of American Statistical Association, 81, 446-451 https://doi.org/10.2307/2289234