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Minimum Chi-Square, not Maximum Likelihood!
Abstract
The sovereignty of MLE is questioned. Minimum $\chi^2_\lambda$ yields the same estimating equations as MLE. For many cases, as illustrated in presented examples, and further algorithmic exploration in progress may show that for all cases, minimum $\chi^2_\lambda$ estimates are available. In this sense minimum $\chi^2$ is the basic principle of estimation. The criterion of asymptotic sufficiency which has been called "second order efficiency" is rejected as a criterion of goodness of estimate as against some loss function such as the mean squared error. The relation between MLE and sufficiency is not assured, as illustrated in an example in which MLE yields $\infty$ as estimate with samples that have different values of the sufficient statistic. Other examples are cited in which minimal sufficient statistics exist but where the MLE is not sufficient. The view is advanced that statistics is a science, not mathematics or philosophy (inference) and as such requires that any claimed attributes of the MLE must be testable by a Monte Carlo experiment.
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“For binned data, the analogue of Least Square regression method consists of minimizing the χ 2 (Chi-square) statistics [23].”
From Estimating the Total Volume of Queries to a Search Engine (2022) · cites this paper as [23]
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Research works printed but not linked: 8
- [4]BERksoN, J. (1951). Relative precision of minimum chi-square and maximum likelihood estimates of regression coefficients. Proc. Second Berkeley Symp. Math. Statist. Probability 471-479. Univ. California Press.
- [10]BERksoN, J. (1966). Examination of randomness of α-particle emissions. In Festschrift for J. Neyman Research Papers in Statistics. (F. N. David, ed.) 37–53. Wiley, London, New York, Sydney.
- [11]BERksoN, J. (1968). Application of minimum logit χ² estimate to a problem of Grizzle with a notation on the problem of "no interaction," with discussion. In The Future of Statistics. 175-200. Academic Press, New York and London.
- [17]BERKsoN, J. AND HoDGEs, J. L., JR. (1961). A minimax estimator for the logistic function. Proc. Fourth Berkeley Symp. Math. Statist. Probability 4 77–86.
- [21]CRAMÉR, H. (1946). Mathematical Methods of Statistics. Princeton Univ. Press.
- [23]FisHER, R. A. (1922). On the mathematical foundations of theoretical statistics. Phil. Trans. Roy. Soc. London, Ser. A 222 309–368.
- [24]FisHER, R. A. (1925). Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22 700–725. likelihood, inverse probability and the method of moments". Ann. Eugenics 8 146–151.
- [34]RAo, C. R. (1961b). Asymptotic efficiency and limiting information. Proc. Fourth Berkeley Symp. Math. Statist. Probability 1 531–545.
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Estimating the Total Volume of Queries to a Search Engine
Lillo, F., & Ruggieri, S. (2022) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2021.3054668 · Cited by 4
Metadata
- Authors
- Berkson, J.
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1980-05-01
- Volume
- 8
- Issue
- 3
- Pages
- 457-487
- ISSN
- 0090-5364
- 2168-8966
- Cited by
- 193
- References
- 0
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