Student’s t-Statistic under Unimodal Densities

Authors

  • Harrie Hendriks Radboud University, Nijmegen, The Netherlands
  • Pieta C. IJzerman-Boon NV Organon, Oss, The Netherlands
  • Chris A.J. Klaassen Universiteit van Amsterdam, Amsterdam, The Netherlands

DOI:

https://doi.org/10.17713/ajs.v35i2&3.361

Abstract

Consider an unknown distribution with a symmetric unimodal density and the induced location-scale family. We study confidence intervals for the location parameter based on Student’s t-statistic, and we conjecture that the uniform distribution is least favorable in that it leads to confidence intervals that are largest given their coverage probability, provided the nominal confidence level is large enough. This conjecture is supported by an argument based on second order asymptotics in the sample size and on asymptotics in
the length of the confidence interval, by a finite sample inequality, and by simulation results.

References

Benjamini, Y. (1983). Is the t test really conservative when the parent distribution is longtailed? Journal of the American Statistical Association, 78, 645-654.

Bowman, K. O., Beauchamp, J. J., and Shenton, L. R. (1977). The distribution of the t-statistic under non-normality. International Statistical Reviews, 45, 233-242.

Chung, K. L. (1946). The approximate distribution of Student’s statistic. Annals of Mathematical Statistics, 17, 447-465.

Edelman, D. (1990a). An inequality of optimal order for the tail probabilities of the t statistic under symmetry. Journal of the American Statistical Association, 85, 120-122.

Edelman, D. (1990b). A note on uniformly most powerful two-sided tests. The American Statistician, 3, 219-220.

Efron, B. (1969). Student’s t-test under symmetry conditions. Journal of the American Statistical Association, 64, 1278-1302.

Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507-521.

Gayen, A. K. (1949). The distribution of ’Student’s’ t in random samples of any size drawn from non-normal universes. Biometrika, 36, 353-369.

Geary, R. C. (1936). The distribution of ”Student’s” ratio for non-normal samples. Supplement Journal of the Royal Statistical Society, 3, 178-184.

Hall, P. (1987). Edgeworth expansion for Student’s t statistic under minimal moment conditions. Annals of Probability, 15, 920-931.

Hotelling, H. (1961). The behavior of some standard statistical tests under nonstandard conditions. In J. Neyman (Ed.), Proceedings of the fourth berkeley symposium on mathematical statistics and probability (Vol. I, p. 319-359). Berkeley: University

of California.

Hyrenius, H. (1950). Distribution of ’Student’-Fisher’s t in samples from compound normal functions. Biometrika, 37, 429-442.

Klaassen, C. A. J., Mokveld, P. J., and van Es, A. J. (2000). Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions. Statistics and Probability Letters, 50, 131-135.

Laderman, J. (1939). The distribution of ”Student’s” ratio for samples of two items drawn from non-normal universes. Annals of Mathematical Statistics, 10, 376-379.

Lehmann, E. L. (1999). “Student” and small-sample theory. Statistical Science, 14, 418-426.

Pearson, K. (1916). Mathematical contributions to the theory of evolution, XIX; Second supplement to a memoir on skew variation. Philos. Trans. Roy. Soc. London Ser. A, 216, 429-457.

Perlo, V. (1933). On the distribution of Student’s ratio for samples drawn from a rectangular distribution. Biometrika, 25, 203-204.

Rider, P. (1929). On the distribution of the ratio of mean to standard deviation in small samples from non-normal universes. Biometrika, 21, 124-143.

Student. (1908). The probable error of a mean. Biometrika, 6, 1-25.

Zwet, W. van. (1964a). Convex transformations: A new approach to skewness and kurtosis. Statist. Neerlandica, 18, 433-441.

Zwet, W. van. (1964b). Convex transformations of random variables (Mathematical Centre Tracts 7 ed.). Amsterdam: Mathematical Centre.

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Published

2016-04-03

How to Cite

Hendriks, H., IJzerman-Boon, P. C., & Klaassen, C. A. (2016). Student’s t-Statistic under Unimodal Densities. Austrian Journal of Statistics, 35(2&3), 131–141. https://doi.org/10.17713/ajs.v35i2&3.361

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