Annals of the Institute of Statistical Mathematics

, Volume 39, Issue 3, pp 627–635

# On the logistic midrange

• E. Olusegun George
• Cecil C. Rousseau
Article

## Summary

It is well-known that for a large family of distributions, the sample midrange is asymptotically logistic. In this article, the logistic midrange is closely examined. Its distribution function is derived using Dixon's formula (Bailey (1935,Generalized Hypergeometric Series, Cambridge University Press, p. 13)) for the generalized hypergeometric function with unit argument, together with appropriate techniques for the inversion of (bilateral) Laplace transforms. Several relationships in distribution are established between the midrange and sample median of the logistic and Laplace random variables. Possible applications in testing for outliers are also discussed.

## Key words and phrases

Characteristic functions sample median logistic distribution Laplace distribution

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## References

1. 
Amemiya, T. (1980). Then −2-order mean squared errors of the maximum chi-square estimator,Ann. Statist.,8, 488–503.
2. 
Bailey, W. N. (1935).Generalized Hypergeometric Series, Cambridge University Press, London.
3. 
Berkson, J. (1944). Application of the logistic function to bio-assay,J. Amer. Statist. Ass.,39, 357–365.Google Scholar
4. 
Cox, D. R. (1970).Analysis of Binary Data, Methuen, London.
5. 
Galambos, J. (1978).The Asymptotic Theory of Extreme Order Statistics, Wiley, New York.
6. 
George, E. O. and Rousseau, C. C. (1986). A moment generating function, Problem 85-22 in Problems and Solutions,Soc. Industr. Appl. Math. (Review). (to appear).Google Scholar
7. 
Gumbel, E. J. (1944). Ranges and midranges,Ann. Math. Statist.,15, 414–422.
8. 
Karlin, S. (1968).Total Positivity, 1, Stanford University Press, Stanford, California.
9. 
Knuth, D. E. (1973).The Art of Computer Programming, 1, Fundamental Algorithms, Addison-Wesley, Reading, Massachusetts.Google Scholar
10. 
Pearl, R. and Reed, L. J. (1920). On the rate of growth of population of the United States since 1790 and its mathematical representation,Proc. Nat. Acad. Sci. USA,6, 275–288.
11. 
Plackett, R. L. (1959). The analysis of life test data,Technometrics,1, 9–19.
12. 
Verhulst, P. F. (1845). Recherches mathematiques sur la loi d'accroisement de la population,Nouveaux memoirs de l'ac Bruxelles,18, 1–38.Google Scholar

© The Institute of Statistical Mathematics, Tokyo 1987

## Authors and Affiliations

• E. Olusegun George
• 1
• Cecil C. Rousseau
• 1
1. 1.Memphis State UniversityUSA