本篇論文主要在研究秩序統計量(order statistic)之費雪訊息。對於任何參數之實數值函數之不偏估計量,其變異數,不論正規條件是否滿足,必存在一下界,亦即為訊息不等式(information inequality)。由此進而可評估此不偏估計量之好壞,當然對參數是為多維度時亦可成立。
在此論文中,我們將說明為何統計量所包涵的訊息數越大,則由此統計量來估計未知參數,將可得較好之結果,
同時本文主要討論,秩序統計量之訊息數,或訊息矩陣。
此外由所得之訊息數或訊息矩陣來了解秩序統計量所包涵之訊息之大小,亦即當樣本數固定時,何處之秩序統計量相對的包涵較多之訊息數。

The purpose of this research is to consider the Fisher information of order statistic. For any unbaised estimator T of real-value function there exists a lower bound for var(T), namely information inequality under regularity assumptions. Also, a similar lower bound exists when these regularity assumptions do not hold. Without loss of generality, we can extend this inequality to multiparameter case. In this paper, we discuss that why more accurately that real-value function of unknow parameter can be estimated when unbaised estimator has larger information. Mainly, we derive the Fisher information or information matrix of order statistics under the family of exponential distribution, for example I_{X_{i:n}},
I_{(X_{r_{1}:n},X_{r_{1}+1:n},cdots,X_{r_{2}:n})} ... etc. Some of them, provide optimal informations.

[1] Balakrishnan, N. and Basu, Asit P. (1995) , The Exponential Distribution:Theory, Methods and Applications, AW Amsterdam, Netherlands.
[2] Chapman, D. G. and Robbins, H. (1951) , ”Minimum variance estimation without regularity assumptions.” Ann. Math. Statist. 22 ,581-586.
[3] Cram´er, H. (1946a) , Mathematical Methods of statistics, Princeton University Press, Princeton.
[4] Cram´er, H. (1946b) , ”A contribution to the theory of statistical estimation.”Skand. Akt. Tidskr. 29 ,85-94.
[5] David, H. A. (1981) , Order statistics, Second edition, John Wiley & Sons, New York.
[6] Darmois, G. (1945) , ”Sur les lois limites de la dispersion de certaines estimations.” Rev. Inst. Int. Statist. 13 ,9-15.
[7] Edgeworth, F. Y. (1908,1909) ”On the probable errors of frequency constants.”J. R. Statist. Soc. 71 ,381-397,499-512,651-678; 72 , 81-90.
[8] Fisher, R. A. (1922) , ”On the mathematical foundations of theoretical statistics.” Philos. Trans. R. Soc. London, Ser. A 222 ,309-368.
[9] Fr´echet, M. (1943) , ”Sur l’extension de certaines evaluations statistiques de petits echantillons.” Rev. Int. Statist. 11 ,182-205.
[10] Johnson, N. L. , Kotz, S. and Balakrishnan, N. (1994) , Continuous Univariate Distribution-Volume 1, Second edition, John Wiley & Sons, New York.
[11] Johnson, N. L. , Kotz, S. and Balakrishnan, N. (1995) , Continuous Univariate Distribution-Volume 2, Second edition, John Wiley & Sons, New York.
[12] Lehmann, E. L. (1983) , Theory of Point Estimation, John Wiley & Sons, New York.
[13] Rao, C. R. (1945) , ”Information and accuracy attainable in the estimation of statistical parameters.” Bull. Calc. Math. Soc. 37 ,81-91.
[14] Rao, C. R. (1947) , ”Minimum variance and the estimation of serveral parameters.” Proc. Camb. Phil. Soc. 43 ,280-283.
[15] Rao, C. R. (1949) , ”Sufficient statistical and minimum variance estimates.”Proc. Camb. Phil. Soc. 45 ,213-218.
[16] Robert V. Hogg and Allen T. Craig (1995) , Introduction To Mathematical Atatistics, Fifth edition, A Simon & Schuster Company, New Jersey.
[17] Shao, J. (1999) , Mathematical Statistics, Springer , New York.
[18] Savage, L. J. (1954,1972) , The foundations of statistics., Wiley, New York. Rev. ed.,Dover Pulications.
[19] Savage, L. J. (1976) , ”On rereading R.A. Fisher (with discussion).”Ann. Statist. 4 ,441-500.
[20] Sen, P. K. and Ghosh, B. K. (1976) , ”Comparison of some bounds in estimation theory.” Ann. Math. Statist. 4 ,755-765
[21] Sukhatem, P. V. (1937) , ”Tests of significance for sample of χ2-population with two degree of freedom.” Ann. of Euge. 8 ,52-56.