分佈刻劃數十年來一直是統計理論的一門重要課題。雖然已經有許多大家熟悉的結果，對於在應用上扮演重要角色的分佈，例如常態及gamma，新的刻劃結果依然相當具有吸引力。在實際應用上，有時我們會猜測觀測到的資料究竟服從什麼分佈，有時會利用刻劃上的特性去推測分佈。本文我們將先探討gamma分佈及利用刻劃上的特性所做的參數估計方法。其次，我們利用電腦模擬來進行一些分析研究。

Characterization of distributions has been an important topic in statistical theory for decades. Although there have been many well known results already developed, it is still of great interest to find new characterizations of commonly used distributions in application, such as normal or gamma distribution. In practice, sometimes we make guesses on the distribution to be fitted to the data observed, sometimes we use the characteristic properties of those distributions to do so. In this paper we will restrict our attention to the characterizations of gamma distribution as well as some related studies on the corresponding parameter estimation based on the characterization properties. Some simulation studies are also given.

Chapter 1: Introduction and Review
Chapter 2: Characterizations of the Gamma Distribution via Conditional Moments
Chapter 3: Characterizations of the Bivariate Gamma distribution
Chapter 4: Characterizations of the Gamma and Generalized Inverse Gaussian distributions
Chapter 5: Characterizations of the Gamma Distribution via Conditional Expectations
Chapter 6: On Parameter Estimation of two Independent Gamma Population With the Same Scale Parameter
Chapter 7: Conclusion

item[]Alzaid, A.A., Al-Osh, M.A. 1991. Characterizations of
probability distributions based on the relation $Xstackrel{
m
d}{=}U(X_{1}+X_{2}).$ extit{Sankhy={a} B} extbf{53}, 188-190.
item[]Andersen, P.K., Gill, R.D. 1982. Cox's regression model for counting processes:
a large sample study. extit{Ann. Stat.} extbf{10}, 1100-1120.
item[]Bar-Lev, S.K., Bshouty, D., Enis, P., Letac, G., Lu, I. , and
Richards, D. 1994. The diagonal multivariate natural exponential
families and their classification. extit{J. Theor. Prob.}
extbf{4}, 883-929.
item[]Bobecka, K. 2002. Regression versions of Lukacs type
characterizations for the bivariate gamma distribution.
extit{J. Appl. Stat. Sci.} extbf{11}, 213-233.
item[]Bobecka, K., Wesolowski, J.
2002a. Three dual regression schemes for the Lukacs theorem.
extit{Metrika} extbf{56}, 55-72.
item[]Bobecka, K., Wesolowski, J. 2002b. The
Lukacs-Olkin-Rubin theorem without invariance of the "quotient".
extit{Studia Math.} extbf{152}, 147-160.
item[]Bolger, E.M., Harkness, W. L.
1965. A characterization of some distributions by conditional
moments. extit{Ann. Math. Stat.} extbf{36}, 703-705.
item[]Casalis , M.,
Letac, G. 1996. The Lukacs-Olkin-Rubin characterization of
Wishart distributions on symmetric cones. extit{Ann. Stat.}
extbf{24}, 763-786.
item[]Chhikara, R.S., Folks, J.L. 1989. extit{The Inverse
Gaussian Distribution: Theory, Methodology and Applications.}
Marcel Dekker Inc., New York.
item[]Chung, K.L. 2001. extit{A course in probability theory},
3rd ed. Academic Press.
item[]Cox , D.R. 1972. Regression models and life tables (with discussion).
extit{J. Roy. Stat. Soc. B} extbf{34}, 187-220.
item[]Hall, W.J., Simons, G. 1969. On characterizations of the
gamma distribution. extit{Sankhy={a} A} extbf{31}, 385-390.
item[]Huang, W.J., Chen, L.S. 1989. Note on a characterization
of gamma distributions. extit{Stat. Prob. Lett.} extbf{8},
485-487.
item[]Huang, W.J., Chen, L.S. 1991. On a study of certain power
mixtures. extit{Chinese J. Math.} extbf{19}, 95-104.
item[]Huang, W.J., Su, J.C. 1997. On a study of renewal process
connected with certain conditional moments. extit{Sankhy={a}
A} extbf{59}, 28-41.
item[]Hwang, T.Y., Hu, C.Y. 1999. On a characterization of the Gamma
distribution: the independence of the sample mean and the sample
coefficient of variation. extit{Ann. Inst. Stat. Math.}
extbf{51}, 749-753.
item[]Hwang, T.Y., Hu, C.Y. 2000. On some characterizations of
population distributions. extit{Taiwanese J. Math.} extbf{4},
427-437.
item[]Hwang, T.Y., Huang, P.H. 2002. On new moment estimation of
parameters of gamma distribution using it's characterization.
extit{Ann. Inst. Stat. Math.} extbf{54}, 840-847.
item[]Johnson, N.L., Kotz, S. 1994. extit{Distributions in Statistics:
Continuous Univariate Distributions} I, 2nd edition. John Wiley
& Sons, New York.
item[]Kagan, A.M., Linnik, Y.V., Rao, C.R. 1973. extit{Characterization
problems in mathematical statistics.} Wiley, New York (translated
by B. Ramachandran).
item[]Laha, R.G. 1964. On a problem connected with beta and gamma
distributions, extit{Transactions of American Mathematical
Society} extbf{113}, 287-298.
item[]Letac, G., Massam, H. 1998. Quadratic and inverse regressions
for Wishart distributions. extit{Ann. Stat.}
extbf{26}, 573-595.
item[]Letac, G., Wesolowski, J. 2000. An independence property for the
product of GIG and gamma laws. extit{Ann. Prob.}
extbf{28}, 1371-1383.
item[]Li, S.H., Huang, W.J., Huang, M.N.L. 1994. Characterizations
of the Poisson process as a renewal process via two conditional
moments. extit{Ann. Inst. Stat. Math.} extbf{46}, 351-360.
item[]Lukacs, E. 1955. A characterization of the gamma distribution.
extit{Ann. Math. Stat.} extbf{26}, 319-324.
item[]Matsumoto, H., Yor, M. 2003. Interpretation via Brownian motion
of some independence properties between GIG and gamma variables.
extit{Stat. Prob. Lett.} extbf{61}, 253-259.
item[]Olkin, I., Rubin, H. 1962. A characterization of the Wishart
distributions. extit{Ann. Math. Stat.} extbf{33}, 1272-1280.
item[]Pakes, A.G. 1992a. A characterization of gamma mixtures of
stable laws motivated by limit theorems. extit{Statistica
Neerlandica} extbf{2-3}, 209-218.
item[]Pakes, A.G. 1992b. On characterizations through mixed sums.
extit{Austral. J. Stat.} extbf{34}, 323-339.
item[]Pakes, A.G., Khattree, R. 1992. Length-biasing,
characterizations of laws and the moment problem.
extit{Austral. J. Stat.} extbf{34}, 307-322.
item[]Pakes, A.G. 1994. Necessary conditions for characterization of
laws via mixed sums. extit{Ann. Inst. Stat. Math.}
extbf{46}, 797-802.
item[]Patil, G.P., Seshadri, V. 1964. Characterization theorems for
some univariate probability distributions. extit{J. Roy. Stat.
Soc. B} extbf{26}, 286-292.
item[]Pusz, J. 1997. Regressional characterization of the generalized
inverse Gaussian population. extit{Ann. Inst. Stat. Math.}
extbf{49}, 315-319.
item[]Pusz, J. 2002. Some extensions on the bivariate populations of
the Laha-Lukacs characterizations. extit{Technical report,
Wydzial Matematyki I Nauk Informacyjnych, Politechnika Warszawska,
Warszawa, Poland.}
item[] Redheffer, R.M. 1992 extit{Introduction to
Differential Equations.} Jones and Bartlett Publishers, Inc.
Boston.
item[]Roussas, G.G. 1997. extit{A Course in Mathematical
Statistics,} 2nd edition. Academic Press, New York.
item[]Seshadri, V. 1999. extit{The Inverse Gaussian Distribution,
Statistical Theory and Applications.} Springer-Verlag Inc., New
York.
item[]Seshadri, V., Wesolowski, J. 2001. Mutual characterizations of
the gamma and generalized inverse Gaussian laws by constancy of
regression. extit{Sankhy={a} A} extbf{63}, 107-112.
item[]Seshadri, V., Wesolowski, J. 2003. Constancy of regressions for
beta distributions. extit{Sankhy={a}} extbf{65}, 284-291.
item[]Shih, J.H., Louis, T.A. 1995. Inference on the association parameter
in copula models for bivariate survival data. extit{Biometrics}
extbf{51}, 1384-1399.
item[]Tweedie, M.C.K. 1945. Inverse statistical variates. extit{Nature}
extbf{155}, 453-453.
item[]Wang, Y.H. 1981. extit{Extensions of Lukacs' characterization of the
gamma distribution. In: Analytical Methods in Probability Theory,
Lecture Notes in Mathematics} extbf{861}, Springer, New York,
166-177.
item[]Wesolowski, J. 1989. A characterization of the gamma process by
conditional moments. extit{Metrika} extbf{36}, 299-309.
item[]Wesolowski, J. 1990. A constant regression characterization of
the gamma law. extit{Adv. Appl. Prob.} extbf{22}, 488-490.
item[]Wesolowski, J. 2002a. On a functional equation related to an
independence property for beta distributions. extit{Aequationes
Math.} extbf{63}, 245-250.
item[]Wesolowski, J. 2002b. The Matsumoto-Yor independence property for
GIG and Gamma laws, revisited. extit{Math. Proc. Camb. Phil.
Soc.} extbf{133}, 153-161.
item[]Yeo, G.F., Milne, R.K. 1991. On characterizations of beta and
gamma distributions. extit{Stat. Prob. Lett.}
extbf{11}, 239-242.