Kotz分布的一些序关系及其相关不等式
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摘要
Kotz分布是一类重要的对称椭球分布。它是由Kotz，S．于1975年首次引进的。作为多维正态分布的推广形式，Kotz分布解决了许多以正态分布假设为前提的模型所不能解决的问题。此外，Kotz分布在经济数学，重复测量学等学科中也有着广泛的应用。
对随机序的研究是目前概率统计领域中的一个热点问题。研究内容主要是比较两个随机向量之间关于某个函数类的数学期望的大小。无论是在理论还是在应用方面，有关随机序关系的研究都占据了重要的地位。
本文的主要研究结果就是得出了一系列有关Kotz分布的随机序的充分及(或)必要条件。同时我们也将正态分布满足的一些不等式推广到Kotz分布的情形。
在绪论中我们综述了对Kotz分布的研究现状，对随机序关系进行研究的重要意义，以及与正态分布相关的不等式的研究现状。
在第二章中，作者以引入参数λ，将Kotz分布的各个参数线性化的思想为核心，给出了一个关于Kotz分布的重要公式。这一公式是研究Kotz分布随机序关系的基础。
在第三章中，作者先定义了各类重要的随机序关系，然后根据第二章得出的公式，给出了Kotz分布满足这些随机序关系的充分及(或)必要条件。
在第四章中，我们研究了Kotz分布的Peak序。由于其在参数k≠1时不满足单峰的条件，故不能直接运用Anderson定理得出结论。利用第二章的结论，我们给出了一维Kotz分布在参数k≠1时满足Peak序的充分必要条件。
在最后一章中，作者验证了一类特殊的Kotz分布的对数凹的性质，并利用这一性质，得到Kotz分布的相关不等式。
Kotz type distribution is an important kind of elliptically symmetric distributions. It was first introcuced by Kotz, S. in 1975. As a generalization of multinormal distribution, Kotz type distributon constructs many models to which the usual normality assumption is not applicable. In addition, Kotz type distribution also has an extensive application in the fields such as economic mathematics, repeated measurement, and so on.
Stochastic order is a hot research topic in the field of probability which mainly compares the expectations of two random vectors about some function groups. Therefore, it plays an important role in both theoretical and applied fields.
The main achievement of this dissertation is to have obtained a series of sufficiency and/or necessity conditions of stochastic orders about Kotz type distribution. This dissertation also extends the correlation inequalities of multinormal distribution to that of Kotz type distribution.
The introduction first has a literature review of Kotz type distribution. Then it discusses the significance of the research of stochastic orders, and the research development of the correlation inequalities of multinormal distribution.
The second chapter based on introducing the parameter A and lining each of the parameters, gives an important identity for Kotz type distribution, which becomes the basis of the research of stochastic orders for Kotz type distribution.
The third chapter first gives the definition of some important stochastic orders, and then gives the sufficiency and/or necessary conditions of corresponding stochastic orders according to the identity given in chapter two.
The fourth chapter deals with the Peak order of Kotz type distribution. We can not obtain the conclusion using Anderson Theory directly because the unimodal condition could not be satisfied if the parameter k doesn't equal one. However, with the identity given in chapter two, we can give the sufficiency and necessity condition of Peak order for uni-dimensional Kotz type distribution even k 1.
In the last chapter, the author verifies the log-concave property of Kotz type distribution and uses it to obtain the correlation inequalities of Kotz type distribution.
对随机序的研究是目前概率统计领域中的一个热点问题。研究内容主要是比较两个随机向量之间关于某个函数类的数学期望的大小。无论是在理论还是在应用方面，有关随机序关系的研究都占据了重要的地位。
本文的主要研究结果就是得出了一系列有关Kotz分布的随机序的充分及(或)必要条件。同时我们也将正态分布满足的一些不等式推广到Kotz分布的情形。
在绪论中我们综述了对Kotz分布的研究现状，对随机序关系进行研究的重要意义，以及与正态分布相关的不等式的研究现状。
在第二章中，作者以引入参数λ，将Kotz分布的各个参数线性化的思想为核心，给出了一个关于Kotz分布的重要公式。这一公式是研究Kotz分布随机序关系的基础。
在第三章中，作者先定义了各类重要的随机序关系，然后根据第二章得出的公式，给出了Kotz分布满足这些随机序关系的充分及(或)必要条件。
在第四章中，我们研究了Kotz分布的Peak序。由于其在参数k≠1时不满足单峰的条件，故不能直接运用Anderson定理得出结论。利用第二章的结论，我们给出了一维Kotz分布在参数k≠1时满足Peak序的充分必要条件。
在最后一章中，作者验证了一类特殊的Kotz分布的对数凹的性质，并利用这一性质，得到Kotz分布的相关不等式。
Kotz type distribution is an important kind of elliptically symmetric distributions. It was first introcuced by Kotz, S. in 1975. As a generalization of multinormal distribution, Kotz type distributon constructs many models to which the usual normality assumption is not applicable. In addition, Kotz type distribution also has an extensive application in the fields such as economic mathematics, repeated measurement, and so on.
Stochastic order is a hot research topic in the field of probability which mainly compares the expectations of two random vectors about some function groups. Therefore, it plays an important role in both theoretical and applied fields.
The main achievement of this dissertation is to have obtained a series of sufficiency and/or necessity conditions of stochastic orders about Kotz type distribution. This dissertation also extends the correlation inequalities of multinormal distribution to that of Kotz type distribution.
The introduction first has a literature review of Kotz type distribution. Then it discusses the significance of the research of stochastic orders, and the research development of the correlation inequalities of multinormal distribution.
The second chapter based on introducing the parameter A and lining each of the parameters, gives an important identity for Kotz type distribution, which becomes the basis of the research of stochastic orders for Kotz type distribution.
The third chapter first gives the definition of some important stochastic orders, and then gives the sufficiency and/or necessary conditions of corresponding stochastic orders according to the identity given in chapter two.
The fourth chapter deals with the Peak order of Kotz type distribution. We can not obtain the conclusion using Anderson Theory directly because the unimodal condition could not be satisfied if the parameter k doesn't equal one. However, with the identity given in chapter two, we can give the sufficiency and necessity condition of Peak order for uni-dimensional Kotz type distribution even k 1.
In the last chapter, the author verifies the log-concave property of Kotz type distribution and uses it to obtain the correlation inequalities of Kotz type distribution.
引文
[1] Anderson, T.W. (1955), The integral of a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6. 170-176.
[2] Cambanis, S., Huang, S., Simon, G. (1981), On the theory of elliptically contoured distributions, J. Mult. Anal. 11,368-385.
[3] Cambanis, S., Simons, G. (1982), Probability and expectation inequalities, Z. Wahr. verw. Geb. 59, 1-25.
[4] Cottle, R.W., Habetler, G.J. Lemke, C.E, (1970), On classes of copositive matrices, Linear Algebra and Its Applications, 3, 295-310.
[5] Fang, K.T., Kotz. S., Ng, K.W. (1990), Symmetric Multivariate and Related Distributions, Chapman and Hall Ltd.
[6] Das Gupta, S., Eaton, M., Olkin, I., Perlman, M., Savage, L. and Sobel, M. (1972), Ineaualities on the probability content of convex region for elliptically contoured distributions, Proc. Sixth Berkeley Symp. Math. Statist. Prob. 3, 241-264.
[7] Hargé,(1998), Une inégalité de décorrélation pour la measure gaussienne, C. R. Acad. Sci. Paris Sér. 326, 1325-1328.
[8] Houdré, C. (1998), Comparison and deviation form a representation formula, in: Karatzas, S., Rajput, B.S., Taqqu, M.S., eds. Stochastic processes and related topics, (Birkhuser, Boston) pp. 207-218.
[9] Houdré, C., Pérez-Abreu, V., Surgailis, D. (1998), Interpolation, Correlation Identities and Inequalities for Infinitely Divisible Variables, J. Fourier Anal. Appl. 4, no. 6, 651-668.
[10] Iycngar, S., Tong, Y.L.(1989), Convexity properties of elliptically contoured distributions, Sankhya, Series A 51, 13-29.
[11] Jogdeo, K.(1970), A simle proof of an inequality for multivariate normal probabilities of rectangles, Ann Math. Statist. 41, 1357-1359.
[12] Khatri, C.G.(1967), On certain inequalities for normal distributions
and their applications to simultaneous confidence bounds, Ann. Math. Stat. 38, 1853-1867.
[13] Kotz, S.(1975), Multivariate distributions at a cross-road, in: Patil, G.P., Kotz, S., Ord, J.K., eds. Statistical Distributions in Scientific Work, (D. Reidel Publishing Company, Dordrecht) pp.247-270.
[14] Kotz, S., Nadarajah, S. (2001), Letter to the editor Communications in Statistics, 30, no.6, 987-992.
[15] Kotz, S., Ostrovskii, I. (1994), Characterstic functions of a class of elliptical distributions, Multivariate Anal, 49, 164-178.
[16] Koutras, M.(1986), On the generalized noncentral chi-squared distribution induced by an elliptical gamma law, Biometrika 73, no.2, 528-532.
[17] Lancaster, P. and Tismenetsky, M.(1985), The Theoty of Matrices, Academic Press, New York.
[18] Li, W.V.(1999a). A Gaussian correlation inequality and its applications to small ball probabilities, Elect. Comm. in Probab., 4, 111-118.
[19] Li, W.V., Shao, Q.M., Gaussian Processes: Inequalities, Small Ball Probabilities and Applications, Stochastic processes: Theory and methods, Handbook of Statistics, Vol.19, Edited by C.R. Rao and D.Shanbhag.
[20] Marshal, A.W., Olkin, I. (1979), Inequalities: theory of majorization and its applications (Academic Press, New York).
[21] Müller, A. (2001b), Stochastic ordering of multivariate normal distributions, Ann. Inst. Statist. Math. 53, 567-575.
[22] Müller, A., Scarsini, M. (2001), Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, 723-740.
[23] Müller, A., Stoyan, D. (2002), Comparison Methods for Stochastic Models and Risks (John Wiley and Sons: New York).
[24] Pitt, L.(1977), A Gaussian correlation inequality for symmetric convex sets, Ann. Probab. 470-474.
[25] Shaked, M., Shanthikumar, J.G. (1994), Stochastic Orders and Their Applications (Academic Press: New York).
[26] Schechtman, G., Schlumprecht, T. and Zinn, J.(1998), On the Gaussian measure of the intersection, Ann. Probab. 26, 346-357.
[27] Shao, Q.M.(1999), A Gaussian correlation inequality and its applications to the existence of small ball constant, preprint.
[28] Sidak, Z.(1967), Rectangular confidence regions for the means of multivariate normal distributions, J. Amer. Statist.Assoc. 62, 626-633.
[29] Sidak, Z.(1968), On multivariate normal probabilities of rectangles: their dependence on correlations, Ann. Math. Stat. 39, 1425-1434.
[30] Streit, F.(1991), On the characteristic functions of the Kotz type distributions, La. Soc. Roy. Canda L'Acad. Sci. C.R. Math. 13, 121-124.
[31] Szekli, R.(1995), Stochastic ordering and dependence in applied probability, Lecture Notes in Statistics, Vol. 97, Springer, New York,
[32] Tong, Y.L.(1980), Probability inequalities in Multi-variate Distributions. Academic Press, New York.
[33] Vitale, R.(1999), Majorization and Gaussian Processes, Ann. Probab. 24, 2174-2178.
[2] Cambanis, S., Huang, S., Simon, G. (1981), On the theory of elliptically contoured distributions, J. Mult. Anal. 11,368-385.
[3] Cambanis, S., Simons, G. (1982), Probability and expectation inequalities, Z. Wahr. verw. Geb. 59, 1-25.
[4] Cottle, R.W., Habetler, G.J. Lemke, C.E, (1970), On classes of copositive matrices, Linear Algebra and Its Applications, 3, 295-310.
[5] Fang, K.T., Kotz. S., Ng, K.W. (1990), Symmetric Multivariate and Related Distributions, Chapman and Hall Ltd.
[6] Das Gupta, S., Eaton, M., Olkin, I., Perlman, M., Savage, L. and Sobel, M. (1972), Ineaualities on the probability content of convex region for elliptically contoured distributions, Proc. Sixth Berkeley Symp. Math. Statist. Prob. 3, 241-264.
[7] Hargé,(1998), Une inégalité de décorrélation pour la measure gaussienne, C. R. Acad. Sci. Paris Sér. 326, 1325-1328.
[8] Houdré, C. (1998), Comparison and deviation form a representation formula, in: Karatzas, S., Rajput, B.S., Taqqu, M.S., eds. Stochastic processes and related topics, (Birkhuser, Boston) pp. 207-218.
[9] Houdré, C., Pérez-Abreu, V., Surgailis, D. (1998), Interpolation, Correlation Identities and Inequalities for Infinitely Divisible Variables, J. Fourier Anal. Appl. 4, no. 6, 651-668.
[10] Iycngar, S., Tong, Y.L.(1989), Convexity properties of elliptically contoured distributions, Sankhya, Series A 51, 13-29.
[11] Jogdeo, K.(1970), A simle proof of an inequality for multivariate normal probabilities of rectangles, Ann Math. Statist. 41, 1357-1359.
[12] Khatri, C.G.(1967), On certain inequalities for normal distributions
and their applications to simultaneous confidence bounds, Ann. Math. Stat. 38, 1853-1867.
[13] Kotz, S.(1975), Multivariate distributions at a cross-road, in: Patil, G.P., Kotz, S., Ord, J.K., eds. Statistical Distributions in Scientific Work, (D. Reidel Publishing Company, Dordrecht) pp.247-270.
[14] Kotz, S., Nadarajah, S. (2001), Letter to the editor Communications in Statistics, 30, no.6, 987-992.
[15] Kotz, S., Ostrovskii, I. (1994), Characterstic functions of a class of elliptical distributions, Multivariate Anal, 49, 164-178.
[16] Koutras, M.(1986), On the generalized noncentral chi-squared distribution induced by an elliptical gamma law, Biometrika 73, no.2, 528-532.
[17] Lancaster, P. and Tismenetsky, M.(1985), The Theoty of Matrices, Academic Press, New York.
[18] Li, W.V.(1999a). A Gaussian correlation inequality and its applications to small ball probabilities, Elect. Comm. in Probab., 4, 111-118.
[19] Li, W.V., Shao, Q.M., Gaussian Processes: Inequalities, Small Ball Probabilities and Applications, Stochastic processes: Theory and methods, Handbook of Statistics, Vol.19, Edited by C.R. Rao and D.Shanbhag.
[20] Marshal, A.W., Olkin, I. (1979), Inequalities: theory of majorization and its applications (Academic Press, New York).
[21] Müller, A. (2001b), Stochastic ordering of multivariate normal distributions, Ann. Inst. Statist. Math. 53, 567-575.
[22] Müller, A., Scarsini, M. (2001), Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, 723-740.
[23] Müller, A., Stoyan, D. (2002), Comparison Methods for Stochastic Models and Risks (John Wiley and Sons: New York).
[24] Pitt, L.(1977), A Gaussian correlation inequality for symmetric convex sets, Ann. Probab. 470-474.
[25] Shaked, M., Shanthikumar, J.G. (1994), Stochastic Orders and Their Applications (Academic Press: New York).
[26] Schechtman, G., Schlumprecht, T. and Zinn, J.(1998), On the Gaussian measure of the intersection, Ann. Probab. 26, 346-357.
[27] Shao, Q.M.(1999), A Gaussian correlation inequality and its applications to the existence of small ball constant, preprint.
[28] Sidak, Z.(1967), Rectangular confidence regions for the means of multivariate normal distributions, J. Amer. Statist.Assoc. 62, 626-633.
[29] Sidak, Z.(1968), On multivariate normal probabilities of rectangles: their dependence on correlations, Ann. Math. Stat. 39, 1425-1434.
[30] Streit, F.(1991), On the characteristic functions of the Kotz type distributions, La. Soc. Roy. Canda L'Acad. Sci. C.R. Math. 13, 121-124.
[31] Szekli, R.(1995), Stochastic ordering and dependence in applied probability, Lecture Notes in Statistics, Vol. 97, Springer, New York,
[32] Tong, Y.L.(1980), Probability inequalities in Multi-variate Distributions. Academic Press, New York.
[33] Vitale, R.(1999), Majorization and Gaussian Processes, Ann. Probab. 24, 2174-2178.
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