Let I be the set of all infinitely divisible random variables with finite second moments,I_(0)={X∈I;Var(X)>0},P_(I)=inf_(x∈I)P{|X-E[X]|≤√Var(X)}and P_(I_(0))=inf P{|X-E[X]|<√Var(X)}.Firstly,we prove that P_...Let I be the set of all infinitely divisible random variables with finite second moments,I_(0)={X∈I;Var(X)>0},P_(I)=inf_(x∈I)P{|X-E[X]|≤√Var(X)}and P_(I_(0))=inf P{|X-E[X]|<√Var(X)}.Firstly,we prove that P_(I)≥P_(I_(0))>0.Secondly,we find_(x∈I_(0))the exact values of inf P{|X-E[X]|≤√Var(X)}and inf P{|X-E[X]|<√Var(X)}for the cases that J is the set of all geometric random variables,symmetric geometric random variables,Poisson random variables and symmetric Poisson random variables,respectively.As a consequence,we obtain that P_(I)≤e^(-1)^(∞)∑_(k=0)1/2^(2k)(k!)^(2)≈0.46576 and P_(I_(0))≤e^(-1)≈0.36788.展开更多
Generalized method of moments based on probability generating function is considered. Estimation and model testing are unified using this approach which also leads to distribution free chi-square tests. The estimation...Generalized method of moments based on probability generating function is considered. Estimation and model testing are unified using this approach which also leads to distribution free chi-square tests. The estimation methods developed are also related to estimation methods based on generalized estimating equations but with the advantage of having statistics for model testing. The methods proposed overcome numerical problems often encountered when the probability mass functions have no closed forms which prevent the use of maximum likelihood (ML) procedures and in general, ML procedures do not lead to distribution free model testing statistics.展开更多
GMM inference procedures based on the square of the modulus of the model characteristic function are developed using sample moments selected using estimating function theory and bypassing the use of empirical characte...GMM inference procedures based on the square of the modulus of the model characteristic function are developed using sample moments selected using estimating function theory and bypassing the use of empirical characteristic function of other GMM procedures in the literature. The procedures are relatively simple to implement and are less simulation-oriented than simulated methods of inferences yet have the potential of good efficiencies for models with densities without closed form. The procedures also yield better estimators than method of moment estimators for models with more than three parameters as higher order sample moments tend to be unstable.展开更多
The main purpose of this paper is to extend the Zolotarev's problem concerning with geometric random sums to negative binomial random sums of independent identically distributed random variables.This extension is ...The main purpose of this paper is to extend the Zolotarev's problem concerning with geometric random sums to negative binomial random sums of independent identically distributed random variables.This extension is equivalent to describing all negative binomial infinitely divisible random variables and related results.Using Trotter-operator technique together with Zolotarev-distance's ideality,some upper bounds of convergence rates of normalized negative binomial random sums(in the sense of convergence in distribution)to Gamma,generalized Laplace and generalized Linnik random variables are established.The obtained results are extension and generalization of several known results related to geometric random sums.展开更多
In this paper, based on the recent results of Cozlan and Leonard we give optimal transportation- entropy inequalities for several usual distributions on R, such as Bernoulli, Binomial, Poisson, Gamma distributions and...In this paper, based on the recent results of Cozlan and Leonard we give optimal transportation- entropy inequalities for several usual distributions on R, such as Bernoulli, Binomial, Poisson, Gamma distributions and infinitely divisible distributions with positive or negative jumps.展开更多
This paper presents two conceptual models of displacement transfer,reverse sym-metry model and infinitely equal division model,based on the fault-bend folding theory.If the fault shape is held constant in the trend,th...This paper presents two conceptual models of displacement transfer,reverse sym-metry model and infinitely equal division model,based on the fault-bend folding theory.If the fault shape is held constant in the trend,then the distribution of slip magnitude,geometry of imbricate structures and its axial surface map all display reverse symmetry on the process of displacement transfer,as called reverse symmetry model in this paper.However,if the ramp height of thrust fault decreases gradually along its strike,the displacement is postulated to be equally and infinitely divided to every thrust that is formed subsequently,this kinematic process is described using infinitely equal division model.In both models,displacement transfer is characterized by the regular changes of imbricate thrusting in the trend.Geometric analysis indicates that the displacement transfer grads can be estimated using the tangent of deflective angle of hinterland structures.Displacement transfer is often responsible for the distortion and branching of the surface anticlines,especially in the region where the multi-level detachment structures is developed.We also present some examples from the frontal structures of the Southern Tianshan fold-and-thrust belt,Xinjiang,China.Displacement transfer between deep imbricate thrusts in the middle segment of Qiulitag anticline zone causes the Kuqatawu and Southern Qiulitag deep an-ticlines left-lateral echelon.The region,where these two deep anticlines overlap,is characterized by duplex structures,and extends about 18 km.The shallow anticline is migrated southward displaying obvious“S”form in this area.展开更多
We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type （a, b, 0） class distributions and family of equilibrium distributions of arbitrary birth-death process. Beside...We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type （a, b, 0） class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity （infinite divisibility）, pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein＇s identity are given for theoretical properties. COM- negative binomial distribution was applied to overdispersion and ultrahigh zeroinflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.展开更多
基金supported by the National Natural Science Foundation of China(12161029,12171335)the National Natural Science Foundation of Hainan Province(121RC149)+1 种基金the Science Development Project of Sichuan University(2020SCUNL201)the Natural Sciences and Engineering Research Council of Canada(4394-2018).
文摘Let I be the set of all infinitely divisible random variables with finite second moments,I_(0)={X∈I;Var(X)>0},P_(I)=inf_(x∈I)P{|X-E[X]|≤√Var(X)}and P_(I_(0))=inf P{|X-E[X]|<√Var(X)}.Firstly,we prove that P_(I)≥P_(I_(0))>0.Secondly,we find_(x∈I_(0))the exact values of inf P{|X-E[X]|≤√Var(X)}and inf P{|X-E[X]|<√Var(X)}for the cases that J is the set of all geometric random variables,symmetric geometric random variables,Poisson random variables and symmetric Poisson random variables,respectively.As a consequence,we obtain that P_(I)≤e^(-1)^(∞)∑_(k=0)1/2^(2k)(k!)^(2)≈0.46576 and P_(I_(0))≤e^(-1)≈0.36788.
文摘Generalized method of moments based on probability generating function is considered. Estimation and model testing are unified using this approach which also leads to distribution free chi-square tests. The estimation methods developed are also related to estimation methods based on generalized estimating equations but with the advantage of having statistics for model testing. The methods proposed overcome numerical problems often encountered when the probability mass functions have no closed forms which prevent the use of maximum likelihood (ML) procedures and in general, ML procedures do not lead to distribution free model testing statistics.
文摘GMM inference procedures based on the square of the modulus of the model characteristic function are developed using sample moments selected using estimating function theory and bypassing the use of empirical characteristic function of other GMM procedures in the literature. The procedures are relatively simple to implement and are less simulation-oriented than simulated methods of inferences yet have the potential of good efficiencies for models with densities without closed form. The procedures also yield better estimators than method of moment estimators for models with more than three parameters as higher order sample moments tend to be unstable.
文摘The main purpose of this paper is to extend the Zolotarev's problem concerning with geometric random sums to negative binomial random sums of independent identically distributed random variables.This extension is equivalent to describing all negative binomial infinitely divisible random variables and related results.Using Trotter-operator technique together with Zolotarev-distance's ideality,some upper bounds of convergence rates of normalized negative binomial random sums(in the sense of convergence in distribution)to Gamma,generalized Laplace and generalized Linnik random variables are established.The obtained results are extension and generalization of several known results related to geometric random sums.
基金Supported by the National Natural Science Foundation of China (No.11001208)the Fundamental Research Funds for the Central Universities
文摘In this paper, based on the recent results of Cozlan and Leonard we give optimal transportation- entropy inequalities for several usual distributions on R, such as Bernoulli, Binomial, Poisson, Gamma distributions and infinitely divisible distributions with positive or negative jumps.
基金the National Natural Science Foundation of China(Grant No.49972077).
文摘This paper presents two conceptual models of displacement transfer,reverse sym-metry model and infinitely equal division model,based on the fault-bend folding theory.If the fault shape is held constant in the trend,then the distribution of slip magnitude,geometry of imbricate structures and its axial surface map all display reverse symmetry on the process of displacement transfer,as called reverse symmetry model in this paper.However,if the ramp height of thrust fault decreases gradually along its strike,the displacement is postulated to be equally and infinitely divided to every thrust that is formed subsequently,this kinematic process is described using infinitely equal division model.In both models,displacement transfer is characterized by the regular changes of imbricate thrusting in the trend.Geometric analysis indicates that the displacement transfer grads can be estimated using the tangent of deflective angle of hinterland structures.Displacement transfer is often responsible for the distortion and branching of the surface anticlines,especially in the region where the multi-level detachment structures is developed.We also present some examples from the frontal structures of the Southern Tianshan fold-and-thrust belt,Xinjiang,China.Displacement transfer between deep imbricate thrusts in the middle segment of Qiulitag anticline zone causes the Kuqatawu and Southern Qiulitag deep an-ticlines left-lateral echelon.The region,where these two deep anticlines overlap,is characterized by duplex structures,and extends about 18 km.The shallow anticline is migrated southward displaying obvious“S”form in this area.
基金The proposed COM-negative binomial distribution of this work was as early as conceptualized in December, 2014 when the authors saw the online version of [15]. The authors want to thank Prof. R. KShler for mailing the valuable encyclopedia of discrete univariate distributions [39] to them. This work was partly supported by the National Natural Science Foundation of China （Grant No. 11201165）.
文摘We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type （a, b, 0） class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity （infinite divisibility）, pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein＇s identity are given for theoretical properties. COM- negative binomial distribution was applied to overdispersion and ultrahigh zeroinflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.
