10 August 2006, Volume 22 Issue 3

    

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  Some Properties of the Absolute Moment

  AN Honzhi

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 233-236.

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  We discuss several properties of the absolute moment of sums of independent random variables, including the expression of $\ep|X+Y|-\ep|X-Y|$, where $X$ and $Y$ are i.i.d. random variables.
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  Estimate a Stationary Process under Left Censoring

  HE Shuyuan, AI Mingyao, SUN Xinli

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 237-244.

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  Let $\{X_t\}$ be a stationary signal process interfered by an white noise $\{Y_t\}$. The signal $X_t$ is detected and observed only when $X_t>Y_t$, otherwise only the white noise $Y_t$ is observed. This model is called the left censored model and the observation is called the left censored observation. In this paper we use the nonparametric MLE of the marginal distributions of $X_t$ and $Y_t$ to construct estimates of the mean, autocovariance and autocorrelation functions of the original signal process $\{X_t\}$. The strong consistency of the estimates is derived. When $\{X_t\}$ is a causal autoregression process, consistent estimates of the autoregression parameters are provided.
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  On Varying Coefficient Mixed-Effects Model

  CHAI Genxiang, SUN Yan

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 245-251.

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  In this paper we consider the following varying coefficient mixed-effects model:
  $y_{ij}=z_{ij}^{\tau}b_{i}+x_{ij}^{\tau}\beta(w_{ij})+\xe_ij},\;i=1,\cdots,m;
  \;j=1,\cdots,n_i$, where $b_{i}$ is i.i.d. random effects with mean vector $\theta$ and covariance matrix $\sigma_{b}^{2}I_{q}$, $\xe_{ij}$ is i.i.d. random errors with zero mean and finite variance. The local polynomial estimator of the function coefficient vector $\beta(\cdot)$ is proposed. The method for estimating the mean of random effects, variances of random effects and random errors are also given. Asymptotic normality and consistency for the estimators are established, which give useful insight into the reliability of these general estimation methods.
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  Influence Analysis in ZI Longitudinal Count Data Models

  WEI Bocheng, XIE Fengchang

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 252-262.

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  Based on the EM algorithm and Laplace approximation, this paper presents a method of influence analysis for zero inflated longitudinal count data models. To detect the influential observations in clustered count data with excess zeros, we regard the random effects as the missing data and put certain weight to the data with zero values in ZI longitudinal data models. According to this fact, we develop the influence method for the model based on the conditional expectation of the
  complete-data log-likelihood function and the associated $Q$-distance function
  under the EM algorithm. The Laplace approximation is also employed for integral
  computing in E-step. Then the case-deletion model and the local influence analysis are investigated for the model and several diagnostic measures are obtained. Finally, a numerical example of the real count data is given to illustrate the results in this paper.
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  On Spectral Decomposition Estimates in Mixed Linear Models

  Wang Songgui

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 263-272.

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  This paper gives a survey of the recent developments on parameter estimation in linear mixed model. The covariance matrix in balanced analysis of variance mixed linear models has a specific structure. For this model, [1] proposed a new approach, spectral decomposition method, to estimate parameters. The merits of the approach is to provide independent estimates of fixed effects and variance components simultaneously, the former is linear and late quadratic. [2--9] established some further properties of the new estimates and corresponding estimates of covariance matrix with risk function. These papers also obtained some relations among the analysis of variance estimate, maximum likelihood estimate, restricted maximum likelihood estimate, minimum norm quadratic unbiased estimate and new estimates. Finally, some open problems are proposed.
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  The Iterated Algorithms of Estimating Parameter

  ZHENG Zukang

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 273-280.

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  Two iterated algorithms of estimating parameter in exponential distribution from
  interval censored data are introduced. Under certain conditions the procedures are convergent.
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  A Kind of Urn Models in Clinical Trials With Multi-Outcome

  CHENG Guijing, HE Shuhe, HONG Shengyan

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 281-287.

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  A kind of adaptive designs in clinical trials with multi-out come have been constructed using urn models some limit theorems have been shown in such models, some examples are given, which indicate that the better treatment will be allocated to patients with large chance, and the statistical efficiency in these designs is still keeping better. So these adaptive designs are feasible.
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  Weak Consistency of Quasi-Maximum Likelihood　Estimates in Multivariate Generalized Linear Models

  LAO Yuan, ZHANG Shanguo, XUE Hongqi

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 288-294.

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  In this paper, we study quasi-likelihood equation $\tsm_{i=1}^nX_i(y_i-\mu(X_i'\beta))=0$　for multivariate generalized linear models (GLMs). Under mild conditions, we prove the　asymptotic existence of the solution $\wh{\beta}_n$ to the above equation and present　its convergence rate, that is $\wh{\beta}_n-\beta_0=O_p(\underline{\lambda}_n^{-1/2})$,　where $\beta_0$ is the true value of parameter $\beta$ and $\underline{\lambda}_n$　denotes the smallest eigenvalue of the matrix $S_n=\tsm_{i=1}^nX_iX_i'$.
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  Multivariate Stochastic Orderings of Spacings of Generalized Order Statistics

  FANG Zhaoben, HU Taizhong, WU Yaohua, ZHUANG Weiwei,

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 295-303.

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  In this paper, we investigate conditions on the underlying distribution function
  and the parameters on which the generalized order statistics are based, to obtain　stochastic comparisons of spacing vectors of generalized order statistics in the　multivariate likelihood ratio and the usual multivariate stochastic orders. Some　applications of the main results are also given.
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  On the Subtrees of Random Binary Search Trees

  SU Chun, MIAO Baiqi, FENG Qunqiang

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 304-310.

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  The subtrees of various sizes and patterns in random binary search trees are
  investigated in this paper. The expectations and variances of their numbers are
  first derived from an essential recursive distributional equation. Applying the
  contraction method, we show both of their asymptotic distributions are normal.
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  Precise Rates in the Law of the Logarithm under Minimal Conditions

  ZHANG Lixin, LIN Zhengyan

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 311-320.

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  Let $X_1,X_2,\cdots$ be i.i.d. random variables, and set $S_n=X_1+\cdots+X_n$,
  $M_n=\max\limits_{k\le n}|S_k|$, $n\ge 1$. The precise rates in the law of the
  logarithm for $M_n$ are obtained under sufficient and necessary conditions.
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  $T$-type Estimators and EM Algorithm in Linear Model and Linear Errors-in-Variables Model

  CUI Hengjian

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 321-328.

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  This paper defines $t$-type regression estimator for the linear functional
  errors-in-variables (EV) models, the EM algorithm for $t$-type estimators in
  the ordinary linear model and linear functional EV model is given. Moreover,
  the consistency of $t$-type estimators for linear EV model is also obtained.
  Simulations are shown that performance of the t-type estimators by EM algorithm
  is quite well.
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  Vertical Density Representation and Its Applications

  YANG Zhenghai, CHENG Weihu

  CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST. 2006, 22(3): 329-336.

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  In this paper, the Trout's conception of VDR is called as Type I VDR and Type II VDR is proposed. The results on Type I are derived by using Type II VDR. We construct a new distribution family by VDR, which is one family of center similar distributions and called as multi-logist distribution with the property whose marginal distribution is multi-logist distribution too.