On the KOLMOGOROV-SMIRNOV Test for the POISSON Distribution with Unknown Mean

The standard tables for the KOLMOGOROV -SMIRNOV test are valid only in the case of testing whether a set of observations is from a completely specified cumulative distribution, F₀(X), with all parameters known. If the parameters are unknown and must be estimated from the sample, then the tables are not valid. A table is given in this paper for use with the KOLMOGOROV -SMIRNOV statistic in the case of testing whether a set of observations is from the POISSON distribution with an unknown mean that must be estimated from the sample. The table is obtained from a Monte Carlo calculation.     

Correlation and Linear Regression if the Random Variables are Subject to Errors or Fluctuations

For the model x=a+e, y=b+d estimators of Pearson's coefficient of correlation and of the line of regression between a and b are presented. The problem of prediction is dealt with.     

Characterizations of the Exponential Distribution by Local Maximum Random Variables

Let {X_n, n≧1} be a sequence of independent and identically distributed random variables. Suppose n(K)=min n for which X_n>X_k, with K fixed and n>K. We will characterize the exponential distribution by considering the distributional properties of X_k and X_k and X_n(k).     

Nearest Neighbor Classification Rule for Mixtures of Discrete and Continuous Random Variables

In this paper very simple nonparametric classification rule for mixtures of discrete and continuous random variables is described. It is based on the method of nearest neighbor proposed by Cover and Hart (1967). The bounds on the limit of the nearest neighbor rule risks are given. Both lower and upper bound depend on the Bayes risk and the loss function. Finally the method is compared with other existing methods on some practical data set.     

Missing Data and the Mixtures of Discrete and Continuous Random Variables

In many practical applications we deal with a problem of estimation of a density function of a vector x some components of which are discrete, while the remaining ones are continuous. Among many models that can be used in this case the most useful are the location model and the kernel model. The problem arises when the observed data contain missing values i.e. on some individuals some of the variables have not been observed with no particular pattern of missingness. An application of the EM algorithm will allow us to estimate the parameters of the location model from incomplete data. The method is described in Section 2. In Section 3 some suggestions how to deal with incompleteness when the kernel model is used are made. Finally, Section 4 contains an example.     

Bayesian Estimation of the Distribution Function of the Poisson Model

This paper presents the Bayes estimators of the Poisson distribution function based on complete and truncated data under a natural conjugate prior. Laplace transform of the incomplete gamma function and the Gauss hypergeometric function have been employed in order to overcome the intractability of the integrals. Numerical examples from biosciences are given to illustrate the results. A Monte Carlo study has been carried out to compare Bayes estimators under complete data with the corresponding maximum liklihood estimators.     

关于数量化I方法的自变量之间交互作用计算

自变量之间存在交互作用时,在应用数量化方法Ｉ时如何计算自变量间的交互作用建立数量化方程．     

Impact of Random Variables Probability Distribution on Public Health Risk Assessment from Contaminated Soil

Probabilistic methods are now being applied increasingly to public health risk assessment instead of the deterministic, conservative, point estimates. An essential part of the probabilistic methods is the selection of probability distribution functions to represent the uncertainty of the random variables considered. We study the effect of selection of different probability distribution functions on the probabilistic outcome using the first-order reliability method (FORM). An example of cancer risk resulting from dermal contact with benzo(a)pyrene (BaP)-contaminated soil is given. Cancer potency factor, soil concentration, and fraction of skin area exposed were assigned normal, lognormal, and uniform probability distribution functions, and the effect of probability of exceeding a target risk level (termed the probability of failure) and sensitivity measures were studied. We investigated the question: what happens when one assumes different distribution shapes with the same mean and standard deviation? The results indicate that the selection of a probability distribution function for the random variables had a moderate impact on the probability of failure when the target risk is at the 50th percentile level, while the impact was much larger for a 95th target risk percentile. We conclude that the probability distribution will have a large impact because in most cases the regulatory threshold risk is at the tail end of the risk distribution. The impact of the distributions on probabilistic sensitivity, however, showed a reversed trend, where the impact was slightly more appreciable for the 50th percentile than for the 95th percentile. The selection of distribution shape did not, however, alter the order of probabilistic sensitivity of the basic random variables.     

Prediction of Random Effects in Linear and Generalized Linear Models under Model Misspecification

Summary Statistical models that include random effects are commonly used to analyze longitudinal and correlated data, often with the assumption that the random effects follow a Gaussian distribution. Via theoretical and numerical calculations and simulation, we investigate the impact of misspecification of this distribution on both how well the predicted values recover the true underlying distribution and the accuracy of prediction of the realized values of the random effects. We show that, although the predicted values can vary with the assumed distribution, the prediction accuracy, as measured by mean square error, is little affected for mild‐to‐moderate violations of the assumptions. Thus, standard approaches, readily available in statistical software, will often suffice. The results are illustrated using data from the Heart and Estrogen/Progestin Replacement Study using models to predict future blood pressure values.     

Estimation of Linear Regression Model with Random Coefficients Ensuring Almost Non-Negativity of Variance Estimators

For a linear regression model with random coefficients, this paper considers the estimation of the mean of coefficient vector which, in turn, involves the estimation of variances of random coefficients. The conventional estimation methods for it sometimes provides negative estimates. In order to circumvent this kind of difficulty, a proposal is forwarded and is examined in the light of existing ones.     

Inferences Concerning a Mean Vector when the Variables are Grouped into Subsets

In experiments involving multivariate measurements biological considerations often indicate a grouping of the variables into subsets such that all the variables in a subset are equally important and the subsets may be arranged in a decreasing order of importance (JENSEN, 1972). This paper deals with a method of analysis appropriate in such cases, to solve multivariate problems such as profile analysis, comparison of two mean vectors, or univariate problems such as analysis of SCHEFFÉ'S mixed effects model with interaction (SCHEFFÉ, 1959; MILLER, 1966), usually treated using HOTELLING'S T². An invariance structure underlying this procedure is examined and used to obtain noncentrality parameters of the power function, and derive some of its properties. The multiple comparisons procedure in MUDHOLKAR and SUBBAIAH (1976) is extended and illustrated in terms of the data from a clinical trial.     

Some Approximations of the Borel-Tanner or the Generalized Poisson Distribution

This paper considers some approximations for the Borel-Tanner (Generalized Poisson) sums by using (i) Gram-Charlier Poisson expansion, (ii) Mixture of two Poisson distributions, (iii) Variance stabilizing technique, and (iv) negative binomial distribution. It has been found that the approximation obtained by using the negative binomial distribution seems to be more efficient than the other approximation.     

A Semiparametric Estimator of the Crossing Point in the Two‐Sample Linear Shift Function: Application to Crossing Lifetime Distributions

Let X and Y be two random variables with continuous distribution functions F and G. Consider two independent observations X₁, … , X_m from F and Y₁, … , Y_n from G. Moreover, suppose there exists a unique x* such that F(x) > G(x) for x < x* and F(x) < G(x) for x > x* or vice versa. A semiparametric model with a linear shift function (Doksum, 1974) that is equivalent to a location‐scale model (Hsieh, 1995) will be assumed and an empirical process approach (Hsieh, 1995) is used to estimate the parameters of the shift function. Then, the estimated shift function is set to zero, and the solution is defined to be an estimate of the crossing‐point x*. An approximate confidence band of the linear shift function at the crossing‐point x* is also presented, which is inverted to yield an approximate confidence interval for the crossing‐point. Finally, the lifetime of guinea pigs in days observed in a treatment‐control experiment in Bjerkedal (1960) is used to demonstrate our procedure for estimating the crossing‐point. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heterogeneity in dynamic species abundance models: the selective effect of extinction processes

There is often large variation in traits across the species of a community. In particular, variation in life history traits affecting population dynamics is likely to affect the species abundance distribution. Applying a dynamic and heterogeneous species abundance model we study how differences in extinction time for species in a community act as a force changing the distribution of dynamic parameters across species. This process may generate communities that are more heterogeneous then the heterogeneity measured as the species enter the community. Analytical results for some versions of the lognormal and gamma species abundance model are given as exemplifications of this process, together with stochastic simulations demonstrating the temporal changes in number of species and community heterogeneity through time.     

On the Estimation of the Mean with Random Number Of Observations. (Comments on a Paper by M. Singh and V. K. Gupta)

In this paper we study the problem of estimating the mean when the number of observations taken on each unit is a random variable. The estimator proposed by M. Singh and V. K. Gupta (1980) is discussed and modified. It is argued, however, that the arithmetic mean is a more appropriate choice of estimator.     

Convergence to Normality of the Asymptotic Quasi‐Score Function on a Linear Model

The asymptotic quasi‐likelihood method is considered for the model y_t = f_t(θ) + M_t, t = 0,1, …,T where f_tθ) is a linear predictable process of the parameter of interest θ, M_t is a martingale difference, and the nature of E(M_t² | ℱ_t–1) is unknown. This paper is concerned with the limiting distribution of the asymptotic quasi‐score function of such a model. Confidence intervals and hypothesis testing of θ is derived from the limiting distribution. Comparison is made between the estimates obtained through this method and those obtained through the least squares method.     

北京地区植被景观中斑块大小的分布特征

利用GIS软件ARC／INFO将北京地区的1：20万植被图数字化,并提取各斑块的面积信息。该图包含72个基本的斑块类型,它们又分属于森林、灌丛、草地、果园、农田和水体6大类型。这些大类又分别包含20、28、4、7、11和2个基本类型。选用斑块个数、总面积、平均斑块面积、标准差、变异系数、中值、最大斑块面积、最小斑块面积、极差和偏态系数等几个描述统计量,以及厂一分布、对数正态分布、Weibull分布、指数分布和正态分布等5个概率分布来刻画斑块大小的分布特征。结果表明：除具有很少斑块的少数基本类型以外,其它基本类型以及所有6个大类的斑块大小的分布都不是对称的,而是右偏的。因此,普通的正态分布不能对它们加以刻画;其他4种概率分布也只能刻画部分类型,并且服从对数正态分布的类型最多,服从负指数分布的斑块类型最少。