A STOCHASTIC ORDERING FOR RANDOM VARIABLES WITH APPLICATIONS

A new univariate stochastic ordering is introduced. Some characterization results for such an ordering are stated. It is proved that the ordering is an integral stochastic ordering, obtaining a maximal generator. By means of this generator, the main properties of the ordering are deduced. A method for introducing univariate stochastic orderings, suggested by the new ordering, is analysed. Relationships with other stochastic orderings are also developed. To conclude, an example of an application of the new ordering to the field of medicine is proposed.     

Structured probability statements

Structured probability statements are defined in an additive and reduced structured model. Under weak assumptions, an estimating set in a structured probability statement is a confidence region, but the corresponding structured probability may differ from the confidence coefficient. Elementary examples are given to show that this difference is an advantage when some estimating sets are empty or consist of the whole parameter space. A structured distribution (Plante 1979) is an extension of a probability measure closely related to structured probability statements.     

Characterization of an optimal matrix estimator under convex loss function

Characterization of an optimal vector estimator and an optimal matrix estimator are obtained. In each case appropriate convex loss functions are considered. The results are illustrated through the problems of simultaneous unbiased estimation, simultaneous equivariant estimation and simultaneous unbiased prediction. Further an optimality criterion is proposed for matrix unbiased estimation and it is shown that the matrix unbiased estimation of a matrix parametric function and the minimum variance unbiased estimation of its components are equivalent.     

Correlation analysis with additive distortion measurement errors

This paper studies the estimation of correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in a additive fashion by an observed confounding variable. Two estimators, a direct-plug-in estimator and a residual-based estimator, are proposed. Their asymptotical results are obtained, and the residual-based estimator is shown asymptotically efficient. Moreover, we suggest an asymptotic normal approximation and an empirical likelihood-based statistic to construct the confidence interval. The empirical likelihood statistic is shown to be asymptotically chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators. These methods are applied to analyse the Boston housing price data for an illustration.     

Some Results for Uniformizable Semi-Markov Processes

It is well known that finite Markov chains (M.C.s) in continuous time are uniformizable. That is, a finite M. C. in continuous time can be treated as an M. C. in discrete time with random Poisson transition epochs. In this paper, we see to what extent generalization of the uniformization to a class of semi-Markov Processes (S.M.P.s) is possible. A necessary condition under which S.M.P.s are uniformizable is provided. It is shown that, an S.M.P. with dwell-time distributions depending only on the current state is uniformizable if and only if the distributions are compound geometric distributions having the same base distribution. It is also shown that if the distributions are of generalized phase type then an S.M.P. being uniformizable implies that it is an M.C. in continuous time. Some properties that are shared by a uniformizable S.M.P. and the associated M.C. in discrete time are also discussed.     

Estimation of extreme survival probabilities with cox model

We propose an extension of the regular Cox's proportional hazards model which allows the estimation of the probabilities of rare events. It is known that when the data are heavily censored, the estimation of the tail of the survival distribution is not reliable. To improve the estimate of the baseline survival function in the range of the largest observed data and to extend it outside, we adjust the tail of the baseline distribution beyond some threshold by an extreme value model under appropriate assumptions. The survival distributions conditioned to the covariates are easily computed from the baseline. A procedure allowing an automatic choice of the threshold and an aggregated estimate of the survival probabilities are also proposed. The performance is studied by simulations and an application on two data sets is given.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.     

An alternative to model selection in ordinary regression

The weaknesses of established model selection procedures based on hypothesis testing and similar criteria are discussed and an alternative based on synthetic (composite) estimation is proposed. It is developed for the problem of prediction in ordinary regression and its properties are explored by simulations for the simple regression. Extensions to a general setting are described and an example with multiple regression is analysed. Arguments are presented against using a selected model for any inferences.     

Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

Model robust confidence intervals

Confidence intervals are constructed for real-valued parameter estimation in a general regression model with normal errors. When the error variance is known these intervals are optimal (in the sense of minimizing length subject to guaranteed probability of coverage) among all intervals estimates which are centered at a linear estimate of the parameter. When the error variance is unknown and the regression model is an approximately linear model (a class of models which permits bounded systematic departures from an underlying ideal model) then an independent estimate of variance is found and the intervals can then be appropriately scaled.     

A Portmanteau Test for ARMA Processes with Infinite Variance

When a control chart is used in practice, knowledge about several characteristics of the method is important for the judgement of which action is appropriate at an alarm. The probability of a false alarm, the delay of an alarm and the predictive value of an alarm are qualities (besides the usual ARL) which are described by a simulation study for the evaluations. Results for finite time are given for a shift in level of a Gaussian process.

Evaluations are made of the full likelihood ratio method that has two parameters and can be made optimal for both the size and the intensity of a shift     

An extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe

In this paper, an extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe is developed. Main new results are presented in theorems. The primary notions of discrete population are transferred to continuous population. First and second order inclusion probabilities for networks are delivered. Horvitz–Thompson estimator for adaptive cluster sampling in continuous universe is constructed. The unbiasedness of the estimator is proven. Variance and unbiased variance estimator are delivered. Finally, the theory is illustrated with an example.     

Computer generated simple random samples

Three computer methods, namely the Random Draw, the Sequential and the Reservoir Methods, to obtain a simple random sample without replacement are reviewed. Several implementations of each method are given and criteria for the selection of an implementation are discussed. The most promising ones are compared for efficiency in their execution times. It is concluded that for small sampling fractions, Ernvall and Nevalainen's (1982) implementation of the Random Draw Method is most efficient when N, the population size, is known. When N is unknown, an Algorithm Z of Vitter (1985), an implementation of the Reservoir Method, is recommended. For larger sampling the algorithm of Bissell (1986) and Vitter (1984) is quite efficient.     

Density Approximation by Summary Statistics: An Information-theoretic Approach

In the case of exponential families, it is a straightforward matter to approximate a density function by use of summary statistics; however, an appropriate approach to such approximation is far less clear when an exponential family is not assumed. In this paper, a maximin argument based on information theory is used to derive a new approach to density approximation from summary statistics which is not restricted by the assumption of validity of an underlying exponential family. Information-theoretic criteria are developed to assess loss of predictive power of summary statistics under such minimal knowledge. Under these criteria, optimal density approximations in the maximin sense are obtained and shown to be related to exponential families. Conditions for existence of optimal density approximations are developed. Applications of the proposed approach are illustrated, and methods for estimation of densities are provided in the case of simple random sampling. Large-sample theory for estimates is developed.     

Multiphase experiments in practice: A look back

Multiphase experiments are introduced and an overview of their design and analysis as it is currently practised is given via an account of their development since 1955 and a literature survey. Methods that are available for designing and analysing them are outlined, with an emphasis on making explicit the role of the model in their design. The availability of software and its use is described in detail. Overall, while multiphase designs have been applied in areas such as plant breeding, plant pathology, greenhouse experimentation, product storage, gene expression studies, and sensory evaluation, their deployment has been limited.     

Bayesian least squares estimates of univariate regression functions

The regression function R(?) to be estimated is assumed to have an expansion in terms of specified functions, orthogonalized vich respect to values of the explanatory variable. Relative precisions of OBSERVATION are assumed known. The estimate is the posterior linear mean of R(?) given the data. The investigator plots graphs of appropriate functions as an aid in eliciting his prior means and precisions for the coefficients in the expansion. The method is illustrated by an example using simulated data, an example in which effects of various dosages of Vitamin D are estimated, and an example in which a utility function is estimated.     

ASYMPTOTIC NORMALITY OF GOODNESS-OF-FIT STATISTICS FOR SPARSE POISSON DATA

Goodness-of-fit tests for discrete data and models with parameters to be estimated are usually based on Pearson's χ2 or the Likelihood Ratio Statistic. Both are included in the family of Power-Divergence Statistics SDλ which are asymptotically χ2 distributed for the usual sampling schemes. We derive a limiting standard normal distribution for a standardization Tλ of SDλ under Poisson sampling by considering an approach with an increasing number of cells. In contrast to the χ2 asymptotics we do not require an increase of all expected values and thus meet the situation when data are sparse. Our limit result is useful even if a bootstrap test is used, because it implies that the statistic Tλ should be bootstrapped and not the sum SDλ. The peculiarity of our approach is that the models under test only specify associations. Hence we have to deal with an infinite number of nuisance parameters. We illustrate our approach with an application.     

Optimal kernels when estimating non-smooth densities

The derivation of new kernel functions for the kernel estimator of an unknown density function is given. These kernels are shown to be optimal in some sense when the underlying density f is continuous but its derivative f′ is not, and consequently a solu tion is presented for an unsolved problem which was stated by van Eeden (1985). Other attractive features of these kernels are also discussed and a number of graphs are listed.     

Refining binomial confidence intervals

A method for refining an equivariant binomial confidence procedure is presented which, when applied to an existing procedure, produces a new set of equivariant intervals that are uniformly superior. The family of procedures generated from this method constitute a complete class within the class of all equivariant procedures. In certain cases it is shown that this class is also minimal complete. Also, an optimally property, monotone minimaxity, is investigated, and monotone minimax procedures are constructed.     

Computing uncertainty measures of location estimates for autonomous vehicles

Determining the location of an autonomous vehicle is an important problem in navigating the vehicle in an unstructured environment. The vehicle controller estimates the vehicle's location and calculates the covariance matrix as an uncertainty measure for the location estimate. The real-time implementation of the controller makes the calculation of the covariance matrix an important issue. There is no exact method for calculating the covariance matrix because of the nonlinear nature of the location estimator. Approximations are needed. In this article, several approximation methods are compared through simulation. The comparisons are focused on each incremental change and on the cummulative effects of the trajectory-following of a path. The robustness of approximations is also studied by investigating the behavior of approximations under different distributional assumptions for measurement models. The results are useful for finding the scopes and limits of applicability of the approximation