Given samples $(x_1,\ldots,x_m)$ and $(z_1, \ldots,z_n)$ which we believe are independent realizations of random variables $X$ and $Z$ respectively, where we further believe that $Z = X + Y$ with $Y$ independent of $X$, the problem is to estimate the distribution of $Y$. We present a new method for doing this, involving simulation. Experiments suggest that the method provides useful estimates.

This paper introduces an iterative tomogravity algorithm for the estimation of a network traffic matrix based on one snapshot observation of the link loads in the network. The proposed method does not require complete observation of the total load on individual edge links or proper tuning of a penalty parameter as existing methods do. Numerical results are presented to demonstrate that the iterative tomogravity method controls the estimation error well when the link data is fully observed and produces robust results with moderate amount of missing link data.

The analysis of computer and communication networks gives rise to some interesting inverse problems. This paper is concerned with active network tomography where the goal is to recover information about quality-of-service (QoS) parameters at the link level from aggregate data measured on end-to-end network paths. The estimation and monitoring of QoS parameters, such as loss rates and delays, are of considerable interest to network engineers and Internet service providers. The paper provides a review of the inverse problems and recent research on inference for loss rates and delay distributions. Some new results on parametric inference for delay distributions are also developed. In addition, a real application on Internet telephony is discussed.

Network tomography has been regarded as one of the most promising methodologies for performance evaluation and diagnosis of the massive and decentralized Internet. This paper proposes a new estimation approach for solving a class of inverse problems in network tomography, based on marginal distributions of a sequence of one-dimensional linear projections of the observed data. We give a general identifiability result for the proposed method and study the design issue of these one dimensional projections in terms of statistical efficiency. We show that for a simple Gaussian tomography model, there is an optimal set of one-dimensional projections such that the estimator obtained from these projections is asymptotically as efficient as the maximum likelihood estimator based on the joint distribution of the observed data. For practical applications, we carry out simulation studies of the proposed method for two instances of network tomography. The first is for traffic demand tomography using a Gaussian Origin-Destination traffic model with a power relation between its mean and variance, and the second is for network delay tomography where the link delays are to be estimated from the end-to-end path delays. We compare estimators obtained from our method and that obtained from using the joint distribution and other lower dimensional projections, and show that in both cases, the proposed method yields satisfactory results.

Yehuda Vardi introduced the term network tomography and was the first to propose and study how statistical inverse methods could be adapted to attack important network problems (Vardi, 1996). More recently, in one of his final papers, Vardi proposed notions of metrics on networks to define and measure distances between a network's links, its paths, and also between different networks (Vardi, 2004). In this paper, we apply Vardi's general approach for network metrics to a real data network by using data obtained from special data network tools and testing procedures presented here. We illustrate how the metrics help explicate interesting features of the traffic characteristics on the network. We also adapt the metrics in order to condition on traffic passing through a portion of the network, such as a router or pair of routers, and show further how this approach helps to discover and explain interesting network characteristics.

We present a class of sparse generalized linear models that include probit and logistic regression as special cases and offer some extra flexibility. We provide an EM algorithm for learning the parameters of these models from data. We apply our method in text classification and in simulated data and show that our method outperforms the logistic and probit models and also the elastic net, in general by a substantial margin.

DNA microarray experiments, a well-established experimental technique, aim at understanding the function of genes in some biological processes. One of the most common experiments in functional genomics research is to compare two groups of microarray data to determine which genes are differentially expressed. In this paper, we propose a methodology to estimate the proportion of differentially expressed genes in such experiments. We study the performance of our method in a simulation study where we compare it to other standard methods. Finally we compare the methods in real data from two toxicology experiments with mice.

The notion of data depth has long been in use to obtain robust location and scale estimates in a multivariate setting. The depth of an observation is a measure of its centrality, with respect to a data set or a distribution. The data depths of a set of multivariate observations translates to a center-outward ordering of the data. Thus, data depth provides a generalization of the median to a multivariate setting (the deepest observation), and can also be used to screen for extreme observations or outliers (the observations with low data depth). Data depth has been used in the development of a wide range of robust and non-parametric methods for multivariate data, such as non-parametric tests of location and scale [Li and Liu (2004)], multivariate rank-tests [Liu and Singh (1993)], non-parametric classification and clustering [Jornsten, (2004)], and robust regression [Rousseeuw and Hubert (1999)].

Many different notions of data depth have been developed for multivariate data. In contrast, data depth measures for functional data have only recently been proposed [Fraiman and Muniz (1999), López-Pintado and Romo (2006a)]. While the definitions of both of these data depth measures are motivated by the functional aspect of the data, the measures themselves are in fact invariant with respect to permutations of the domain (i.e. the compact interval on which the functions are defined). Thus, these measures are equally applicable to multivariate data where there is no explicit ordering of the data dimensions. In this paper we explore some extensions of functional data depths, so as to take the ordering of the data dimensions into account.

A stratified sampling plan to audit health insurance claims is offered. The stratification is by dollar amount of the claim. The plan is representative in the sense that with high probability for each stratum, the difference in the average dollar amount of the claim in the sample and the average dollar amount in the population, is ``small.'' Several notions of ``small'' are presented. The plan then yields a relatively small total sample size with the property that the overall average dollar amount in the sample is close to the average dollar amount in the population. Three different estimators and corresponding lower confidence bounds for over (under) payments are studied.

The notion of confidence distribution (CD), an entirely frequentist concept, is in essence a Neymanian interpretation of Fisher's Fiducial distribution. It contains information related to every kind of frequentist inference. In this article, a CD is viewed as a distribution estimator of a parameter. This leads naturally to consideration of the information contained in CD, comparison of CDs and optimal CDs, and connection of the CD concept to the (profile) likelihood function. A formal development of a multiparameter CD is also presented.

False discovery rate (FDR) has been widely used as an error measure in large scale multiple testing problems, but most research in the area has been focused on procedures for controlling the FDR based on independent test statistics or the properties of such procedures for test statistics with certain types of stochastic dependence. Based on an approach proposed in Tang and Zhang (2005), we further develop in this paper empirical Bayes methods for controlling the FDR with dependent data. We implement our methodology in a time series model and report the results of a simulation study to demonstrate the advantages of the empirical Bayes approach.

When a sample frequency table is published, disclosure risk arises when some individuals can be identified on the basis of their values in certain attributes in the table called key variables, and then their values in other attributes may be inferred, and their privacy is violated.

On the basis of the sample to be released, and possibly some partial knowledge of the whole population, an agency which considers releasing the sample, has to estimate the disclosure risk.

Risk arises from non-empty sample cells which represent small population cells and from population uniques in particular. Therefore risk estimation requires assessing how many of the relevant population cells are likely to be small. Various methods have been proposed for this task, and we present a method in which estimation of a population cell frequency is based on smoothing using a local neighborhood of this cell, that is, cells having similar or close values in all attributes.

We provide some preliminary results and experiments with this method. Comparisons are made to two other methods: 1. a log-linear models approach in which inference on a given cell is based on a ``neighborhood'' of cells determined by the log-linear model. Such neighborhoods have one or some common attributes with the cell in question, but some other attributes may differ significantly. 2. The Argus method in which inference on a given cell is based only on the sample frequency in the specific cell, on the sample design and on some known marginal distributions of the population, without learning from any type of ``neighborhood'' of the given cell, nor from any model which uses the structure of the table.

The $U, V$ method of estimation provides unbiased estimators or predictors of random quantities. The method was introduced by Robbins and subsequently studied in a series of papers by Robbins and Zhang. (See Zhang). Practical applications of the method are featured in these papers. We demonstrate that for one $U$ function (one for which there is an important application) the $V$ estimator is inadmissible for a wide class of loss functions. For another important $U$ function the $V$ estimator is admissible for the squared error loss function.

We reveal the phenomenon that “naive” multivariate local polynomial regression can adapt to local smooth lower dimensional structure in the sense that it achieves the optimal convergence rate for nonparametric estimation of regression functions belonging to a Sobolev space when the predictor variables live on or close to a lower dimensional manifold.

Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included.

A class of estimating functions is introduced for the regression parameter of the Cox proportional hazards model to allow unknown failure statuses on some study subjects. The consistency and asymptotic normality of the resulting estimators are established under mild conditions. An adaptive estimator which achieves the minimum variance-covariance bound of the class is constructed. Numerical studies demonstrate that the asymptotic approximations are adequate for practical use and that the efficiency gain of the adaptive estimator over the complete-case analysis can be quite substantial. Similar methods are also developed for the nonparametric estimation of the survival function of a homogeneous population and for the estimation of the cumulative baseline hazard function under the Cox model.

This paper derives the nonparametric maximum likelihood estimator (NPMLE) of a distribution function from observations which are subject to both bias and censoring. The NPMLE is obtained by a simple EM algorithm which is an extension of the algorithm suggested by Vardi (Biometrika, 1989) for size biased data. Application of the algorithm to many models is discussed and a simulation study compares the estimator's performance to that of the product-limit estimator (PLE). An example demonstrates the utility of the NPMLE to data where the PLE is inappropriate.

We consider the non-parametric maximum likelihood estimation in the class of Polya frequency functions of order two, viz. the densities with a concave logarithm. This is a subclass of unimodal densities and fairly rich in general. The NPMLE is shown to be the solution to a convex programming problem in the Euclidean space and an algorithm is devised similar to the iterative convex minorant algorithm by Jongbleod (1999). The estimator achieves Hellinger consistency when the true density is a PFF$_2$ itself.

Consider a model parameterized by a scalar parameter of interest and a nuisance parameter vector. Inference about the parameter of interest may be based on the signed root of the likelihood ratio statistic $R$. The standard normal approximation to the conditional distribution of $R$ typically has error of order $O(n^{-1/2})$, where $n$ is the sample size. There are several modifications for $R$, which reduce the order of error in the approximations. In this paper, we mainly investigate Barndorff-Nielsen's modified directed likelihood ratio statistic, Severini's empirical adjustment, and DiCiccio and Martin's two modifications, involving the Bayesian approach and the conditional likelihood ratio statistic. For each modification, two formats were employed to approximate the conditional cumulative distribution function; these are Barndorff-Nielson formats and the Lugannani and Rice formats. All approximations were applied to inference on the ratio of means for two independent exponential random variables. We constructed one and two-sided hypotheses tests and used the actual sizes of the tests as the measurements of accuracy to compare those approximations.

We discuss two research areas dealing respectively with (1) a class of multivariate medians and (2) a symmetrization algorithm for probability measures.

Data miners (minors?) and neural networkers tend to eschew modelling, misled perhaps by misinterpretation of strongly expressed views of John Tukey. I discuss Vardi's views of these issues as well as other aspects of Vardi's work in emision tomography and in sampling bias.