The interval mapping method has been shown to be a powerful tool for mapping QTL. However, it is still a challenge to perform a simultaneous analysis of several linked QTLs, and to isolate multiple linked QTLs. To circumvent these problems, multiple regression analysis has been suggested for experimental species. In this paper, the multiple regression approach is extended to human sib-pair data through multiple regression of the squared difference in trait values between two sibs on the proportions of alleles shared identical by descent by sib pairs at marker loci.We conduct an asymptotic analysis of the partial regression coeffcients, which provide a basis for the estimation of the additive genetic variance and of locations of the QTLs. We demonstrate how the magnitude of the regression coefficients can be used to separate multiple linked QTLs. Further, we shall show that the multiple regression model using sib pairs is identifiable, and our proposed procedure for locating QTLs is robust in the sense that it can detect the number of QTLs and their locations in the presence of several linked (QTLs) in an interval, unlike a simple regression model which may find a “ghost” QTL with no effect on the trait in the interval with several linked QTLs. Moreover, we give procedures for computing the threshold values for prespecified significance levels and for computing the power for detecting (QTLs). Finally, we investigate the consistency of the estimator for QTL locations. Using the concept of epi-convergence and variation analysis theory, we shall prove the consistency of the estimator of map location in the framework of the multiple regression approach. Since the true IBD status is not always known, the multiple regression of the squared sib difference on the estimated IBD sharing is also considered.

This paper describes a parametric deconvolution method (PDPS) appropriate for a particular class of signals which we call spike-convolution models. These models arise when a sparse spike train —Dirac deltas according to our mathematical treatment —is convolved with a fixed point-spread function, and additive noise or measurement error is superimposed.We view deconvolution as an estimation problem, regarding the locations and heights of the underlying spikes, as well as the baseline and the measurement error variance as unknown parameters.Our estimation scheme consists of two parts: model fitting and model selection.To fit a spike-convolution model of a specific order, we estimate peak locations by trigonometric moments, and heights and the baseline by least squares. The model selection procedure has two stages. Its first stage is so designed that we expect a model of a somewhat larger order than the truth to be selected. In the second stage, the final model is obtained using backwards deletion. This results in not only an estimate of the model order, but also an estimate of peak locations and heights with much smaller bias and variation than that found in a direct trigonometric moment estimate. A more efficient maximum likelihood estimate can be calculated from these estimates using a Gauss–Newton algorithm. We also present some relevant results concerning the spectral structure of Toeplitz matrices which play a key role in the estimation. Finally, we illustrate the behavior of these estimates using simulated and real DNA sequencing data.

We consider the problem of estimating $\|s\|^2$ when $s$ belongs to some separable Hilbert space and one observes the Gaussian process $Y(t) = \langles, t\rangle + \sigmaL(t)$, for all $t \epsilon \mathbb{H}$,where $L$ is some Gaussian isonormal process. This framework allows us in particular to consider the classical “Gaussian sequence model” for which $\mathbb{H} = l_2(\mathbb{N}*)$ and $L(t) = \sum_{\lambda\geq1}t_{\lambda}\varepsilon_{\lambda}$, where $(\varepsilon_{\lambda})_{\lambda\geq1}$ is a sequence of i.i.d. standard normal variables. Our approach consists in considering some at most countable families of finite-dimensional linear subspaces of $\mathbb{H}$ (the models) and then using model selection via some conveniently penalized least squares criterion to build new estimators of $\|s\|^2$. We prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter $s$, depending on the family of models from which they are built.In particular, in the context of the Gaussian sequence model, a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles, ellipsoids, $l_p$-bodies or Besov bodies.We take special care to describe the conditions under which the penalized estimator is efficient when the level of noise $\sigma$ tends to zero. Our construction is an alternative to the one by Efroïmovich and Low for hyperrectangles and provides new results otherwise.

We consider methods for kernel regression when the explanatory and/or response variables are adjusted prior to substitution into a conven- tional estimator.This “data-sharpening” procedure is designed to preserve the advantages of relatively simple, low-order techniques, for example, their robustness against design sparsity problems, yet attain the sorts of bias reductions that are commonly associated only with high-order methods.We consider Nadaraya–Watson and local-linear methods in detail, although data sharpening is applicable more widely. One approach in particular is found to give excellent performance. It involves adjusting both the explanatory and the response variables prior to substitution into a local linear estimator. The change to the explanatory variables enhances resistance of the estimator to design sparsity, by increasing the density of design points in places where the original density had been low. When combined with adjustment of the response variables, it produces a reduction in bias by an order of magnitude. Moreover, these advantages are available in multivariate settings. The data-sharpening step is simple to implement, since it is explicitly defined. It does not involve functional inversion, solution of equations or use of pilot bandwidths.

We consider the asymptotic behavior ofregression estimators that minimize the residual sum of squares plus a penalty proportional to $\sum|\beta_j|^{\gamma}$. for some $\gamma > 0$. These estimators include the Lasso as a special case when $\gamma = 1$. Under appropriate conditions, we show that the limiting distributions can have positive probability mass at 0 when the true value of the parameter is 0.We also consider asymptotics for “nearly singular” designs.

A criterion for local estimation and approximation in nonlinear regres- sion and neural network training is introduced and motivated. $N$th-order greedy approximation for the regression (or target) function based on the criterion is shown to converge at rate $O(1/N^{1/2})$ in the nonsampling case.

Let p be a random probability measure chosen by a Dirichlet process whose parameter a is a finite measure with support contained in $[0, +\infty)$ and suppose that $V = \int x^2p(dx)-[\int xp(dx)]^2$ is a (finite)random variable. This paper deals with the distribution of $V$, which is given in a rather general case. A simple application to Bayesian bootstrap is also illustrated.

We characterize priors which asymptotically match the posterior coverage probability of a Bayesian prediction region with the corresponding frequentist coverage probability. This is done considering both posterior quantiles and highest predictive density regions with reference to a future observation. The resulting priors are shown to be invariant under reparameterization. The role of Jeffreys’ prior in this regard is also investigated. It is further shown that, for any given prior, it may be possible to choose an interval whose Bayesian predictive and frequentist coverage probabilities are asymptotically matched.

The asymptotic theory for the sample autocorrelations and extremes of a GARCH (1, 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series.

Fixed case-control studies separately collect a case sample and a control sample withthe two sample sizes being fixed prior to studies and sometimes arbitrarily chosen. This often results in loss of efficiency of case-control designs in terms of cost-saving or time-saving of the studies. We study sequential case-control designs and, in connection with treatment allocation and stochastic approximation, derive a simple sampling rule that leads to optimal case-control designs. Some important issues suchas fixed- width confidence intervals and sequential tests of hypotheses with possible early stopping to save time or costs, whichcannot be answered with fixed case-control designs, are shown to be naturally solved with the derived optimal sequential case-control designs.

The goal of this paper is to develop a general framework for constructing sequential fixed size confidence regions based on maximum likelihood estimates. Asymptotic properties of the sequential procedure for setting up the confidence regions are analyzed under very broad assumptions on the underlying parametric model. It is shown that the proposed sequential procedure is asymptotically optimal in the sense that it approximates the optimal fixed-sample size procedure. It is further shown that the “cost of ignorance” associated with the sequential procedure is bounded. Applications are made to estimation problems arising in prospective and retrospective studies.