Estimation bayesienne des effets dans le modele a effets aleatoires de classification double avec interaction

The problem of estimating the effects in a balanced two-way classification with interaction \documentclass{article}\pagestyle{empty}\begin{document}$i = 1, \ldots ,I;j = 1, \ldots ,J;k = 1, \ldots ,K$\end{document} using a random effect model is considered from a Bayesian view point. Posterior distributions of r_i, c_j and t_ij are obtained under the assumptions that r_i, c_j, t_ij and e_ijk are all independently drawn from normal distributions with zero meansand variances \documentclass{article}\pagestyle{empty}\begin{document}$\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document} respectively. A non informative reference prior is adopted for \documentclass{article}\pagestyle{empty}\begin{document}$\mu ,\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document}. Various features of thisposterior distribution are obtained. The same features of the psoterior distribution for a fixed effect model are also obtained. A numerical example is given.     

Consistency of maximum likelihood estimators in a large class of deconvolution models

We consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada     

Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

A likelihood ratio test for goodness‐of‐fit of recessive and dominant models for case–control studies

Testing goodness‐of‐fit of commonly used genetic models is of critical importance in many applications including association studies and testing for departure from Hardy–Weinberg equilibrium. Case–control design has become widely used in population genetics and genetic epidemiology, thus it is of interest to develop powerful goodness‐of‐fit tests for genetic models using case–control data. This paper develops a likelihood ratio test (LRT) for testing recessive and dominant models for case–control studies. The LRT statistic has a closed‐form formula with a simple $\chi^{2}(1)$ null asymptotic distribution, thus its implementation is easy even for genome‐wide association studies. Moreover, it has the same power and optimality as when the disease prevalence is known in the population. The Canadian Journal of Statistics 41: 341–352; 2013 © 2013 Statistical Society of Canada     

Parallelism,uniqueness, and large‐sample asymptotics for the Dantzig selector

The Dantzig selector (Candès & Tao, 2007) is a popular $\ell^{1}$ ‐regularization method for variable selection and estimation in linear regression. We present a very weak geometric condition on the observed predictors which is related to parallelism and, when satisfied, ensures the uniqueness of Dantzig selector estimators. The condition holds with probability 1, if the predictors are drawn from a continuous distribution. We discuss the necessity of this condition for uniqueness and also provide a closely related condition which ensures the uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for the Dantzig selector, that is, almost sure convergence and the asymptotic distribution, follow directly from our uniqueness results and a continuity argument. The limiting distribution of the Dantzig selector is generally non‐normal. Though our asymptotic results require that the number of predictors is fixed (similar to Knight & Fu, 2000), our uniqueness results are valid for an arbitrary number of predictors and observations. The Canadian Journal of Statistics 41: 23–35; 2013 © 2012 Statistical Society of Canada     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

New recursive estimators of the time-average variance constant

Estimation of the time-average variance constant (TAVC) of a stationary process plays a fundamental role in statistical inference for the mean of a stochastic process. Wu (2009) proposed an efficient algorithm to recursively compute the TAVC with \(O(1)\) memory and computational complexity. In this paper, we propose two new recursive TAVC estimators that can compute TAVC estimate with \(O(1)\) computational complexity. One of them is uniformly better than Wu’s estimator in terms of asymptotic mean squared error (MSE) at a cost of slightly higher memory complexity. The other preserves the \(O(1)\) memory complexity and is better then Wu’s estimator in most situations. Moreover, the first estimator is nearly optimal in the sense that its asymptotic MSE is \(2^{10/3}3^{-2} \fallingdotseq 1.12\) times that of the optimal off-line TAVC estimator.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

On the invariance of conditioning procedures for the specification of prior distributions for nested DAG models

Given a prior distribution for a model , the prior information specified on a nested submodel by means of a conditioning procedure crucially depends on the parameterisation used to describe the model. Regression coefficients represent the most common parameterisation of Gaussian DAG models. Nevertheless, in the specification of prior distributions, invariance considerations lead to the use of different parameterisations of the model, depending on the required invariance class. In this paper we consider the problem of prior specification by conditioning on zero regression coefficients and show that also such a procedure satisfies the property of invariance with respect to a class of parameterisations and characterise such a class.     

Hypotheses testing about the drift parameter in linear stochastic differential equation driven by stable processes

In this paper, we consider the problem of hypotheses testing about the drift parameter \(\theta \) in the process \(\text {d}Y^{\delta }_{t} = \theta \dot{f}(t)Y^{\delta }_{t}\text {d}t + b(t)\text {d}L^{\delta }_{t}\) driven by symmetric \(\delta \)-stable Lévy process \(L^{\delta }_{t}\) with \(\dot{f}(t)\) being the derivative of a known increasing function f(t) and b(t) being known as well. We consider the hypotheses testing \(H_{0}: \theta \le 0\) and \(K_{0}: \theta =0\) against the alternatives \(H_{1}: \theta >0\) and \(K_{1}: \theta \ne 0\), respectively. For these hypotheses, we propose inverse methods, which are motivated by sequential approach, based on the first hitting time of the observed process (or its absolute value) to a pre-specified boundary or two boundaries until some given time. The applicability of these methods is illustrated. For the case \(Y^{\delta }_{0}=0\), we are able to calculate the values of boundaries and finite observed times more directly. We are able to show the consistencies of proposed tests for \(Y^{\delta }_{0}\ge 0\) with \(\delta \in (1,2]\) and for \(Y^{\delta }_{0}=0\) with \(\delta \in (0,2]\) under quite mild conditions.     

A unified method for constructing expectation tolerance intervals

Given a random sample of size \(n\) with mean \(\overline{X} \) and standard deviation \(s\) from a symmetric distribution \(F(x; \mu , \sigma ) = F_{0} (( x- \mu ) / \sigma ) \) with \(F_0\) known, and \(X \sim F(x;\; \mu , \sigma )\) independent of the sample, we show how to construct an expansion \( a_n^{\prime } = \sum _{i=0}^\infty \ c_i \ n^{-i} \) such that \(\overline{X} - s a_n^{\prime } < X < \overline{X} + s a_n^{\prime } \) with a given probability \(\beta \) . The practical value of this result is illustrated by simulation and using a real data set.     

A discrepancy bound for deterministic acceptance-rejection samplers beyond $$N^{-1/2}$$ in dimension 1

In this paper we consider an acceptance-rejection (AR) sampler based on deterministic driver sequences. We prove that the discrepancy of an N element sample set generated in this way is bounded by \(\mathcal {O} (N^{-2/3}\log N)\), provided that the target density is twice continuously differentiable with non-vanishing curvature and the AR sampler uses the driver sequence \(\mathcal {K}_M= \{( j \alpha , j \beta ) ~~ mod~~1 \mid j = 1,\ldots ,M\},\) where \(\alpha ,\beta \) are real algebraic numbers such that \(1,\alpha ,\beta \) is a basis of a number field over \(\mathbb {Q}\) of degree 3. For the driver sequence \(\mathcal {F}_k= \{ ({j}/{F_k}, \{{jF_{k-1}}/{F_k}\} ) \mid j=1,\ldots , F_k\},\) where \(F_k\) is the k-th Fibonacci number and \(\{x\}=x-\lfloor x \rfloor \) is the fractional part of a non-negative real number x, we can remove the \(\log \) factor to improve the convergence rate to \(\mathcal {O}(N^{-2/3})\), where again N is the number of samples we accepted. We also introduce a criterion for measuring the goodness of driver sequences. The proposed approach is numerically tested by calculating the star-discrepancy of samples generated for some target densities using \(\mathcal {K}_M\) and \(\mathcal {F}_k\) as driver sequences. These results confirm that achieving a convergence rate beyond \(N^{-1/2}\) is possible in practice using \(\mathcal {K}_M\) and \(\mathcal {F}_k\) as driver sequences in the acceptance-rejection sampler.     

Transformation approaches of linear random-effects models

Assume that a linear random-effects model \(\mathbf{y}= \mathbf{X}\varvec{\beta }+ \varvec{\varepsilon }= \mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \varvec{\varepsilon }\) is transformed as \(\mathbf{T}\mathbf{y}= \mathbf{T}\mathbf{X}\varvec{\beta }+ \mathbf{T}\varvec{\varepsilon }= \mathbf{T}\mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \mathbf{T}\varvec{\varepsilon }\) by pre-multiplying a given matrix \(\mathbf{T}\) of arbitrary rank. These two models are not necessarily equivalent unless \(\mathbf{T}\) is of full column rank, and we have to work with this derived model in many situations. Because predictors/estimators of the parameter spaces under the two models are not necessarily the same, it is primary work to compare predictors/estimators in the two models and to establish possible links between the inference results obtained from two models. This paper presents a general algebraic approach to the problem of comparing best linear unbiased predictors (BLUPs) of parameter spaces in an original linear random-effects model and its transformations, and provides a group of fundamental and comprehensive results on mathematical and statistical properties of the BLUPs. In particular, we construct many equalities for the BLUPs under an original linear random-effects model and its transformations, and obtain necessary and sufficient conditions for the equalities to hold.     

On Bahadur representation for sample quantiles under α-mixing sequence

In this paper, by relaxing the mixing coefficients to α(n) = O(n ^?β), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as ${O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}$ . Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n ^?β ), ${\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}$ , we have the rate as ${O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ . Specifically, if ${\delta=\frac{\sqrt{41}-5}{4}}$ and ${\beta > \frac{\sqrt{41}+7}{2}}$ , then the rate is presented as ${O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ .     

On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .     

The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

On comparison of dispersion matrices of estimators under a constrained linear model

We introduce some new mathematical tools in the analysis of dispersion matrices of the two well-known OLSEs and BLUEs under general linear models with parameter restrictions. We first establish some formulas for calculating the ranks and inertias of the differences of OLSEs’ and BLUEs’ dispersion matrices of parametric functions under the general linear model \({\mathscr {M}}= \{\mathbf{y}, \ \mathbf{X }\pmb {\beta }, \ \pmb {\Sigma }\}\) and the constrained model \({\mathscr {M}}_r = \{\mathbf{y}, \, \mathbf{X }\pmb {\beta }\, | \, \mathbf{A }\pmb {\beta }= \mathbf{b}, \ \pmb {\Sigma }\}\), where \(\mathbf{A }\pmb {\beta }= \mathbf{b}\) is a consistent linear matrix equation for the unknown parameter vector \(\pmb {\beta }\) to satisfy. As applications, we derive necessary and sufficient conditions for many equalities and inequalities of OLSEs’ and BLUEs’ dispersion matrices to hold under \({\mathscr {M}}\) and \({\mathscr {M}}_r\).     

Consistency of a nonparametric conditional mode estimator for random fields

Given a stationary multidimensional spatial process $\left\{ Z_{\mathbf{i}}=\left( X_{\mathbf{i}},\ Y_{\mathbf{i}}\right) \in \mathbb R ^d\right. \left. \times \mathbb R ,\mathbf{i}\in \mathbb Z ^{N}\right\} $ , we investigate a kernel estimate of the spatial conditional mode function of the response variable $Y_{\mathbf{i}}$ given the explicative variable $X_{\mathbf{i}}$ . Consistency in $L^p$ norm and strong convergence of the kernel estimate are obtained when the sample considered is a $\alpha $ -mixing sequence. An application to real data is given in order to illustrate the behavior of our methodology.