L'effet de l'erreur d'arrondi sur l'estimateur d'une densité par la méthode du noyau

In a model for rounded data suppose that the random sample X₁,.,.,X_n,. i.i.d., is transformed into an observed random sample X_1Δ,.,.,X_nΔ, where X_iΔ = 2vΔ if X_i, ∈ (2vΔ - Δ, 2vΔ + Δ), for i = 1,.,.,n. We show that the precision Δ of the observations has an important effect on the shape of the kernel density estimator, and we identify important points for the graphical display of this estimator. We examine the IMSE criteria to find the optimal window under the rounded-data model.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

La méthode de vraisemblance pour les processus linéaires spatiaux définis sur un treillis triangulaire

Spatial linear processes {X_s, s ? T} where T is a triangular lattice in R² are considered. Special attention is given to the class of spatial moving-average processes. Precisely, for each site s T, the variable X_s is defined as a linear combination of real-valued random shocks located at the vertices of regular concentric hexagons centered at s. For Gaussian random shocks, the process is also Gaussian, and estimates of its parameters are obtained by maximizing the exact likelihood. For non-Gaussian random shocks, the exact likelihood is difficult to obtain; however, the Gaussian likelihood is still used giving the pseudo-Gaussian likelihood estimates. The behaviour of these estimates is analyzed through the study of asymptotic properties and some simulation experiments based on an isotropic model defined with one coefficient.     

On an improved Erdő-Rényi-type law for increments of partial sums

Let X₁, X₂,… be an independently and identically distributed sequence with ξX₁ = 0, ξ exp (tX₁ < ∞ (t ≧ 0) and partial sums S_n = X₁ + … + X_n. Consider the maximum increment D₁ (N, K) = max_{0≤n ≤ N - K} (S_{n + K} - S_n)of the sequence (S_n) in (0, N) over a time K = K_N, 1 ≦ K ≦ N. Under appropriate conditions on (K_N) it is shown that in the case K_N/log N → 0, but K_N/(log N)^1/2 → ∞, there exists a sequence (α_N) such that K_-1/2 D₁ (N, K) - α_N converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erd?s-Rényi-type laws stated by Csörg? and Steinebach (1981).     

A Note on the Calculation of Pr{X 1 < X 2 < ··· < Xk }

Suppose that X_i are independent random variables, and that X_i has cdf F_i (x), 1 ≤ i ≤ k. Many statistical problems involve the probability Pr{X ₁ < X ₂ < ··· < X_k }. In this note a numerical method is proposed for computing this probability.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Deux méthodes d'estimation pour les paramètres de processus moyenne mobile spatiaux

We consider moving average processes, {X_s, s ∈ ??}, where ?? is a triangular lattice in the plane R^2. To estimate the parameters of such processes, Adjengue & Moore (1993) have considered likelihood and gaussian pseudo-likelihood methods. We consider here two other methods. The first one is based on the estimation of the correlations and the relation between these correlations and the parameters of the process. The second relies on a linear approximation of the process. The asymptotic properties of the proposed estimators are analyzed and compared. A simulation study allows us to compare the estimators for fixed sample sizes.     

CALCULATION OF MULTIVARIATE NORMAL PROBABILITIES—ANOTHER SPECIAL CASE

The problem of calculating orthant probabilities for sets of variables (X₁,., X_n) is considered in the case where they are jointly normally distributed with zero means and a correlation matrix such that the correlation between X_i and X_i is zero if |i-j|> 1. An effective method is given which works for quite large n when the correlations between X_i and X_i+1 have the values 1/2, 2/5, 3/10, 4/17, 5/26,. and more approximate methods are given for other values. The accuracy is investigated numerically.     

De l'unicité des estimateurs robustes en régression lorsque le paramètre d'échelle et le paramètre de la régression sont estimés simultanément

Several methods have been suggested to calculate robust M- and G-M -estimators of the regression parameter β and of the error scale parameter σ in a linear model. This paper shows that, for some data sets well known in robust statistics, the nonlinear systems of equations for the simultaneous estimation of β, with an M-estimate with a redescending ψ-function, and σ, with the residual median absolute deviation (MAD), have many solutions. This multiplicity is not caused by the possible lack of uniqueness, for redescending ψ-functions, of the solutions of the system defining β with known σ; rather, the simultaneous estimation of β and σ together creates the problem. A way to avoid these multiple solutions is to proceed in two steps. First take σ as the median absolute deviation of the residuals for a uniquely defined robust M-estimate such as Huber's Proposal 2 or the L₁-estimate. Then solve the nonlinear system for the M-estimate with σ equal to the value obtained at the first step to get the estimate of β. Analytical conditions for the uniqueness of M and G-M-estimates are also given.     

Random Weighting Estimation for Quantile Processes and Negatively Associated Samples

This article presents new theories of random weighting estimation for quantile processes and negatively associated samples. Under the condition that X ₁, X ₂,…, X _n are independent random variables with a common distribution, the consistency for random weighting estimation of quantile processes is rigorously proved. When X ₁, X ₂,…, X _n are not independent of each other, random weighting estimation of sample mean is established for negatively associated samples.     

Prediction of Variables Via Their Order Statistics in Bivariate Elliptical Distributions with Application in the Financial Markets

Assuming that (X₁, X₂) has a bivariate elliptical distribution, we obtain an exact expression for the joint probability density function (pdf) as well as the corresponding conditional pdfs of X₁ and X₍₂₎ ? max?{X₁, X₂}. The problem is motivated by an application in financial markets. Exchangeable random variables are discussed in more detail. Two special cases of the elliptical distributions that is the normal and the student’s t models are investigated. For illustrative purposes, a real data set on the total personal income in California and New York is analyzed using the results obtained. Finally, some concluding remarks and further works are discussed.     

Likelihood ratio test for independence with partial multivariate normal data

This article derives the likelihood ratio statistic to test the independence between (X ₁,…,X _r ) and (X _r+1,…,X _k ) under the assumption that (X ₁,…,X _k ) has a multivariate normal distribution and that a sample of size n is available, where for N observation vectors all components are available, while for M = (n + N) observation vectors, the data on the last q components, (X_k-q+1,…,X _k ) are missing (k+q≥r).     

A Bayesian non‐inferiority test for two independent binomial proportions

In drug development, non‐inferiority tests are often employed to determine the difference between two independent binomial proportions. Many test statistics for non‐inferiority are based on the frequentist framework. However, research on non‐inferiority in the Bayesian framework is limited. In this paper, we suggest a new Bayesian index τ = P(π₁ > π₂ ? Δ₀ | X₁,X₂), where X₁ and X₂ denote binomial random variables for trials n₁ and n₂, and parameters π₁ and π₂, respectively, and the non‐inferiority margin is Δ₀ > 0. We show two calculation methods for τ, an approximate method that uses normal approximation and an exact method that uses an exact posterior PDF. We compare the approximate probability with the exact probability for τ. Finally, we present the results of actual clinical trials to show the utility of index τ. Copyright © 2013 John Wiley & Sons, Ltd.     

Sums, products and ratios of generalized beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products and ratios of independent beta random variables. In this paper, we extend professor Pham-Gia's results when X ₁ and X ₂ are independent random variables distributed according to two generalized beta distributions. For each of these distributions, we derive exact expressions for the densities of S=X ₁+X ₂, D=X ₁−X ₂, P=X ₁ X ₂, and R=X ₂/X ₁. The expressions turn out to involve the incomplete beta function as well as the hypergeometric functions of one and two variables.     

The parametric hypothesis test of multiple uniform populations

X₁, X₂, …, X_k are k(k ? 2) uniform populations which each X_i follows U(0, θ_i). This note shows the test statistic for the null hypothesis H₀: θ₁ = θ₂ = ??? = θ_k by using the order statistics.     

On the strong Kotz approximation of Dirichlet random vectors

Let (X ₁, X ₂) be a bivariate L _p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α₁,α₂. If p=α₁=α₂=2, then (X ₁, X ₂) is a spherical random vector. The estimation of the conditional distribution of Z _u *:=X ₂ | X ₁>u for u large is of some interest in statistical applications. When (X ₁, X ₂) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z _u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z _u :=X ₂| X ₁=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z _u * and Z _u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.     

Testing retrospective hypotheses

This paper develops a conditional approach to testing hypotheses set up after viewing the data. For example, suppose X_i are estimates of location parameters θ_i, i = 1,…n. We show how to compute p-values for testing whether θ₁ is one of the three largest θ_i after observing that X₁ is one of the three largest X_i, or for testing whether θ₁ > θ₂ > … > θ_n after observing X₁ >X₂> … >X_n.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

Characterization of the mixtures of Rayleigh distributions by conditional expectation of order statistics

The mixture of Rayleigh random variables X ₁and X ₂ are identified in terms of relations between the conditional expectation of ( X_2:2² -X_1:2²)^r{\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}} given X _1:2 (or X_2:2^2k{X_{2:2}^{2k}} given X_1:2,"k ￡ r){X_{1:2},\forall k\leq r)} and hazard rate function of the distribution, where X _1:2 and X _2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions. Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used to illustrate our results.