Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

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.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

Contre-exemple au théorème de P.C. Consul sur la factorisation des distributions de Poisson généralisées

This note exhibits two independent random variables on integers, X₁ and X₂, such that neither X₁ nor X₂ has a generalized Poisson distribution, but X₁ + X₂ has. This contradicts statements made by Professor Consul in his recent book.     

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as X_t = α X_t-1 + Y_t . Until a specified time T, Y_t are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Y_t, t≥T, is a Tukey mixture T(εσ) = (1 – ε)N(0,1) + εN(0, σ²). Bias of LSE as a function of α and ε, and σ² is considered. A rather unexpected fact is revealed: given α and ε, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with ε (“the amount of contaminants”).     

Nonlinear regression in a trivariate elliptical distribution via a linear combination of order statistics

In this paper, by assuming that (X, Y ₁, Y ₂)^T has a trivariate elliptical distribution, we derive the exact joint distribution of X and a linear combination of order statistics from (Y ₁, Y ₂)^T and show that it is a mixture of unified bivariate skew-elliptical distributions. We then derive the corresponding marginal and conditional distributions for the special case of t kernel. We also present these results for an exchangeable case with t kernel and illustrate the established results with an air-pollution data.     

On the Failure Probability of Used Coherent Systems

In analyzing the lifetime properties of a coherent system, the concept of “signature” is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s₁, s₂, …, s_n), such that s_i=P(T=X_i:n), where, X_i:n, i=1, 2, …, n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as p_i(t)=P(T=X_i:n|T>t) and investigate its properties as a function of time.     

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.     

A note on L1 consistent estimation

Let (??, ??) be a space with a σ-field, M = {P_s; s o} a family of probability measures on A, Θ arbitrary, X₁,…,X_n independently and identically distributed P random variables. Metrize Θ with the L₁ distance between measures, and assume identifiability. Minimum-distance estimators are constructed that relate rates of convergence with Vapnik-Cervonenkis exponents when M is “regular”. An alternative construction of estimates is offered via Kolmogorov's chain argument.     

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 ?}.     

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.     

ON A SHARED ALLELE TEST OF RANDOM MATING

A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, n_ijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n₀, n₁ and n₂ where n_p is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?_o where n₀, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X² compares observed (n_s) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X² is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in _e. This paper gives two theorems which show that x is indeed the approximate distribution of X² for large n and K₁“, provided that no allele type over-dominates the others.     

Recursive estimation of the mode of a multivariate density

Let f be an unknown possibly multimodal density on R^d and let X₁, X₂, … be a sequence of independent random vectors with density f. Several recursive estimates of the mode of f are proposed, and sufficient conditions ensuring their weak and strong consistency are established.     

Lp convergence and complete convergence for weighted sums of AANA random variables

Let {X_n, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {d_ni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the L^p convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.     

FIRST-EXIT TIMES FOR POISSON SHOT NOISE

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T _b at which a given level b > 0 is exceeded. An integral equation for the joint density of T _b and X(T _b ) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT of the function ? P(T _b > t) in terms of certain LTs derived from the distribution function H(x; t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T _b > t, X(t) < u), that is the joint distribution function of sup_{0 ≤ s ≤ t} X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).