In this paper we deal with a locally asymptotic stringent test for a general class of nonlinear time series heteroscedastic models. Based on the local asymptotic normality (LAN) property of these models, we propose a scoretype test statistic for testing hypotheses on the parameters appearing in the mean and variance functions of the proposed statistical test with and without nuisance parameters. Its asymptotic null distribution is obtained as well as the local power of the test.

One of the problems which appear during comprehension and working of the complex systems is the analysis of the stability of their functioning. These last years, the specialists became aware of the importance of the stability analysis in actuarial sciences and in financial mathematics. In this work, we are interested in the application of the strong stability method in queueing systems and in risk models. We clarify the conditions of equivalence and translation of results between models of these two theories.

In this work we propose a new methodology for the comparison of two regression functions $f_1$ and $f_2$ in the case of homoscedastic error structure and a fixed design. Our approach is based on the empirical Fourier coefficients of the regression functions $f_1$ and $f_2$ respectively. As our main results we obtain the asymptotic distribution of the test statistic under the null hypothesis $f_1 = f_2$ and local and global alternatives. A simulation study is conducted to investigate the finite sample performance of our test.

We propose to study, by two different approaches, Bayesian and classical, the test of the hypothesis of stationary first order autoregressive model against the random walk model. Therefore, we are going to present the classical approach and the Bayesian approach of this test says the unit root test. The principal aim is to improve the unit root test, either with proposing a better statistic, or with proposing an adequate prior in order to make it more powerful.

We set general conditions under which the general poverty index, which summarizes all the available indices, is asymptotically represented with some empirical processes. This representation theorem offers a general key, in most directions, for the asymptotic of the bulk of poverty indices and issues in poverty analysis. Our representation results uniformly hold on a large collection of poverty indices. They enable the continuous measure of poverty with longitudinal data.

In this paper, we are concerned with the stochastic process $$\beta_n (q_t, t) = \beta_n(t) = \frac{1}{\sqrt{n}}\sum_{j=1}^n \{G_{t,n}(Y(t)) - G_t(Y_j(t))\} q_t(Y_j(t)), \tag{A}$$ where for $n \geq 1$ and $T > 0$, the sequences $\{Y_1(t), Y_2(t), \cdots, Y_n(t), t \in [0, T]\}$ are independent observations of some real stochastic process $Y(t), t \in [0,T]$, for each $t \in [0,T], G_t$ is the distribution function of $Y(t)$ and $G_{t,n}$ is the empirical distribution function based on $Y_1(t), Y_2(t), \cdots, Y_n(t)$ and finally $q_t$ is a bounded real function defined on $\mathbb{R}$. This process appears when investigating some time-dependent L-Statistics which are expressed as a function of some functional empirical process and the process (A). Since the functional empirical process is widely investigated in the literature, the process reveals itself as an important key for L-Statistics laws. In this paper, we state an extended study of this process, give complete calculations of the first moments, the covariance function and find conditions for asymptotic tightness.

In this paper, we consider the problem of stability of the estimation in autoregressive models for the finite sample case. A Monte Carlo comparison of the least square estimator and the Hurwicz estimator is performed in various contaminated models. The paper shows that, the least square estimator is very sensitive to contamination and, despite the median-unbiasedness of the Hurwicz estimator, an optimal and stable estimator in small sample case is to be constructed.

We discuss two distinct approaches, for distorting risk measures of sums of dependent random variables, which preserve the property of coherence. The first, based on distorted expectations, operates on the survival function of the sum. The second, simultaneously applies the distortion on the survival function of the sum and the dependence structure of risks, represented by copulas. Our goal is to propose risk measures that take into account the fluctuations of losses and possible correlations between risk components.

We consider a class of nonstationary time series defined by $Y_t = \mu_t + \sum^{\infty}_{k=0} C_{t, k^{\sigma} t-k^{\eta}t-k}$ where $\{\eta_t ; t \in \mathbb{Z}\}$ is sequence of iid random variables with regularly varying tail probabilities, $\sigma_t$ is a scale parameter and $\{C_{t,k.} t \in \mathbb{Z}, K > 0\}$ an infinite array of random variables identically distributed called weights. In this article, the extreme value theory of ${Y_t}$ is studied. Under mild conditions, convergence results for a point process based on the moving averages are proved.

In this paper, we obtain asymptotic confidence bands for both the density and regression functions in the framework of nonparametric estimation. Beforehand, the asymptotic behaviors in probability of the kernel estimator of the density and the Nadaraya-Watson estimator of the regression function are described while local and global optimal smoothing parameters are investigated. A simulation study is conducted, showing the good performance of the confidence bands obtained for small sample.

By transforming a data set with a modification of the Champernowne distribution function, a kernel quantile estimator for heavy-tailed distributions is given. The asymptotic mean squared error (AMSE) of the proposed estimator and related asymptotically optimal bandwidth are evaluated. Some simulations are drawn to show the performance of the obtained results

We are interested in the existence and uniqueness of maximum likelihood estimators of parameters in the two multiplicative regression models, with Poisson or negative binomial probability distributions. Following its work on the multiplicative Poisson model with two factors without repeated measures, Haberman gave a necessary and sufficient condition for existence and uniqueness of the maximum likelihood estimator of this model, furthermore, he provided an explicit expression of this estimator. In this paper, we propose a generalization of these results to a multiplicative Poisson model with repeated measures and more than two factors. We also show that the condition obtained is also a necessary and sufficient condition for the existence and uniqueness of the maximum likelihood estimator in the multiplicative negative binomial model with several factors, with or without repeated measures.