Luís Ramos

In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn), wheregis some smooth function,is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn} are constants such that μn→∞. These bounds allow us to identify a suitable sequence of random variables which is asymptotically of the same type of g(Vn+μn) showing that the rate of convergence for these uniform approximations depends on the ratio of the second derivative to the first derivative ofg. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.

We use the almost normality approach to derive the models we use. Polynomial almost normality is presented in a first chapter. Thus we show that low degree polynomials on independent normal variables with small variation coefficients are very approximately normal distributed. This result is then used do derive models for series of studies assuming normality and independence for the initial observations and low variation coefficients. We then apply this models first for single series and then for matched series of studies. We will assume that the matched series are associated to the treatments and orthogonal design. The special case of prime basis factorials is considered.

When a diffusion is ergodic its transition density converges to its invariant density, see Durrett (1998). This convergence enabled us to introduce a sample partitioning technique that gives in each sub-sample, maximum likelihood estimators. The averages of these being a natural choice as estimators. To compare our estimators with the optimal we obtained from martingale estimating functions, see Sorensen (1998), we used the Ornstein-Uhlenbeck process for which exact simulations can be carried out.

Some asymptotic expansions non necessarily related to the central limit theorem are discussed. After observing that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation, two instances of this observation are presented. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to $(g(X+ μ_n))_{n ın \mathbbm{N}}$, where $g$ is some smooth function, $X$ is a random variable having a moment and a bounded density and $(μ_{n})_{n ın \mathbbm{N}}$ is a sequence going to infinity; the multivariate case as well as the proofs and a complete set of references will be published elsewhere. We next present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas, namely, a generic Laplace's type integral, by the sequence $(μ_n X)_{n ın \mathbbm{N}}$, $X$ being a Gamma distributed random variable. Finally, a simulation study of this last example is presented in order to stress the quality of asymptotic approximations proposed.

The required results of stochastic calculus are introduced as well as sufficient conditions for a diffusin to be ergodic. Invariant densities are obtained for two families of diffusion. These families belong the Ornstein-Uhlenbeck and Cox-Ingersoll & Ross diffusions. The moments of transition density of the Cox-Ingersoll & Ross diffusion were obtained and it was shown that this density converges to the invariant density. Lastly a technique, based on sample partition, is given for parameter estimation for ergodic diffusion. A numerical application of that technique for the Ornstein-Uhlenbeck diffusion is given. Final remarks and comments are included.

Some asymptotic expansions not necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to gX + nn∈, where g is some smooth function, X is a random variable and nn∈ is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace’s type integral, randomized by the sequence nXn∈, X being a Gamma distributed random variable.