Divergence Measures Estimation and its Asymptotic Normality Theory in the Discrete Case

We consider a simple estimator of the density mode using the frequency polygon estimate. We investigate strong consistency of the estimator for strong mixing sequence of real variables under mild assumptions. We study the almost sure rate of convergence and we show that the estimator can achieve optimal almost sure rates of convergence for appropriate choices of the bin widths. The asymptotic normality of the simple estimator is given and a simulation study is performed. Our asymptotic results are obtained without any differentiability condition assumed on the density around the mode.

We present a statistical process depending on a continuous time parameter τ whose each margin provides a Generalized Hill’s estimator. In this paper, the asymptotic normality of the finite-dimensional distributions of this family are completely characterized for τ > 1/2 when the underlying distribution function lies on the maximum domain of attraction. The ratio of two different margins of the statistical process characterizes entirely the whole domain of attraction. Its asymptotic normality is also studied. The results permit in general to build a new family of estimators for the extreme value index whose asymptotic properties can be easily derived. For example, we give a new estimate of the Weibull extreme value index and we study its consistency and its asymptotic normality.

In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence.For this purpose we generalize an elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies, which was recently established by the authors.First we study martingale convergence in L2, and in the setting of the Azuma-Hoeffding inequality.In a second step we study the strong law of large numbers for martingale differences.Applications are the tradeoff for the multicolor generalized Plya urn processes, the generalized Chinese restaurant process, statistical M-estimators, as well as excursion frequencies of the Galton-Watson branching process.

Eweda and Macchi [IEEE Trans. Automat. Control, 29 (1984), pp. 119–127] and Watanabe [IEEE Trans. Inform. Theory, 30 (1984), pp. 134–140] show that thee sequence of random vectors generated by a stochastic gradient adaptive filtering algorithm converges almost surely and in $L_p $ (for p an even integer) to the solution of the associated Wiener–Hopf equation when the driving data process is stationary and weakly dependent. Under strong (i.e., Rosenblatt or $\alpha $) and $\psi $ -mixing conditions, together with various moment bounds on the driving data process, an almost sure functional invariance principle is obtained that approximates the sample paths of the random process generated by the stochastic gradient algorithm with the sample paths of a particular Gauss–Markov process. Almost sure rates of convergence in the form of laws of the iterated logarithm follow from the functional invariance principle. As a byproduct a functional central limit theorem is also obtained for a sequence of processes derived by suitably scaling the sequence of iterations generated by the algorithm.

ON THE ALMOST SURE RATE OF CONVERGENCE OF TEMPORAL-DIFFERENCE LEARNING ALGORITHMS

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. https://doi.org/10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is Hin (0,1) and the drift is mathcal {C}^alpha , alpha in [0,1] and alpha >1-1/(2H), we show the strong L_p and almost sure rates of convergence to be ((1/2+alpha H)wedge 1) -varepsilon , for any varepsilon >0. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. https://doi.org/10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence 1/2-varepsilon of the Euler–Maruyama scheme for mathcal {C}^alpha drift, for any varepsilon ,alpha >0.

We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\frac{T}{n^{6/7}}\to 0} as T → ∞ {T\to\infty} and n → ∞ {n\to\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.

Sample distribution function based goodness-of-fit test for complex surveys

We consider the random continued fraction where s n are independent random variables with the same gamma distribution. Every realization of the sequence defines a Stieltjes function that can be expressed as for some measure σ on the positive half-line. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Padé approximants of the function S . Using the Dyson–Schmidt method for an equivalent one-dimensional disordered system and the results of Marklof et al. , we obtain explicit formulae (in terms of modified Bessel functions) for the almost sure rate of convergence of these approximants, and for the almost sure distribution of their poles.

Procedures based on latent-trait models are described for computation and asymptotic statistical inference for two decision consistency indexes often used in mastery or competency testing. Details are given for the Rasch model. Simulations were conducted for some typical mastery testing situations with tests of 30 to 60 items and for raw score frequency distributions with small to marked levels of negative skewness. The results appear to indicate that, for these situations, the asymptotic normal theory may be safely used with samples of 100 subjects.

Inverse sampling is commonly used for surveying rare (but not clustered) populations. However, when the sample size from the rare group is chosen too small, the customary unbiased estimator of the population mean appears to be highly skewed. In such a case, confidence intervals based on asymptotic normal theory have coverage rate smaller than the nominal level. As an approach to overcome this problem, we propose two resampling methods consisting of with-replacement bootstrap (BWR) and without replacement bootstrap (BWO) to construct confidence intervals for the population mean under simple inverse sampling without replacement. We carried out a simulation study to evaluate the behavior of suggested bootstrap methods, the normal approximation and the logarithmic transformation methods. Our simulation results suggest that the BWO method is preferable, since it provides intervals with coverage rate closer to the nominal level together with more balanced error rate.

In investigations on the behaviour of robust estimators, typically their consistency and their asymptotic normality are studied as a necessity. Their rates of convergence, however, are often given less weight. We show here that the rate of convergence of a multivariate robust estimator to its true value plays an important role when using the estimator in procedures for identifying outliers in multivariate data.

Direct shape from shading with improved rate of convergence

Extremum seeking control (ESC) is an adaptive optimization method originally proposed for static systems but later extended to Hammestein/Wiener-like systems and more recently also to more general dynamic systems. In the latter case the focus has been on proving convergence and stability of solutions in the vicinity of the optimum. The proofs are in general based on combining asymptotic methods like singular perturbations and averaging, which leads to a three time-scale factorization of the problem where the control action is forced to be several orders of magnitude slower than the open-loop dynamics of the plant. This implies that the convergence rate will be impractically slow for many applications. In this paper, we employ Tikhonov theory and averaging to study the rate of convergence while employing only two time-scales. In particular, the analysis places no restrictions on the rate of the gradient estimation and therefore allows for significantly faster control compared to the conventional approach. The plant is approximated as a linear parameter varying system (LPV) which is then used to derive a global quantitative expression for the convergence rate in terms of the ESC parameters and the frequency response of the LPV plant. For Hammerstein/Wiener-like systems, the derived expression is used to show that the ESC loop behaves like a gradient descent method while it has a more complex behavior in the general case. Finally, an isothermal biochemical reactor is used to illustrate the results and some of the difficulties which might arise in the general case, such as the fact that the convergence rate can be low locally even if the gradient of the cost function is large.

UNSTEADY HYDROELASTICITY OF FLOATING PLATES