Search results for "Delta method"
showing 10 items of 13 documents
A ML Estimator of the Correlation Dimension for Left-hand Truncated Data Samples
2002
— A maximum-likelihood (ML) estimator of the correlation dimension d 2 of fractal sets of points not affected by the left-hand truncation of their inter-distances is defined. Such truncation might produce significant biases of the ML estimates of d 2 when the observed scale range of the phenomenon is very narrow, as often occurs in seismological studies. A second very simple algorithm based on the determination of the first two moments of the inter-distances distribution (SOM) is also proposed, itself not biased by the left-hand truncation effect. The asymptotic variance of the ML estimates is given. Statistical tests carried out on data samples with different sizes extracted from populatio…
Functional Principal Components Analysis with Survey Data
2008
This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. FPCA relies on estimations of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and convergent. Under mild assumptions, asymptotic variances are derived for the FPCA’ estimators and convergent estimators of them are proposed. Our approach is illustrated with a simulation study and we…
A novel Stochastic Discretized Weak Estimator operating in non-stationary environments
2012
The task of designing estimators that are able to track time-varying distributions has found promising applications in many real-life problems. A particularly interesting family of distributions are the binomial/multiomial distributions. Existing approaches resort to sliding windows that track changes by discarding old observations. In this paper, we report a novel estimator referred to as the Stochastic Discretized Weak Estimator (SDWE), that is based on the principles of Learning Automata (LA). In brief, the estimator is able to estimate the parameters of a time varying binomial distribution using finite memory. The estimator tracks changes in the distribution by operating on a controlled…
Multivariate equivalence tests for use in pharmaceutical development.
2014
Statistical equivalence analyses are well-established parts of many studies in the biomedical sciences. Also in pharmaceutical development and manufacturing equivalence testing methods are required in order to statistically establish similarities between machines, process components, or complete processes. This article presents a choice of multivariate equivalence testing procedures for normally distributed data as generalizations of existing univariate methods. In all derived methods, variability is interpreted as nuisance parameter. The use of the proposed methods in pharmaceutical development is demonstrated with a comparative analysis of dissolution profiles.
Use of functionals in linearization and composite estimation with application to two-sample survey data
2009
An important problem associated with two-sample surveys is the estimation of nonlinear functions of finite population totals such as ratios, correlation coefficients or measures of income inequality. Computation and estimation of the variance of such complex statistics are made more difficult by the existence of overlapping units. In one-sample surveys, the linearization method based on the influence function approach is a powerful tool for variance estimation. We introduce a two-sample linearization technique that can be viewed as a generalization of the one-sample influence function approach. Our technique is based on expressing the parameters of interest as multivariate functionals of fi…
The asymptotic covariance matrix of the Oja median
2003
The Oja median, based on a sample of multivariate data, is an affine equivariant estimate of the centre of the distribution. It reduces to the sample median in one dimension and has several nice robustness and efficiency properties. We develop different representations of its asymptotic variance and discuss ways to estimate this quantity. We consider symmetric multivariate models and also the more narrow elliptical models. A small simulation study is included to compare finite sample results to the asymptotic formulas.
A Distribution-Free Two-Sample Equivalence Test Allowing for Tied Observations
1999
A new testing procedure is derived which enables to assess the equivalence of two arbitrary noncontinuous distribution functions from which unrelated samples are taken as the data to be analyzed. The equivalence region is defined to consist of all pairs (F, G) of distribution functions such that for independent X ∼F, Y ∼G the conditional probability of {X > Y} given {X ¬= Y} lies in some short interval around 1/2. The test rejects the null hypothesis of nonequivalence if and only if the standardized distance between the U-statistics estimator of P|X > Y | X ¬= Y] and the center of the equivalence interval (1/2 - e 1 , 1/2 + e 2 ) does not exceed a critical upper bound which has to be comput…
Properties of Design-Based Functional Principal Components Analysis.
2010
This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and con…
Variance Estimation and Asymptotic Confidence Bands for the Mean Estimator of Sampled Functional Data with High Entropy Unequal Probability Sampling …
2013
For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson estimator can be approximated by the Hajek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that it asymptotically provides a uniformly consistent estimator of the variance function of the Horvitz-Thompson estimator of the mean function. Rates of convergence to the true variance function are gi…
Establishing some order amongst exact approximations of MCMCs
2016
Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis-Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order …