Variance Estimation for Fractional Brownian Motions with Fixed Hurst Parameters

Some real-world phenomena in geo-science, micro-economy, and turbulence, to name a few, can be effectively modeled by a fractional Brownian motion indexed by a Hurst parameter, a regularity level, and a scaling parameter σ², an energy level. This article discusses estimation of a scaling parameter σ² when a Hurst parameter is known. To estimate σ², we propose three approaches based on maximum likelihood estimation, moment-matching, and concentration inequalities, respectively, and discuss the theoretical characteristics of the estimators and optimal-filtering guidelines. We also justify the improvement of the estimation of σ² when a Hurst parameter is known. Using the three approaches and a parametric bootstrap methodology in a simulation study, we compare the confidence intervals of σ² in terms of their lengths, coverage rates, and computational complexity and discuss empirical attributes of the tested approaches. We found that the approach based on maximum likelihood estimation was optimal in terms of efficiency and accuracy, but computationally expensive. The moment-matching approach was found to be not only comparably efficient and accurate but also computationally fast and robust to deviations from the fractional Brownian motion model.     

Corrected likelihood-ratio tests in logistic regression using small-sample data

Likelihood-ratio tests (LRTs) are often used for inferences on one or more logistic regression coefficients. Conventionally, for given parameters of interest, the nuisance parameters of the likelihood function are replaced by their maximum likelihood estimates. The new function created is called the profile likelihood function, and is used for inference from LRT. In small samples, LRT based on the profile likelihood does not follow χ² distribution. Several corrections have been proposed to improve LRT when used with small-sample data. Additionally, complete or quasi-complete separation is a common geometric feature for small-sample binary data. In this article, for small-sample binary data, we have derived explicitly the correction factors of LRT for models with and without separation, and proposed an algorithm to construct confidence intervals. We have investigated the performances of different LRT corrections, and the corresponding confidence intervals through simulations. Based on the simulation results, we propose an empirical rule of thumb on the use of these methods. Our simulation findings are also supported by real-world data.     

The conditional level of confidence intervals for the normal variance and applications

Suppose we observe two independent random vectors each having a multivariate normal distribution with covariance matrix known up to an unknown scale factor σ . The first random vector has a known mean vector while the second has an unknown mean vector. Interest centers around finding confidence intervals for σ² with confidence coefficient 1 ? α. Standard results show that, when we only observe the first random vector, an optimal (i.e., smallest length) confidence interval C, based on the well-known chi- squared statistic, can be constructed for σ² . When we additionally observe the second random vector, the confidence interval C is no longer optimal for estimating σ². One criterion useful for detecting the non-optimality of a confidence interval C concerns whether C admits positively or negatively biased relevant subsets. This criterion has recently received a good deal of attention. It is shown here that under some conditions the confidence interval C admits positively biased relevant subsets.

Applications of this result to the construction of ‘better‘ unconditional confidence intervals for σ² are presented. Some simulation results are given to indicate the typical extent of improvement attained.     

Bayesian adaptive lasso with variational Bayes for variable selection in high-dimensional generalized linear mixed models

This article describes a full Bayesian treatment for simultaneous fixed-effect selection and parameter estimation in high-dimensional generalized linear mixed models. The approach consists of using a Bayesian adaptive Lasso penalty for signal-level adaptive shrinkage and a fast Variational Bayes scheme for estimating the posterior mode of the coefficients. The proposed approach offers several advantages over the existing methods, for example, the adaptive shrinkage parameters are automatically incorporated, no Laplace approximation step is required to integrate out the random effects. The performance of our approach is illustrated on several simulated and real data examples. The algorithm is implemented in the R package glmmvb and is made available online.     

Estimating population size with truncated sampling

Let X₁, X₂,…,X_n be independent, indentically distributed random variables with density f(x,θ) with respect to a σ-finite measure μ. Let R be a measurable set in the sample space X. The value of X is observable if X ? (X?R) and not otherwise. The number J of observable X’s is binomial, N, Q, Q = 1?P(X ? R). On the basis of J observations, it is desired to estimate N and θ. Estimators considered are conditional and unconditional maximum likelihood and modified maximum likelihood using a prior weight function to modify the likelihood before maximizing. Asymptotic expansions are developed for the [Ncirc]’s of the form [Ncirc] = N + α√N + β + o_p(1), where α and β are random variables. All estimators have the same α, which has mean 0, variance σ² (a function of θ) and is asymptotically normal. Hence all are asymptotically equivalent by the usual limit distributional theory. The β’s differ and Eβ can be considered an “asymptotic bias”. Formulas are developed to compare the asymptotic biases of the various estimators. For a scale parameter family of absolutely continuous distributions with X = (0,∞) and R = (T,∞), special formuli are developed and a best estimator is found.     

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.     

Wavelet Estimation of a Density in a GARCH-type Model

We consider the GARCH-type model: S = σ² Z, where σ² and Z are independent random variables. The density of σ² is unknown whereas the one of Z is known. We want to estimate the density of σ² from n observations of S under some dependence assumption (the exponentially strongly mixing dependence). Adopting the wavelet methodology, we construct a nonadaptive estimator based on projections and an adaptive estimator based on the hard thresholding rule. Taking the mean integrated squared error over Besov balls, we prove that the adaptive one attains a sharp rate of convergence.     

Empirical likelihood for nonlinear models with missing responses

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.     

Two-sided tests for change in level for correlated data

The classical problem of change point is considered when the data are assumed to be correlated. The nuisance parameters in the model are the initial level μ and the common variance σ². The four cases, based on none, one, and both of the parameters are known are considered. Likelihood ratio tests are obtained for testing hypotheses regarding the change in level, δ, in each case. Following Henderson (1986), a Bayesian test is obtained for the two sided alternative. Under the Bayesian set up, a locally most powerful unbiased test is derived for the case μ=0 and σ²=1. The exact null distribution function of the Bayesian test statistic is given an integral representation. Methods to obtain exact and approximate critical values are indicated.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.     

Small-sample behavior of weighted least squares in experimental design applications

In experimental design applications unbiased estimators s_i ² of the variances σ_i ² are possible. These estimators may be used in Weighted Least Squares (WLS) when estimating the parameters β. The resulting small-sample behavior is investigated in a Monte Carlo experiment. This experiment shows that an asymptotically valid covariance formula can be used if s_i ² is based on, say, at least 5 observations. The WLS estimator based on estimators s_i ² gives more accurate estimators of β, provided the σ_i ² differ by a factor, say, 10.     

ON TESTING AGAINST RESTRICTED ALTERNATIVES FOR THE VARIANCES OF GAUSSIAN MODELS

Let π₁,…,π_p be p independent normal populations with means μ₁…, μ_p and variances σ²₁,…, σ²_p respectively. Let X(n_i) be a simple random sample of size n_i from π_i, i = 1,…,p. Given the simple random samples X(n₁),…, X(n_p) from π₁,…,π_p respectively, a test has been proposed for testing the homogeneity of variances H₀: σ²₁=…σ²_p, against the restricted alternative, H₁: σ²₁≥…≥σ²_p, with at least one strict inequality. Some properties of the test are discussed and critical values are tabulated.     

La méthode de vraisemblance pour les processus linéaires spatiaux définis sur un treillis triangulaire

Spatial linear processes {X_s, s ? T} where T is a triangular lattice in R² are considered. Special attention is given to the class of spatial moving-average processes. Precisely, for each site s T, the variable X_s is defined as a linear combination of real-valued random shocks located at the vertices of regular concentric hexagons centered at s. For Gaussian random shocks, the process is also Gaussian, and estimates of its parameters are obtained by maximizing the exact likelihood. For non-Gaussian random shocks, the exact likelihood is difficult to obtain; however, the Gaussian likelihood is still used giving the pseudo-Gaussian likelihood estimates. The behaviour of these estimates is analyzed through the study of asymptotic properties and some simulation experiments based on an isotropic model defined with one coefficient.     

Powerful Parametric Tests Based on Sum‐Functions of Spacings

Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this paper, tests based on sum‐functions of sample spacings are proposed, and large sample theory of the tests are presented under simple null hypotheses as well as under close alternatives. Tests, which are optimal within this class, are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test in regular parametric models. Some examples are given, which show that the proposed tests work also in situations where the likelihood ratio test breaks down. Extensions to more general hypotheses are discussed.     

Empirical likelihood for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically $\chi ^2_{q+r}$ , where $q$ and $r$ are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.     

The generalized inverse Lindley distribution: A new inverse statistical model for the study of upside-down bathtub data

ABSTRACT

In this article, a two-parameter generalized inverse Lindley distribution capable of modeling a upside-down bathtub-shaped hazard rate function is introduced. Some statistical properties of proposed distribution are explicitly derived here. The method of maximum likelihood, least square, and maximum product spacings are used for estimating the unknown model parameters and also compared through the simulation study. The approximate confidence intervals, based on a normal and a log-normal approximation, are also computed. Two algorithms are proposed for generating a random sample from the proposed distribution. A real data set is modeled to illustrate its applicability, and it is shown that our distribution fits much better than some other existing inverse distributions.     

A regression approach for estimating the parameters of the covariance function of a stationary spatial random process

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions, and various methods for estimating the parameters have been proposed in the literature. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h>0$h>0$ which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. Simulation studies show that our method outperforms all well-known least squares-based approaches to the variogram estimation and is comparable to the maximum likelihood estimation of the parameters of the covariance function. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions.     

Estimating and testing generalized linear models under inequality restrictions

We consider maximum likelihood estimation and likelihood ratio tests under inequality restrictions on the parameters. A special case are order restrictions, which may appear for example in connection with effects of an ordinal qualitative covariate. Our estimation approach is based on the principle of sequential quadratic programming, where the restricted estimate is computed iteratively and a quadratic optimization problem under inequality restrictions is solved in each iteration. Testing for inequality restrictions is based on the likelihood ratio principle. Under certain regularity assumptions the likelihood ratio test statistic is asymptotically distributed like a mixture of χ², where the weights are a function of the restrictions and the information matrix. A major problem in theory is that in general there is no unique least favourable point. We present some empirical findings on finite-sample behaviour of tests and apply the methods to examples from credit scoring and dentistry.     

Inference on the Common Variance of Correlated Normal Random Variables

ABSTRACT

Scale equivariant estimators of the common variance σ², of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ² which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ²: a Bayesian estimator for invariant prior as well as two non-Bayesian estimators. We then generalized these three estimators for the case of several variables with multiple unknown correlations. In addition, we developed a system of confidence intervals which produce the desired coverage probability while being efficient in terms of expected length.