Using power transformations when approximating quantiles

Estimators are obtained tor quantiles of survival distributions. This is accomplished by approximating Lritr distribution of the transtorrneri data, where the transformation used is that of Box and Cox (1964). The normal approximation as in Box and Cox and, in addition, the extreme value approximation are considered. More generally, to use the methods given, the approximating distribution must come from a location-scale family. For some commonly used survival random variables T the performance of the above approximations are evaluated in terms of the ratio of the true quantiles of T to the estimated one, in the long run. This performance is also evaluated for lower quantiles using simulated lognormai, Weibull and gamma data. Several examples are given to illustrate the methodology herein, including one with actual data.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

Estimation of gaussian mixtures with rotationally invariant covariance matrices

Homoscedastic and heteroscedastic Gaussian mixtures differ in the constraints placed on the covariance matrices of the mixture components. A new mixture, called herein a strophoscedastic mixture, is defined by a new constraint, This constraint requires the matrices to be identical under orthogonal trans¬formations, where different transformations are allowed for different matrices. It is shown that the M-step of the EM method for estimating the parameters of strophoscedastic mixtures from sample data is explicitly solvable using singular value decompositions. Consequently, the EM-based maximum likelihood estimation algorithm is as easily implemented for strophoscedastic mixtures as it is for homoscedastic and heteroscedastic mixtures. An example of a “noisy” Archimedian spiral is presented.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

A semiparametric maximum likelihood ratio test for the change point in copula models

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. We also discuss the Gaussian-type approximations for the semiparametric likelihood ratio. The asymptotic distribution of the proposed statistic under specified alternatives is shown to be normal, and an approximation to the power function is given. Simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the double exponential and Gaussian-type approximations.     

Correlation in a Bayesian framework

The authors consider the correlation between two arbitrary functions of the data and a parameter when the parameter is regarded as a random variable with given prior distribution. They show how to compute such a correlation and use closed form expressions to assess the dependence between parameters and various classical or robust estimators thereof, as well as between p‐values and posterior probabilities of the null hypothesis in the one‐sided testing problem. Other applications involve the Dirichlet process and stationary Gaussian processes. Using this approach, the authors also derive a general nonparametric upper bound on Bayes risks.     

Confidence curves and improved exact confidence intervals for discrete distributions

The author describes a method for improving standard “exact” confidence intervals in discrete distributions with respect to size while retaining correct level. The binomial, negative binomial, hypergeometric, and Poisson distributions are considered explicitly. Contrary to other existing methods, the author's solution possesses a natural nesting condition: if α < α', the 1 ‐ α' confidence interval is included in the 1 ‐ α interval. Nonparametric confidence intervals for a quantile are also considered.     

A new estimation method for Weibull-type tails based on the mean excess function

Studying the right tail of a distribution, one can classify the distributions into three classes based on the extreme value index γ$\gamma$. The class γ>0$\gamma >0$ corresponds to Pareto-type or heavy tailed distributions, while γ<0$\gamma <0$ indicates that the underlying distribution has a finite endpoint. The Weibull-type distributions form an important subgroup within the Gumbel class with γ=0$\gamma =0$. The tail behaviour can then be specified using the Weibull tail index. Classical estimators of this index show severe bias. In this paper we present a new estimation approach based on the mean excess function, which exhibits improved bias and mean squared error. The asserted properties are supported by simulation experiments and asymptotic results. Illustrations with real life data sets are provided.     

On the stability of the risk hull method for projection estimators

We consider in this paper the regularization by projection of a linear inverse problem Y=Af+εξ$Y=\mathit{Af}+\epsilon \xi$ where ξ$\xi$ denotes a Gaussian white noise, A   a compact operator and ε>0$\epsilon >0$ a noise level. Compared to the standard unbiased risk estimation (URE) method, the risk hull minimization (RHM) procedure presents a very interesting numerical behavior. However, the regularization in the singular value decomposition setting requires the knowledge of the eigenvalues of A$A$. Here, we deal with noisy eigenvalues: only observations on this sequence are available. We study the efficiency of the RHM method in this situation. More generally, we shed light on some properties usually related to the regularization with a noisy operator.