Showing papers in "Statistics and Risk Modeling in 1995"
TL;DR: In this article, a nonparametric partitioning estimate m n (x ), which is a histogram-like mean regression function estimate, is presented, and proved its strong consistency under no smoothness condition on the regression function or on the distribution of X, in the sense that the integrated squared error between the estimate and the regression functions tends to zero almost surely as the sample size η tends to infinity.
Abstract: Let X be a random vector taking values in IR' and let Y be a non-negative bounded random variable . Moreover, assume a right censoring random variable C, with continuous distribution function, operating on Υ, independently of X and Y. In this randomly censored situation, we want to estimate Y based on the vector X of covariates, so that the mean squared error is minimized. For this purpose we construct a nonparametric partitioning estimate m n (x ) , which is a histogram-like mean regression function estimate, and prove its strong consistency under no smoothness condition on the regression function or on the distribution of X , in the sense that the integrated squared error between the estimate and the regression function tends to zero almost surely as the sample size η tends to infinity.
61 citations
TL;DR: In this article, the authors proposed a non-informative prior for posterior credible regions in the presence of nuisance parameters, which can be used for comparative purposes in a Bayesian analysis.
Abstract: Priors ensuring frequentist validity, up to o(n ), of credible regions based on the highest posterior density have been characterized in the presence of nuisance parameters. In this connection, the consequences of an orthogonal parametrization have also been discussed. 1: Introduction In recent years, there has been a revival of interest in problems relating to approximate frequentist validity of posterior credible regions. As noted in Tibshirani (1989), apart from providing a method for constructing accurate frequentist confidence regions, such studies are also helpful in defining noninformative priors which could be potentially useful for comparative purposes in a Bayesian analysis. The results available in the literature in this general area include those related to the approximate frequentist validity of one-sided posterior regions based on posterior quantiles ([18], [13], [16], [17], [12]), posterior regions based on the inversion of likelihood ratio and related statistics ([7], [8]) and highest posterior density (HPD) regions ([14], [9]). We refer to [11] and [15] for further interesting results and references. As for the problem of characterizing priors ensuring approximate frequentist validity of HPD regions, it appears that not much work has as yet been done on models involving nuisance parameters. In consideration of the current interest on such models, in this work we propose to fill up this gap to some extent. As a special case, models with an orthogonal parametrization ([5]) have been considered. An advantage of our approach is that it does not require an explicit evaluation of all the coefficients involved in the (approximate) posterior density of the interest parameter and this keeps the algebra relatively simple (see Section 3). AMS subject classification: 62F15, 62F25, 62E20 Key word· and phrases: highest posterior density, non-informative prior, parametric orthogonality.
33 citations
29 citations
TL;DR: In this paper, the authors considered estimating a bounded normal mean under LINEX loss and showed that a two-point prior is least favorable in case the parameter interval is small enough.
Abstract: Estimating a bounded normal mean is considered under LINEX loss $$L_\alpha (\theta, d) = s \cdot \biggl( \exp \bigl\{ \alpha (d - \theta) \bigr\} - \alpha (d - \theta) - 1 \biggr),$$ where $\alpha
e 0$ and $s > 0$ are fixed constants. It is shown that a two point prior is least favourable in case the parameter interval is small enough. So minimax and $\Gamma$-minimax estimators are determined where $\Gamma$ represents special vague prior information.
17 citations
17 citations
TL;DR: New nonparametric truncated sequential changepoint detection processes are considered via a theorem of Darling and Erdös (1956) and are found to stop faster than other existing procedures for relatively large changes that occur early in the sequence of observations.
Abstract: New nonparametric truncated sequential changepoint detection processes are considered via a theorem of Darling and Erdös (1956). Their asymptotic and finite sample properties are examined and compared to existing procedures. They are found to stop faster than other existing procedures for relatively large changes that occur early in the sequence of observations.
12 citations
12 citations
TL;DR: A new hierarchy of polynomial kernel functions with asymmetric supports is established, and a corrected version for a result that appeared in previous papers on the theory of optimal kernel functions in nonparametric curve estimation is provided.
Abstract: In this note, we provide a corrected version for a result that appeared in previous papers on the theory of optimal (in the sense of minimizing mean squared error) kernel functions in nonparametric curve estimation. In particular, a new version of Lemma 3 in Granovsky and Müller [2] is given. In this context, we establish a new hierarchy of polynomial kernel functions with asymmetric supports. Other new results include local optimality of certain polynomial kernel functions and surprising connections with smooth boundary kernels. 1. Optimal Kernel Functions In a variety of statistical curve estimation settings, including nonparametric regression, nonparametric density estimation and estimation of functionals of curves, the kernel method is a basic paradigm for constructing corresponding estimators, as it is closely related to the approximation of functions by convolution operators. Gasser, Müller and Mammitzsch [1] posed and investigated the general question of optimality of a kernel function. For a comprehensive review on the problem of optimal kernel choice, see Granovsky and Müller [3]. AMS 1980 subject classification: Primary 62G07, Secondary 49A34, 49B36, 42C10
11 citations
10 citations
TL;DR: In this article, the authors considered a class of zero mean second order stochastic processes whose covarlance kernel admlts a Fourier series decomposition, E[A'(s + t)X(3)] ~ e*\"*, and the subdass of these processes for which the coefficient functions ha are the Fourier transforms of complex measures nia, a 6 R. The problem of the asymptotic behavior of the variance, and of the consistency for some natural estimators of 6a(t), and of /«(A) whenever
Abstract: We consider a class of zero mean second order stochastic processes whose covarlance kernel admlts a Fourier series decomposition, E[A'(s + t)X(3)] ~ e*\"*, and the subdass of these processes for which the coefficient functions ha are the Fourier transforms of complex measures nia, a 6 R. These classes of processes which contain the almost periodically correlated processes and the strongly harmonizable processes, are frequently applied in signal analysis. This paper addresses the problem of the asymptotic behavior of the variance, and of the consistency for some natural estimators of 6a(t), and of /«(A) whenever m a ( d X ) = /„(A) dX. We deal with this problem under two different types of hypotheses: first, in terms of conditions on the associated stochastic spectral measure whenever the process is strongly harmonizable and gaussian, next, in terms of mixing conditions.
9 citations
TL;DR: In this paper, an upper bound for the maximum asymptotic bias for the linear regression model with random carriers and intercept is derived for the random carriers, and a recommendation for the choice of a function p is given for MM- and τ-estimators based on these comparisons.
Abstract: Yohai [14] established MM-estimators for robust regression with a high breakdown point and high asymptotic efficiency. An MM-estimator is a special case of an M-estimator for regression and uses a bounded function p and a general scale estimate. In this article we consider S-estimators (Rousseeuw and Yohai [13] ) with a modification of the corresponding ρ-function of the final M-estimator (more precisely: [formula omitted] for some constant k0 which guarantees asymptotic efficiency 95%) for the scale. A similar alternative with the same features are τ-estimators proposed by Yohai and Zamar [16] Within τ-contamination-neighbourhoods of S-estimators for scale, M-estimators for regression with general scale and τ-estimators an upper bound for the maximum asymptotic bias will be derived for the linear regression model with random carriers and intercept. Different possibilities for p will be compared with respect to these upper bounds. A recommendation for the choice of a function p will be given for MM- and τ-estimators based on these comparisons. The recommended function is constant for x ≥ some constant, same as for LS-estimators around the origin and smoothed in between. The practical meaning of results and some features of computing MM- and τ-estimators are discussed. © R. Oldenbourg Verlag, Munchen 1995
Journal Article•
TL;DR: In this article, an upper bound for the maximum asymptotic bias for the linear regression model with random carriers and intercept is derived for the random carriers, and a recommendation for the choice of a function p is given for MM- and τ-estimators based on these comparisons.
Abstract: Yohai [14] established MM-estimators for robust regression with a high breakdown point and high asymptotic efficiency. An MM-estimator is a special case of an M-estimator for regression and uses a bounded function p and a general scale estimate. In this article we consider S-estimators (Rousseeuw and Yohai [13] ) with a modification of the corresponding ρ-function of the final M-estimator (more precisely: [formula omitted] for some constant k0 which guarantees asymptotic efficiency 95%) for the scale. A similar alternative with the same features are τ-estimators proposed by Yohai and Zamar [16] Within τ-contamination-neighbourhoods of S-estimators for scale, M-estimators for regression with general scale and τ-estimators an upper bound for the maximum asymptotic bias will be derived for the linear regression model with random carriers and intercept. Different possibilities for p will be compared with respect to these upper bounds. A recommendation for the choice of a function p will be given for MM- and τ-estimators based on these comparisons. The recommended function is constant for x ≥ some constant, same as for LS-estimators around the origin and smoothed in between. The practical meaning of results and some features of computing MM- and τ-estimators are discussed. © R. Oldenbourg Verlag, Munchen 1995
TL;DR: In this article, the problem of selecting the best (or the worst) out of k(k > 2) available Negative Binomial populations is considered under the indifference zone approach, where the populations are ranked in terms of the success probabilities while the measure of aggregation is assumed to be known.
Abstract: The problem of selecting the best (or the worst) out of k(k > 2) available Negative Binomial populations is considered under the indifference zone approach. The populations are ranked in terms of the success probabilities while the measure of aggregation is assumed to be known. Two distance measures are used simultaneously to overcome the difficulty in obtaining optimal sample size due to dependency among the population parameters. For small samples, a method for finding a less conservative estimate of optimal sample size compared to Mulekar and Young (1993) to satisfy the criteria of minimal expected probability of correct selection is given. Large sample approximation to the probability of correct selection is used to obtain approximation to the optimal size of sample required from each population to meet specification of the probability of correct selection and the distance between the best and the next population. Explicit expression for the upper bound on the optimal sample size is also given. Alternate form of the probability of correct selection that AMS 1991 subject classifications. 62F07, 62F05
Abstract: Let {Κ, : η ä 0} be a sequence of independent and identically distributed random variables with continuous distribution function, and let {N(t) : t > 0} be a renewal process independent of {y„ : η > θ}. In this paper, we consider the renewal paced record process {X(f) : t > 0} based on {y„ : η > 0} and on [N(t) : t > 0} . We establish a strong invariance principle and limit laws for the record times of the process {X(e') : t θ}.
TL;DR: In this article, it was shown that the portion of observations which fall into a truncated region of sufficiently small probability can be neglected without losing asymptotically information without losing the information.
Abstract: Consider an iid sample of an underlying statistical experiment which is asymptotically Gaussian. It is shown that the portion of observations which fall into a truncated region of sufficiently small probability can be neglected without losing asymptotically information. As a special case we consider those observations which exceed a given threshold.
TL;DR: In this paper, the asymptotic efficiency of such a sample circular median as an estimate of the location parameter of a circularly symmetric distribution on the circle was shown.
Abstract: In [15], a ciTcularly Symmetrie directional distribution was obtained by showing that in the class oj circularly Symmetrie distributiona on the circle it is the only distribution for which the sample circular median is a maximum likelihood estimate of the location Parameter. We demonstrate asymptotic efficiency of such a sample circular median as an estimate of the location parameter of this model, tuith asymptotic efficiency being described within the Hdjek-LeCam framework. Accordingly, a suitable notion of local asymptotic normality for distributions on the circle is introduced and a convolution theorem characterizing the limit laws of regulär estimates is proved in the process. AMS Subject Classificationa : 62G20, 62H11
TL;DR: In this paper, a complete orthonormal system is incorporated in the adaptive determination of the score generating function, based on a suitable stopping rule, and various properties of this sequential adaptive procedure and the stopping rule are mentioned.
Abstract: Abstract Adaptive estimators (of parameters in simple regression models) based on signed-rank statistics are constructed and their asymptotic optimalities are studied. A complete orthonormal system is incorporated in the adaptive determination of the score generating function. The proposed sequential procedure is based on a suitable stopping rule. Various properties of this sequential adaptive procedure and the stopping rule are mentioned. Asymptotic linearity results of signed-rank statistics are also established and some rates of the convergence are studied.