Pitman closeness of record values to population quantiles

In this paper, we examine the Pitman closeness of record statistics to the population quantiles of a location-scale family of distributions and study its monotonicity properties. Even though in general it depends on the parent distribution, exact expressions are derived for the required probabilities in the case of Uniform (−1,1) and exponential distributions. For the population median, it is shown that the first upper record is the Pitman-closest among all upper record values. Moreover, for the population median, in the case of symmetric distributions, the Pitman closeness probabilities of records are shown to be distribution-free and explicit expressions are also derived for these probabilities.     

Pitman closeness results for Type-I censored data from exponential distribution

Recent work on Pitman closeness has compared estimators under Type-II censored samples from exponential distribution based on observed number of failures. In this paper, we carry out similar Pitman closeness comparisons for Type-I censored samples from exponential distribution based on time under test.     

Comparisons among some predictors of exponential distributions using Pitman closeness

The bias, mean squared error and likelihood optimality criteria are used frequently to compare estimators and predictors. Recently, the probability of nearness around a statistic (estimator or predictor) has received a considerable attention in the literature. In this article, we adopt the Pitman’s measure of closeness (PMC) as an optimality criterion to compare the maximum likelihood, best linear unbiased, best linear invariant, median unbiased and conditional median predictors of a future ordered statistic based on a type II censored sample from an exponential distribution with unknown scale parameter. Numerical computations of the PMC for all comparisons among these predictors are performed and presented.     

Pitman closeness of k-records from two sequences to progressive Type-II censored order statistics

In this paper, we consider two independent \(k\) -record sequences with the same distribution. We determine the closeness probability of \(k\) -record values to a specific progressive Type-II censored order statistic. With this in mind, we first derive the exact expression for the Pitman closeness of records in general, and some special properties of the closeness probability are presented. Then, we apply the obtained results for the standard uniform and exponential distributions and exact expressions for the Pitman closeness are obtained. Finally, numerical results are displayed in figures.     

Risk and Pitman closeness properties of feasible generalized double k-class estimators in linear regression models with non-spherical disturbances under balanced loss function

In this article, a family of feasible generalized double k-class estimator in a linear regression model with non-spherical disturbances is considered. The performance of this estimator is judged with feasible generalized least-squares and feasible generalized Stein-rule estimators under balanced loss function using the criteria of quadratic risk and general Pitman closeness. A Monte-Carlo study investigates the finite sample properties of several estimators arising from the family of feasible double k-class estimators.     

Bayesian prediction based on a class of shrinkage priors for location-scale models

A class of shrinkage priors for multivariate location-scale models is introduced. We consider Bayesian predictive densities for location-scale models and evaluate performance of them using the Kullback–Leibler divergence. We show that Bayesian predictive densities based on priors in the introduced class asymptotically dominate the best invariant predictive density.     

Optimal reparametrization and large sample likelihood inference for the location-scale skew-normal model

Motivated by results in Rotnitzky et al. (2000), a family of parametrizations of the location-scale skew-normal model is introduced, and it is shown that, under each member of this class, the hypothesis H ₀: ?? = 0 is invariant, where ?? is the asymmetry parameter. Using the trace of the inverse variance matrix associated to a generalized gradient as a selection index, a subclass of optimal parametrizations is identified, and it is proved that a slight variant of Azzalini??s centred parametrization is optimal. Next, via an arbitrary optimal parametrization, a simple derivation of the limit behavior of maximum likelihood estimators is given under H ₀, and the asymptotic distribution of the corresponding likelihood ratio statistic for this composite hypothesis is determined.     

On the closeness of trajectories for model quasi-gasdynamic equations

On a model example of a linear hyperbolic equation with small parameter multiplying the highest time derivative it is shown that the closeness of individual trajectories to the dynamics of the limiting parabolic equation essentially depends on the Fourier spectra of the initial data. The trajectories stay close if the higher modes decay sufficiently fast. If the initial data are irregular and there are relatively high modes, then the convergence of the trajectories becomes non-uniform. Namely, the boundary layer is formed and there exist small moments of time such that the difference of the solutions reaches in the mean a finite value as the coefficient of the highest time derivative tends to zero. These results reflect the difficulties that may arise in the analysis of the systems of non-linear quasi-gasdynamic equations.     

Asymptotic behavior of polynomial Pitman estimators

The asymptotic normality of polynomial Pitman estimators for the location parameter is proved, the principal term of the formula for the variance of these estimators is obtained, and certain characterization problems are considered that occur in the study of polynomial Pitman estimators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 43, pp. 30–39, 1974.     

Some characterization and ordering results based on entropies of current records

The aim of this paper is to investigate the entropy properties of current records. Several characterizations are obtained based on the entropy of these statistics. It is proved that the equality of the entropy of the endpoints of record coverage is a characteristic property of symmetric distribution. It is shown that the parent distribution can be identified uniquely by the entropy properties of the endpoints of record coverage. Considering the records coming from distribution with decreasing (increasing) density, the monotonicity of entropy of current records is discussed. We also prove that the dispersive ordering of the parent distributions implies the entropy ordering of their respective current records.     

Local asymptotic minimax properties of Pitman estimates

Summary It is shown that for Euclidean parameter spaces every sequence of Pitman estimates has local asymptotic minimax properties. The result generalizes a previous result of Hájek, 1972, which has been proved under the condition of local asymptotic normality. In the present paper it is only assumed that a sequence of experiments E _n, n, converges weakly to a translation invariant limit experiment. According to LeCam, 1973 b, this is nearly the most general case which may occur. There are two main results. The first result states that Pitman estimates are minimax for translation invariant experiments. This improves a theorem of Girshick and Savage, 1951, which is restricted to location parameter experiments. In the second result we prove that the distributions of Pitman estimates for E _n, n, converge weakly to the distribution of the Pitman estimate for the limit experiment. This improves previous assertions of this kind due to Ibragimov and Has'minskii, 1973, Inagaki and Ogata, 1975, or Großmann, 1979, since the condition of weak convergence of experiments used here is considerably weaker than the invariance principles for likelihood processes used by these authors.     

Estimates for the closeness of successive convolutions of multidimensional symmetric distributions

Summary Let ξ₁, ξ₂,... be i.i.d random vectors in ℝ ^k with a common distribution ℒ(ξ_i),... = F, i = 1, 2,.... Let S _n = ξ₁+...+ξ _n . We investigate how small is the difference between ℒ(S _n ) and ℒ(S _{n+ m} ) in the case when ξ _i have symmetric distributions.     

The closeness and closability criteria for infinitesimal operators of certain semigroups

We consider generators of certain operator semigroups and study the closeness and closability properties of certain generators (in the operator class). We obtain the necessary and sufficient conditions for these properties.     

Improved estimation under Pitman's measure of closeness

Stein-type and Brown-type estimators are constructed for general families of distributions which improve in the sense of Pitman closeness on the closest (in a class) estimator of a parameter. The results concern mainly scale parameters but a brief discussion on improved estimation of location parameters is also included. The loss is a general continuous and strictly bowl shaped function, and the improved estimators presented do not depend on it, i.e., uniform domination is established with respect to the loss. The normal and inverse Gaussian distributions are used as illustrative examples. This work unifies and extends previous relevant results available in the literature.     

Bandwidth selection for the estimation of transition probabilities in the location-scale progressive three-state model

Times between consecutive events are often of interest in medical studies. Usually the events represent different states of the disease process and are modeled using multi-state models. This paper introduces and studies a feasible estimation method for the transition probabilities in a progressive three-state model. We assume that the vector of gap times $(T_1,T_2)$ satisfies a nonparametric location-scale regression model $T_2=m(T_1)+\sigma (T_1)\epsilon $ , where the functions $m$ and $\sigma $ are ‘smooth’, and $\epsilon $ is independent of $T_1$ . Under this model, Van Keilegom et al. (J Stat Plan Inference 141:1118–1131, 2011) proposed estimators of the transition probabilities. However, the important issue of automatic bandwidth choice in this setting has not been examined, making the analysis of real datasets rather difficult. In this paper, we study the performance of their estimator in practice, we propose some modifications and study practical issues related to the implementation of the estimator, which involves the choice of an appropriate bandwidth. In an extensive simulation study the good performance of the method is shown. Simulations also demonstrate that the proposed estimator compares favorably with alternative estimators. Furthermore, the proposed methodology is illustrated with a real database on breast cancer.     

Asymptotic closeness to limiting shapes for expanding embedded plane curves

We show that for embedded or convex plane curves expansion, the difference u(x,t)-r(t) in support functions between the expanding curves γ_t and some expanding circles C_t (with radius r(t)) has its asymptotic shape as t→∞. Moreover the isoperimetric difference L²-4πA is decreasing and it converges to a constant if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then L²-4πA decreases to and there exists an asymptotic center of expansion for γ_t. Mathematics Subject Classification (2000) 35K15, 35K55     

On the closeness of eigenvalues and singular values for almost normal matrices

It is proved that a matrix is almost normal if and only if its singular values are close to the absolute values of its eigenvalues. In the special case when the spectral norm and spectral radius are close, it is proved that the dominating eigenvalue is well conditioned. A refinement of a perturbation theorem by Henrici is proved, and its numerical behavior is compared with adaptations of the Gerschgorin theorem. It is specially devised for almost triangular matrices.     

On a topological closeness of perturbed Mandelbrot sets

In the present work we propose a numerical and visual tool for the study of the deformation of the Mandelbrot sets of perturbed Mandelbrot maps by noise in comparison with the original Mandelbrot set. Further, by employing these numerical tools, we support the invariance of the Mandelbrot set of a noise-perturbed Mandelbrot map under different noise realizations. Finally, we provide evidence for the non-fractal structure of the Mandelbrot set of a noise-perturbed Mandelbrot map.