On piecewise linear density estimators

We study piecewise linear density estimators from the L ₁ point of view: the frequency polygons investigated by S cott (1985) and J ones et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L ₁ strongly consistent. We derive large deviation inequalities. For twice differentiable densities with compact support their expected L ₁ error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.     

Nonparametric density estimation in compound Poisson processes using convolution power estimators

Consider a compound Poisson process which is discretely observed with sampling interval $\Delta $ until exactly $n$ nonzero increments are obtained. The jump density and the intensity of the Poisson process are unknown. In this paper, we build and study parametric estimators of appropriate functions of the intensity, and an adaptive nonparametric estimator of the jump size density. The latter estimation method relies on nonparametric estimators of $m$ th convolution powers density. The $L^2$ -risk of the adaptive estimator achieves the optimal rate in the minimax sense over Sobolev balls. Numerical simulation results on various jump densities enlight the good performances of the proposed estimator.     

Convergence rates of density estimators for sums of powers of observations

Densities of functions of independent and identically distributed random observations can be estimated by a local U-statistic. It has been shown recently that, under an appropriate integrability condition, this estimator behaves asymptotically like an empirical estimator. In particular, it converges at the parametric rate. The integrability condition is rather restrictive. It fails for the sum of powers of two observations when the exponent is at least two. We have shown elsewhere that for exponent equal to two the rate of convergence slows down by a logarithmic factor on the support of the squared observation and is still parametric outside this support. For exponent greater than two, and on the support of the exponentiated observation, the estimator behaves like a classical density estimator: The bias is not negligible and the rate depends on the bandwidth. Outside the support, the rate is again parametric.     

The density of the parameter estimators when the observations are distributed exponentially

We present the probability density of parameter estimators whenN independent variables are observed, each of them distributed according to the exponential low (with some parameters to be estimated). The numberN is supposed to be small. Typically, such an experimental situation arises in problems of software reliability, another case is a small sample in the GLIM modeling. The considered estimator is defined by the maximum of the posterior probability density; it is equal to the maximum likelihood estimator when the prior is uniform. The exact density is obtained, and its approximation is discussed in accordance with some information-geometric considerations. The main body of the paper has been prepared during the author’s visit in LMC/IMAG Grenoble, France, on the invitation of Université Joseph Fourier in January 1994.     

Felix Bernstein

Felix Bernstein was a mathematician and statistician who is remembered for the "Schröder-Bernstein Theorem" of set theory. He was appointed director in 1907 of what he was to develop into the Göttingen Institute for Mathematical Statistics, and in 1911 was involved in the formulation and proof of what became known as the Borel-Cantelli lemmas. Bernstein worked out in 1924 the correct hypothesis for the genetic transmission of human bloodgroups, on the basis of the available statistical material.     

Predictive density estimators for daily volatility based on the use of realized measures

The main objective of this paper is to propose a feasible, model free estimator of the predictive density of integrated volatility. In this sense, we extend recent papers by Andersen et al. [Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P., 2003. Modelling and forecasting realized volatility. Econometrica 71, 579–626], and by Andersen et al. [Andersen, T.G., Bollerslev, T., Meddahi, N., 2004. Analytic evaluation of volatility forecasts. International Economic Review 45, 1079–1110; Andersen, T.G., Bollerslev, T., Meddahi, N., 2005. Correcting the errors: Volatility forecast evaluation using high frequency data and realized volatilities. Econometrica 73, 279–296], who address the issue of pointwise prediction of volatility via ARMA models, based on the use of realized volatility. Our approach is to use a realized volatility measure to construct a non-parametric (kernel) estimator of the predictive density of daily volatility. We show that, by choosing an appropriate realized measure, one can achieve consistent estimation, even in the presence of jumps and microstructure noise in prices. More precisely, we establish that four well known realized measures, i.e. realized volatility, bipower variation, and two measures robust to microstructure noise, satisfy the conditions required for the uniform consistency of our estimator. Furthermore, we outline an alternative simulation based approach to predictive density construction. Finally, we carry out a simulation experiment in order to assess the accuracy of our estimators, and provide an empirical illustration that underscores the importance of using microstructure robust measures when using high frequency data.     

Riesz estimators

We consider properties of estimators that can be written as vector lattice (Riesz space) operations. Using techniques widely used in economic theory and functional analysis, we study the approximation properties of these estimators paying special attention to additive models. We also provide two algorithms RIESZVAR(i-ii) for the consistent parametric estimation of continuous multivariate piecewise linear functions.     

Unbiased product estimators

Summary The paper presents a comparative study of product estimators proposed byRobson [1957] andMurthy [1964]. It is seen that the Robson's estimator gives a better performance.     

Risk-efficient nonparametric sequential estimators

Summary Letx ₁,x ₂,x ₃, ... be a sequence of independent identically distributed random variables andτ an estimable parameter of their distribution. We want to estimateτ by the correspondingU-statisticu _n with loss function (u _n −τ)² +cn. We derive a stopping time and prove its risk-efficiency in the sense of Starr (1966) without any assumption on the nature of the distribution function other than the existence of some moments. Research supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 72, at the Universit?t Bonn.     

Reinterpreting polynomial regression

Recent work by Stimson et al. (1978) and Jagodzinski and Weede (1981) has explored a translation technique intended to circumvent some of the attendant problems of polynomial regression in sociological research. This piece specifies some of the limitations of the translation technique while explicating a straightforward method of using it when appropriate. An alternate technique is then developed which avoids the earlier pitfalls. The application of the alternative technique is exemplified in an analysis of alternative occupational scales.     

Dominating estimators for minimum-variance portfolios

In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets d≥4$d\ge 4$ and number of observations n≥d+2$n\ge d+2$. The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed d$d$ but n→∞$n\to \infty$ and n,d→∞$n,d\to \infty$ but n/d→q≤∞$n/d\to q\le \infty$ are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.     

Comparison of tail index estimators

We compare various estimators for the index of distribution functions with regularly varying tails by calculating their asymptotic mean squared errors after choosing the optimal number of upper order statistics involved (which is different for different estimators).     

Regularization of nonparametric frontier estimators

In production theory and efficiency analysis, we estimate the production frontier, the locus of the maximal attainable level of an output (the production), given a set of inputs (the production factors). In other setups, we estimate rather an input (or cost) frontier, the minimal level of the input (cost) attainable for a given set of outputs (goods or services produced). In both cases the problem can be viewed as estimating a surface under shape constraints (monotonicity, …). In this paper we derive the theory of an estimator of the frontier having an asymptotic normal distribution. It is based on the order-m$m$ partial frontier where we let the order m$m$ to converge to infinity when n→∞$n\to \infty$ but at a slow rate. The final estimator is then corrected for its inherent bias. We thus can view our estimator as a regularized frontier. In addition, the estimator is more robust to extreme values and outliers than the usual nonparametric frontier estimators, like FDH and than the unregularized order-_mn$m_n$ estimator of Cazals et al. (2002) converging to the frontier with a Weibull distribution if _mn→∞$m_n\to \infty$ fast enough when n→∞$n\to \infty$. The performances of our estimators are evaluated in finite samples and compared to other estimators through some Monte-Carlo experiments, showing a better behavior (in terms of robustness, bias, MSE and achieved coverage of the resulting confidence intervals). The practical implementation and the robustness properties are illustrated through simulated data sets but also with a real data set.     

Asymptotic linearity of minimax estimators

The celebrated local asymptotic minimax (LAM) theorem due to HÁjek (1972) also includes the statement that a LAM estimator Is necessarily asymptotically linear. A similar result. is true for semi-parametric models, but Hájek's result doesn't apply to this case as the efficient influence function is often not contained in the (proper) tangent space. This note gives a simple, elementary proof of both the LAM theorem and the necessity of asymptotic linearity of a LAM estimator sequence.     

On construction of admissible quadratic estimators

A random linear model for spatially located sensors measured intensity of a source of signals in discrete instants of time is considered. A basis of a quadratic subspace useful in quadratic estimation of a function of model parameters is given. Received: December 1999     

Recursions for the two-stage least-squares estimators

This paper presents recursion formulae for the two-stage least-squares estimators of the structural coefficients in a simultaneous equation model and for the residual sum of squares used in estimating the asymptotic covariance matrix. Included are formulae for updating estimates when a new set of observations is obtained and for revising estimates when a set of observations is discarded. The recursion formulae should prove to be of both practical and theoretical interest to econometricians.     

Minimax estimators of a normal variance

In the estimation problem of unknown variance of a multivariate normal distribution, a new class of minimax estimators is obtained. It is noted that a sequence of estimators in our class converges to the Stein's truncated estimator. Received: March 1998     

Approximations for densities of sufficient estimators

A simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described. It is an extension of the one developed by O. Barndorff-Nielsen and D.R. Cox (1979) for exponential families. A series expansion in powers of n^?1 is derived of which the first term has an error of order n^?1 which can effectively be reduced to $\text{n}{}^{\mathrm{<mtext>-</mtext><mtext>?3</mtext><mtext>2</mtext>}}$ by renormalization. The results obtained are similar to those given by H.E. Daniels's (1954) saddlepoint method but the derivations are simpler. A brief treatment of approximations to conditional densities is given. Theorems are proved which extend the validity of the multivariate Edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors. These extensions permit the treatment of problems arising in time series analysis. The technique is used by J. Durbin (1980) to obtain approximations to the densities of partial serial correlation coefficients.     

Data-based ranking of realised volatility estimators

This paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process. I provide conditions under which the relative accuracy of competing estimators can be consistently estimated (as T→∞), and show that forecast evaluation tests may be adapted to the problem of ranking these estimators. The proposed methods avoid making specific assumptions about microstructure noise, and facilitate comparisons of estimators that would be difficult using methods from the extant literature, such as those based on different sampling schemes. An application to high frequency IBM data between 1996 and 2007 illustrates the new methods.