Multivariate density estimation using dimension reducing information and tail flattening transformations for truncated or censored data

This paper introduces a multivariate density estimator for truncated and censored data with special emphasis on extreme values based on survival analysis. A local constant density estimator is considered. We extend this estimator by means of tail flattening transformation, dimension reducing prior knowledge and a combination of both. The asymptotic theory is derived for the proposed estimators. It shows that the extensions might improve the performance of the density estimator when the transformation and the prior knowledge is not too far away from the true distribution. A simulation study shows that the density estimator based on tail flattening transformation and prior knowledge substantially outperforms the one without prior knowledge, and therefore confirms the asymptotic results. The proposed estimators are illustrated and compared in a data study of fire insurance claims.     

Variable location and scale kernel density estimation

Variable (bandwidth) kernel density estimation (Abramson (1982,Ann. Statist.,10, 1217–1223)) and a kernel estimator with varying locations (Samiuddin and El-Sayyad (1990,Biometrika,77, 865–874)) are complementary ideas which essentially both afford bias of orderh ⁴ as the overall smoothing parameterh 0, sufficient differentiability of the density permitting. These ideas are put in a more general framework in this paper. This enables us to describe a variety of ways in which scale and location variation may be extended and/or combined to good theoretical effect. This particularly includes extending the basic ideas to provide new kernel estimators with bias of orderh ⁶. Technical difficulties associated with potentially overly large variations are fully accounted for in our theory.     

Reducing asymptotic bias of weak instrumental estimation using independently repeated cross-sectional information

In this paper, we consider the instrumental variable estimation (the two-stage least squares estimator and the limited information maximum likelihood estimator) using weak instruments in a repeated measurements or a panel data model. We show that independently repeated cross-sectional data can reduce the asymptotic bias of the instrumental variable estimation when instruments are weakly correlated with endogenous variables. When the number of repeated measurements tends to infinity, we can achieve consistent instrumental variable estimation with weak instruments.     

Asymptotically unbiased estimation of the second order tail parameter

We introduce a class of asymptotically unbiased estimators for the second order parameter in extreme value statistics. The estimators are constructed by means of an appropriately chosen linear combination of two simple, but biased, kernel estimators for the second order parameter. Asymptotic normality is proven under a third order condition on the tail behavior, some conditions on the kernel functions and for an intermediate number of upper order statistics. A specific member from the proposed class, obtained with power kernel functions, is derived and its finite sample behavior studied in a small simulation experiment.     

Confident estimation for density of a biological population based on line transect sampling

Line transect sampling is a very useful method in survey of wildlife population. Confident interval estimation for density D of a biological population is proposed based on a sequential design. The survey area is occupied by the population whose size is unknown. A stopping rule is proposed by a kernel-based estimator of density function of the perpendicular data at a distance. With this stopping rule, we construct several confidence intervals for D by difference procedures. Some bias reduction techniques are used to modify the confidence intervals. These intervals provide the desired coverage probability as the bandwidth in the stopping rule approaches zero. A simulation study is also given to illustrate the performance of this proposed sequential kernel procedure.     

Approximating conditional density functions using dimension reduction

We propose to approximate the conditional density function of a random variable Y given a dependent random d-vector X by that of Y given θ^τX, where the unit vector θ is selected such that the average Kullback-Leibler discrepancy distance between the two conditional density functions obtains the minimum. Our approach is nonparametric as far as the estimation of the conditional density functions is concerned. We have shown that this nonparametric estimator is asymptotically adaptive to the unknown index θ in the sense that the first order asymptotic mean squared error of the estimator is the same as that when θ was known. The proposed method is illustrated using both simulated and real-data examples.     

Robust and asymptotically unbiased estimation of extreme quantiles for heavy tailed distributions

A robust and asymptotically unbiased extreme quantile estimator is derived from a second order Pareto-type model and its asymptotic properties are studied under suitable regularity conditions. The finite sample properties of the proposed estimator are investigated with a small simulation experiment.     

Non–parametric probability density function estimation on a bounded support: Applications to shape classification and speech coding

In statistics, it is usually difficult to estimate the probability density function from N independent samples X₁,X₂, …?, X_N identically distributed. A lot of work has been done in the statistical literature on the problem of probability density estimation (e.g. Cencov, 1962; Devroye and Gyorfi, 1981; Hall, 1980 and 1982; Hominal, 1979; Izenman, 1991; Kronmal and Tarter, 1968; Parzen, 1962; Rosenblatt, 1956). In this paper, we consider random variables on bounded support. Orthogonal series estimators, studied in detail by Kronmal and Tarter (1968), by Hall (1982) and by Cencov (1962), show that there is a disadvantage related to the Gibbs phenomenon on the bias of these estimators. We suggest a new method for the non–parametric probability density function estimation based on the kernel method using an appropriately chosen regular change of variable. The new method can be used for several problems of signal processing applications (scalar or vector quantization, speech or image processing, pattern recognition, etc.). Applications to shape classification and speech coding are given.     

Smooth estimation of survival and density functions for a stationary associated process using Poisson weights

Let {X_n,n≥1} be a sequence of stationary non-negative associated random variables with common marginal density f(x). Here we use the empirical survival function as studied in Bagai and Prakasa Rao (1991) and apply the smoothing technique proposed by Gawronski (1980) (see also Chaubey and Sen, 1996) in proposing a smooth estimator of the density function f and that of the corresponding survival function. Some asymptotic properties of the resulting estimators, similar to those obtained in Chaubey and Sen (1996) for the i.i.d. case, are derived. A simulation study has been carried out to compare the new estimator to the kernel estimator of a density function given in Bagai and Prakasa Rao (1996) and the estimator in Buch-Larsen et al. (2005).