Jan Mielniczuk

Estimating the density of a copula function

This paper deals with estimation of the density of a copula function as well as with that of the Radon-Nikodym derivative of a bivariate distribution function with respect to the product of its marginal distribution functions. Strong uniform consistency and asymptotic normality of kernel-type estimators are proved under various conditions on the bandwidth and on the smoothness of the kernel. As an application, the estimation of Neyman-Pearson curves in the testing of independence problem is discussed.

Estimation of Hurst exponent revisited

In order to estimate the Hurst exponent of long-range dependent time series numerous estimators such as based e.g. on rescaled range statistic (R/S) or detrended fluctuation analysis (DFA) are traditionally employed. Motivated by empirical behaviour of the bias of R/S estimator, its bias-corrected version is proposed. It has smaller mean squared error than DFA and behaves comparably to wavelet estimator for traces of size as large as 2^1^5 drawn from some commonly considered long-range dependent processes. It is also shown that several variants of R/S and DFA estimators are possible depending on the way they are defined and that they differ greatly in their performance.

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

SummaryConsider the stationary sequenceX1=G(Z1),X2=G(Z2),..., whereG(·) is an arbitrary Borel function andZ1,Z2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z1Zk+1) satisfyingr(0)=1 and ∑k=1∞ |r(k)|m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {Xn}, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetFn(·) be the empirical distribution function ofX1,...,Xn. We prove that, asn→∞, the empirical processn1/2{Fn(·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

Random-design regression under long-range dependent errors

We consider the random-design nonparametric regression model with long-range dependent errors that may also depend on the independent and identically distributed explanatory variables. Disclosing a smoothing dichotomy, we show that the finite-dimensional distributions of the Nadaraya-Watson kernel estimator of the regression fimunction converge either to those of a degenerate process with completely dependent marginals or to those of a Gaussian white-noise process. The first case occurs when the bandwidths are large enough in a specified sense to allow long-range dependence to prevail. The second case is for bandwidths that are small in the given sense, when both the required norming sequence and the limiting process are the same as if the errors were independent. This conclusion is also derived for all bandwidths if the errors are short-range dependent. The borderline situation results in a limiting convolution of the two cases. The main results contrast with previous findings for deterministic-design regression.

Local linear regression estimation for time series with long-range dependence

Consider the nonparametric estimation of a multivariate regression function and its derivatives for a regression model with long-range dependent errors. We adopt local linear fitting approach and establish the joint asymptotic distributions for the estimators of the regression function and its derivatives. The nature of asymptotic distributions depends on the amount of smoothing resulting in possibly non-Gaussian distributions for large bandwidth and Gaussian distributions for small bandwidth. It turns out that the condition determining this dichotomy is different for the estimates of the regression function than for its derivatives; this leads to a double bandwidth dichotomy whereas the asymptotic distribution for the regression function estimate can be non-Gaussian whereas those of the derivatives estimates are Gaussian. Asymptotic distributions of estimates of derivatives in the case of large bandwidth are the scaled version of that for estimates of the regression function, resembling the situation of estimation of cumulative distribution function and densities under long-range dependence. The borderline case between small and large bandwidths is also examined.

Data-dependent bandwidth choice for a grade density kernel estimate

The method of choosing a smoothing parameter for a grade density kernel estimate g is proposed. It consists in estimating the minimizer of the asymptotic MISE for two main terms in the expansion of g. The behaviour of the estimates incorporating proposed bandwidths is investigated in the variety of parametric models and compared with that of estimates using bandwidths suitable for the complete observability case. They are shown to perform well for unimodal densities and moderately well for multimodal ones.

Estimating density ratio with application to discriminant analysis

The estimator ĥ for the ratio of densities based on grade transformation is introduced. Under certain conditions the strong uniform consistency of ĥ is proved and its asymptotic law is determined. The result is applied to study the behaviour of divergence curve used in discriminant analysis.

Distant long-range dependent sums and regression estimation

Consider a stationary sequence G(Z0), G(Z1), ..., where G(·) is a Borel function and Z0, Z1, ... is a sequence of standard normal variables with covariance function E(Z0Zj) = j-[alpha]L(j), j = 1, 2, ..., where E(G(Z0)) = 0, E(G2(Z0)) 0 and sequences of gap-lengths l1,n, ..., lk,n such that l1,n --> [infinity] and lj,n - lj-1,n --> [infinity], j = 2, ..., k, arbitrary slowly, the vector process , 0

Density estimation in the simple proportional hazards model

In the simple proportional hazards model of random right censorship the limiting variance vACL2(x) at x of the kernel density estimator based on the Abdushukurov-Cheng-Lin estimator is shown to be equal to the corresponding variance pertaining to the Kaplan-Meier estimator times the expected proportion p of uncensored observations. More surprisingly, for appropriate p, vACL2(x) is smaller than the asymptotic variance of the classical kernel estimator based on a complete sample, for any x below the (1 - e-1)-quantile.

On the asymptotic mean integrated squared error of a kernel density estimator for dependent data

Hall and Hart (1990) proved that the mean integrated squared error (MISE) of a marginal kernel density estimator from an infinite moving average process X1, X2, ... may be decomposed into the sum of MISE of the same kernel estimator for a random sample of the same size and a term proportional to the variance of the sample mean. Extending this, we show here that the phenomenon is rather general: the same result continues to hold if dependence is quantified in terms of the behaviour of a remainder term in a natural decomposition of the densities of (X1, X1+i), I = 1, 2, ....