We study a rank based univariate two-sample distribution-free test. The test statistic is the difference between the average of between-group rank distances and the average of within-group rank ...distances. This test statistic is closely related to the two-sample Cramér–von Mises criterion. They are different empirical versions of a same quantity for testing the equality of two population distributions. Although they may be different for finite samples, they share the same expected value, variance and asymptotic properties. The advantage of the new rank based test over the classical one is its ease to generalize to the multivariate case. Rather than using the empirical process approach, we provide a different easier proof, bringing in a different perspective and insight. In particular, we apply the Hájek projection and orthogonal decomposition technique in deriving the asymptotics of the proposed rank based statistic. A numerical study compares power performance of the rank formulation test with other commonly-used nonparametric tests and recommendations on those tests are provided. Lastly, we propose a multivariate extension of the test based on the spatial rank.
By extending the methods in Peligrad et al. (2014a, b), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for ...studying nonparametric regression with random field errors and strong limit theorems.
In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to ...the domain of attraction of a stable law with index \(0<\alpha\leq2\) under the condition that the innovations are centered if \(1<\alpha\leq2\) and are symmetric if \(\alpha=1\). We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.
In this paper, we establish a local limit theorem for linear fields of random variables constructed from independent and identically distributed innovations each with finite second moment. When the ...coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, i.e. for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.
In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (2002) studied the polynomial transformations of Gaussian FARIMA(0,d,0) processes by ...applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (1996, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p,d,q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.
In this paper we propose a variable bandwidth kernel regression estimator for \(i.i.d.\) observations in \(\mathbb{R}^2\) to improve the classical Nadaraya-Watson estimator. The bias is improved to ...the order of \(O(h_n^4)\) under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.
J. Appl. Probab. 56 (2019) 223-245 We study the Cram\'er type moderate deviation for partial sums of random
fields by applying the conjugate method. The results are applicable to the
partial sums of ...linear random fields with short or long memory and to
nonparametric regression with random field errors.
We study the mutual information estimation for mixed-pair random variables. One random variable is discrete and the other one is continuous. We develop a kernel method to estimate the mutual ...information between the two random variables. The estimates enjoy a central limit theorem under some regular conditions on the distributions. The theoretical results are demonstrated by simulation study.
The best mean square error that the classical kernel density estimator achieves if the kernel is non-negative and f has only two continuous derivatives, is of the order of special characters omitted. ...If negative kernels are allowed, then this rate can be improved depending on the smoothness of f and the order of the kernel. Abramson and others modified the classical kernel estimator, assumed non-negative, by allowing the bandwidth hn to depend on the data. The last and best result in the literature is Hall, Hu and Marron who show that under suitable assumptions on a non-negative kernel K and the density f, |fˆn( t) − f(t)| = O P(special characters omitted) for fixed t. The main result of this thesis states that special characters omitted |fˆn(t) − f(t)| = OP((special characters omitted)4/9) where Dn and fˆn(t) are purely data driven and Dn can be taken as close as desired to the set { t : f(t) > 0}. This rate is best possible for estimating a density in the sup norm. The data driven fˆn(t) and Dn have 'ideal' counterparts that depend on f, and for the ideal estimator, slightly sharper results are proven.
Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$
in the central limit theorem for martingales with some restrictive conditions.
In the present paper a modification of the ...methods, developed by Bolthausen
(Ann. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), is
applied for obtaining the same convergence rate for a class of more general
martingales. An application to linear processes is discussed.
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