Eckhard Liebscher

Semiparametric density estimators using copulas

ABSTRACT In this article we establish semiparametric density estimators by employing the ideas of copulas and density weighting functions. Results on asymptotic normality and uniform strong consistency are derived. The formula for the asymptotic mean square error is used to get the optimal bandwidth. Moreover, applications to several families of copulas are discussed.

Erratum to “Construction of asymmetric multivariate copulas” [J. Multivariate Anal. 99 (2008) 2234-2250

Assume that C1, . . . , Ck : [0, 1]d → [0, 1] are copulas. Let gji : [0, 1] → [0, 1] for j = 1, . . . , k, i = 1, . . . , d be functions with the property that each of them is strictly increasing or is identically equal to 1. Suppose that ∏k j=1 gji(v) = v for v ∈ [0, 1], i = 1, . . . , d, and limv→0+0 gij(v) = gji(0) for j = 1, . . . , k, i = 1, . . . , d. Here we consider the two-dimensional product copula:

Copula-based dependence measures

Abstract The aim of the present paper is to examine two wide classes of dependence coefficients including several well-known coefficients, for example Spearman’s ρ, Spearman’s footrule, and the Gini coefficient. There is a close relationship between the two classes: The second class is obtained by a symmetrisation of the coefficients in the former class. The coefficients of the first class describe the deviation from monotonically increasing dependence. The construction of the coefficients can be explained by geometric arguments. We introduce estimators of the dependence coefficients and prove their asymptotic normality.

Strong convergence of estimators in nonlinear autoregressive models

In the paper we prove rates of strong convergence of M-estimators for the parameters in a general nonlinear autoregressive model. In the proofs we utilize a variational principle from stochastic optimization theory which was proved by Shapiro (Ann. Oper. Res. 30 (1991) 169). The application of the general theory is illustrated in the case of continuous threshold models.

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.

Investigation of the suitability of tissue phantoms for testing the constancy of ultrasonic transducer arrays in quality assurance.

PURPOSE To ensure high quality ultrasound diagnostics, proper functioning of the devices used is a necessary prerequisite. Ultrasound transducers have proven to be the most failure-prone part of the signal chain. Their technical monitoring is possible in principle with the help of tissue phantoms. The background of the present study is to determine which type of phantoms and which measurement parameters are best suited to a consistency test as part of routine quality assurance of ultrasound imaging systems. MATERIALS AND METHODS A classic wire-type phantom (ATS Mod. 539, ATS Labs Bridgeport, USA) and a 3 D cyst phantom (TCC, Timelkam, Austria) were used for the studies and comparative tests were conducted between intact transducers and those in which faults had been simulated. The collected measurement data show a relatively large scatter. Therefore, statistical analysis methods were used, and the discrimination analysis proved to be a useful tool. RESULTS Local failures which arise, e. g. due to the breakdown of individual piezoelectric elements or element groups in the transducer array, can be detected with the help of the gray value targets of the ATS phantom, but only in those cases in which the error-affected sound field part actually overlaps with the target under consideration. The TCC phantom is not suitable for the detection of such errors. Global transducer failures, i. e. those that affect the entire array, can even be detected with both types of phantoms. CONCLUSION When the emphasis of quality assessment is on the detection of local defects in the array that make up the largest part of the transducer faults, studies with conventional phantoms are only of limited value.

Quick test of ultrasonic transducer arrays radiating in air using B-mode-images.

Abstract This paper describes a statistical procedure working on ultrasound B-mode images that were recorded with the transducer array radiating in air. From the digital image data, a gray value profile was generated using a computer program. A statistical procedure was developed to examine if there are significant deviations from the mean. They occur, for example, when single piezoelements are severely damaged or completely out of function. The entire technique was tested using an ultrasonic imaging system (SonixTouch, Ultrasonix Medical Corporation, Richmond, Canada) in which individual elements or groups of elements can be turned off via firmware. A failure of as large as two out of 128 elements was detectable.

Consistency check of diagnostic ultrasound transducer arrays using tissue-equivalent phantoms

Measurements at two different types of Phantoms - a classic thread phantom and a 3D cyst phantom - were carried out to assess the technical quality of ultrasound B-scan systems. Using statistical methods we examined whether the phantoms are suitable for a consistency test of the transducer arrays. Based on simulated transducer failures it was found that in particular the detection of local of element failures in arrays turned out to be problematic.

Copula-Based Dependence Measures For Piecewise Monotonicity

Abstract The aim of the present paper is to develop and examine association coefficients which can be helpfully applied in the framework of regression analysis. The construction of the coeffiecients is connected with the well-known Spearman coeffiecient and extensions of it (see Liebscher [5]). The proposed coeffiecient measures the discrepancy between the data points and a function which is strictly increasing on one interval and strictly decreasing in the remaining domain.We prove statements about the asymptotic behaviour of the estimated coeffiecient (convergence rate, asymptotic normality).

Goodness-of-Approximation of Copulas by a Parametric Family

In the paper we introduce a measure for goodness of approximation based on the Cramer von Mises-statistic. In place of the unknown parameter of interest, a minimum-distance estimator of the parameter is plugged in. We prove asymptotic normality of this statistic and establish a test on goodness-of-approximation.