Browsing by Subject "Weak convergence"

Filter results by typing the first few letters
Now showing 1 - 5 of 5
  • Thumbnail Image
    ItemOpen Access
    Bivariate distribution and the hazard functions when a component is randomly truncated
    (Elsevier, 1997-01) Gürler, Ü.
    In random truncation models one observes the i.i.d. pairs (Ti≤Yi), i=1, ..., n. If Y is the variable of interest, then T is another independent variable which prevents the complete observation of Y and random left truncation occurs. Such a type of incomplete data is encountered in medical studies as well as in economy, astronomy, and insurance applications. Let (Y, Y) be a bivariate vector of random variables with joint distribution function F(y, x) and suppose the variable Y is randomly truncated from the left. In this study, nonparametric estimators for the bivariate distribution and hazard functions are considered. A nonparametric estimator for F(y, x) is proposed and an a.s. representation is obtained. This representation is used to establish the consistency and the weak convergence of the empirical process. An expression for the variance of the asymptotic distribution is presented and an estimator is proposed. Bivariate "diverse-hazard" vector is introduced which captures the individual and joint failure behaviors of the random variables in opposite "time" directions. Estimators for this vector are presented and the large sample properties are discussed. Possible applications and a moderate size simulation study are also presented. © 1997 Academic Press.
  • Thumbnail Image
    ItemUnknown
    Bivariate estimation with right-truncated data
    (Taylor & Francis, 1996) Gürler, Ü.
    Bivariate estimation with survival data has received considerable attention recently; however, most of the work has focused on random censoring models. Another common feature of survival data, random truncation, is considered in this study. Truncated data may arise if the time origin of the events under study precedes the observation period. In a random right-truncation model, one observes the iid samples of (Y, T) only if (Y ≤ T), where Y is the variable of interest and T is an independent variable that prevents the complete observation of Y. Suppose that (V, X) is a bivariate vector of random variables, where Y is subject to right truncation. In this study the bivariate reverse-hazard vector is introduced, and a nonparametric estimator is suggested. An estimator for the bivariate survival function is also proposed. Weak convergence and strong consistency of this estimator are established via a representation by iid variables. An expression for the limiting covariance function is provided, and an estimator for the limiting variance is presented. Alternative methods for estimating the bivariate distribution function are discussed. Obtaining large-sample results for the bivariate distribution functions present more technical difficulties, and thus their performances are compared via simulation results. Finally, an application of the suggested estimators is presented for transfusion-related AIDS (TR-AIDS) data on the incubation time.
  • Thumbnail Image
    ItemOpen Access
    A comparison of hazard rate estimators for left truncated and right censored data
    (1992) Uzunoğulları, Ü.; Wang, J.-L.
    SUMMARY: Left truncation and right censoring arise frequently in practice for life data. This paper is concerned with the estimation of the hazard rate function for such data. Two types of nonparametric estimators based on kernel smoothing methods are considered. The first one is obtained by convolving a kernel with a cumulative hazard estimator. The second one is in the form of a ratio of two statistics. Local properties including consistency, asymptotic normality and mean squared error expressions are presented for both estimators. These properties facilitate locally adaptive bandwidth choice. The two types of estimators are then compared based on their theoretical and empirical performances. The effect of overlooking the truncation factor is demonstrated through the Channing House data.
  • Thumbnail Image
    ItemUnknown
    Relaxation of multidimensional variational problems with constraints of general form
    (Pergamon Press, 2001) Hüsseinov, F.
    A relaxation of multidimensional variational problems with constraint of rather general form on gradients of admissible functions is investigated. It is assumed that the gradient of an admissible function belongs to an arbitrary bounded set. This relaxation involves as a class of admissible function the closure of the class of admissible functions of the original problem in the topology of uniform convergence, and uses a theorem characterizing this closure.
  • Thumbnail Image
    ItemUnknown
    Toeplitz operators on arveson and dirichlet spaces
    (Birkhaeuser Science, 2007) Alpay, D.; Kaptanoǧlu, H. T.
    We define Toeplitz operators on all Dirichlet spaces on the unit ball of CN and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. © Birkhäuser Verlag Basel/Switzerland 2007.