Browsing by Subject "Bivariate distribution"

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Open Access
Bivariate density estimation with randomly truncated data
(Elsevier, 2000) Gürler, Ü.; Prewitt, K.
In this study bivariate kernel density estimators are considered when a component is subject to random truncation. In bivariate truncation models one observes the i.i.d. samples from the triplets (T, Y, X) only if T less than or equal to 1: In this set-up, Y is said to be left truncated by T and T is right truncated by Y. We consider the estimation of the bivariate density function of (Y, X) via nonparametric kernel methods where Y is the variable of interest and X a covariate. We establish an i.i.d, representation of the bivariate distribution function estimator and show that the remainder term achieves an improved order of O(n(-1) In n), which is desirable fur density estimation purposes. Expressions are then provided for the bias and the variance of the estimators. Finally some simulation results are presented. (C) 2000 Academic Press
Open 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.
Open Access
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.
Open Access
Covariance function of a bivariate distribution function estimator for left truncated and right censored data
(Institute of Statistical Science, Academia Sinica, 1998) Gijbelsi I.; Gürler, Ü.
In left truncation and right censoring models one observes i.i.d. samples from the triplet (T, Z, δ) only if T ≤ Z, where Z = min(Y, C) and δ is one if Z = Y and zero otherwise. Here, Y is the variable of interest, T is the truncating variable and C is the censoring variable. Recently, Gürler and Gijbels (1996) proposed a nonparametric estimator for the bivariate distribution function when one of the components is subject to left truncation and right censoring. An asymptotic representation of this estimator as a mean of i.i.d. random variables with a negligible remainder term has been developed. This result establishes the convergence to a two time parameter Gaussian process. The covariance structure of the limiting process is quite complicated however, and is derived in this paper. We also consider the special case of censoring only. In this case the general expression for the variance function reduces to a simpler formula.
Open Access
Variance of the bivariate density estimator for left truncated right censored data
(Elsevier, 1999) Prewitt, K.; Gürler, Ü.
In this study the variance of the bivariate kernel density estimators for the left truncated and right censored (LTRC) observations are considered. In LTRC models, the complete observation of the variable Y is prevented by the truncating variable T and the censoring variable C. Consequently, one observes the i.i.d, samples from the triplets (T,Z,delta) only if T less than or equal to Z, Z=min(Y, C) and delta is one if Z=Y and zero otherwise. Gurler and Prewitt (1997, submitted for publication) consider the estimation of the bivariate density function via nonparametric kernel methods and establish an i.i.d. representation of their estimators. Asymptotic variance of the i.i.d, part of their representation is developed in this paper. Application of the results are also discussed for the data-driven and the least-squares cross validation bandwidth choice procedures. (C) 1999 published by Elsevier Science B.V. All rights reserved.