Estimating a Distribution Function with Truncated Data

2013

Astrophys Space Sci

The classical 1/V a and P C methods of constructing binned luminosity functions (LFs) are revisited and compared by graphical analysis. Using both theoretical analysis and illustration with an example, we show why the two methods give different results for the bins which are crossed by the flux limit curves L = L lim (z). Based on a combined sample simulated by a Monte Carlo method, the estimate φ of two methods are compared with the input model LFs. The two methods give identical and ideal estimate for the high luminosity points of each redshift interval. However, for the low luminosity bins of all the redshift intervals both methods give smaller estimate than the input model. We conclude that once the LF is evolving with redshift, the classical binned methods will unlikely give an ideal estimate over the total luminosity range. Page & Carrera (2000) noticed that for objects close to the flux limit φ 1/Va nearly always to be too small. We believe this is due to the arbitrary choosing of redshift and luminosity intervals. Because φ 1/Va is more sensitive to how the binning are chosen than φ P C . We suggest a new binning method, which can improve the LFs produced by the 1/V a method significantly, and also improve the LFs produced by the P C methods. Our simulations show that after adopting this new binning, both the 1/V a and P C methods have comparable results.

Section: Non-parametric Methodsmentioning

confidence: 99%

A graphical analysis of the systematic error of classical binned methods in constructing luminosity functions

Yuan

2013

Astrophys Space Sci

“…We adopt Woodroofe's (1985) notation throughout the presentation. The cumulative hazard function of F (or X) is: J0 x (2.1)…”

Section: Kernel Hazard Estimatesmentioning

confidence: 99%

Nonparametric estimation of hazard functions and their derivatives under truncation model

Gürler

1993

Ann Inst Stat Math

A b s t r a c t . Nonparametric kernel estimators for hazard functions and their derivatives are considered under the random left truncation model. The estimator is of the form of sum of identically distributed but dependent random variables. Exact and asymptotic expressions for the biases and variances of the estimators are derived. Mean square consistency and local asymptotic normality of the estimators are established. Adaptive local bandwidths are obtained by estimating the optimal bandwidths consistently.

“…Most existing research works (Woodroofe 1985;Keiding and Gill 1990;Wang 1991; van der Laan 1996b) focused on bivariate survival analysis under either censoring or truncation. Some other existing methods dealt with bivariate survival function estimation under the scenario where one component is censored and truncated, but the other one is fully observed (Gürler 1997;Gijbels and Gürler 1998).…”

Section: Introductionmentioning

confidence: 99%

A class of nonparametric bivariate survival function estimators for randomly censored and truncated data

Dai

Restaino

Journal of Nonparametric Statistics

2016

This paper proposes a class of nonparametric estimators for the bivariate survival function estimation under both random truncation and random censoring. In practice, the pair of random variables under consideration may have certain parametric relationship. The proposed class of nonparametric estimators uses such parametric information via a data transformation approach and thus provides more accurate estimates than existing methods without using such information. The large sample properties of the new class of estimators and a general guidance of how to find a good data transformation are given. The proposed method is also justified via a simulation study and an application on an economic data set.