Agnieszka Rossa

A SIMPLE IMPROVEMENT OF THE KAPLAN-MEIER ESTIMATOR

ABSTRACT Though widely used, the celebrated Kaplan-Meier estimator suffers from a disadvantage: it may happen, and in small and moderate samples it often does, that even if the difference between two consecutive times t 1 and t 2 ( ) is considerably large, for the values of the Kaplan-Meier estima-tor KM(t 1) and KM(t 2) we may have KM . Although that is a general problem in estimating a smooth and monotone distribution function from small or moderate samples, in the context of estimating survival probabilities the disadvantage is particularly annoying. In the paper we discuss a local smoothing of the Kaplan-Meier estimator based on an approximation by the Weibull distribution function. It appears that Mean Square Error and Mean Absolute Deviation of the smoothed estimator is significantly smaller. It follows also from the Pitman Closeness Criterion that the new version of the estimator can be recommended.

Estimation of Survival Distributions Under Right-Censoring When Sample Size Is Random

Abstract The paper deals with two classes of unbiased nonparametric estimators of survival and cumulative hazard functions in a population subject to right-censoring. Both classes of estimators are based on a sequential sampling scheme, and are similar to the well-known Kaplan–Meier and Nelson–Aalen estimators.

Fuzzy Mortality Model Based on Banach Algebra

The problem of the determining the best mortality models is one of the basic fields in the forecasting strategy of insurance companies. For a given age group x at year t the mortality rate, m_x(t) can be expressed in the form of so-called Lee-Carter stochastic mortality model (LC). However, the LC-model assumes the homoscedasticity of error terms which is not adequate to the real life. Koissi and Shapiro have formulated the fuzzy version of the LC-model (FLC), where the model coefficients are assumed to be fuzzy numbers with the symmetric triangular membership function (STMF). To make the inference based on improved FLC more precise and elegant, we apply the Banach algebra of fuzzy numbers, i.e., OFN-algebra introduced by Kosinski et al. [8].

ON THE ESTIMATION OF SURVIVAL FUNCTION UNDER RANDOM CENSORSHIP

ABSTRACT The paper deals with an improvement of the well-known Kaplan–Meier estimator of survival function when the censoring mechanism is random and independent of the failure times. Small sample size properties of the new estimator, as well as the original Kaplan–Meier estimator are inspected by means of Monte Carlo simulations. It follows from the simulations that the proposed estimator prevails with respect to some basic statistical characteristics.

The Fuzzy Mortality Model based on Quaternion Theory

The mortality models are of fundamental importance in many areas, such as the pension plans, the care of the elderly, the provision of health service, etc. In the paper, we propose a new class of mortality models based on a fuzzy version of the well-known Lee–Carter model (1992). Theoretical backgrounds are based on the algebraic approach to fuzzy numbers (Ishikawa, 1997, Kosiński, Prokopowicz, Ślęzak, 2003, Rossa, Socha, Szymański, 2015, Szymański, Rossa, 2014). The essential idea in our approach focuses on representing a membership function of a fuzzy number as an element of quaternion algebra. If the membership function μ(z) of a fuzzy number is strictly monotonic on two disjoint intervals, then it can be decomposed into strictly decreasing and strictly increasing functions Φ(z), Ψ(z), and the inverse functions f(u)=Φ−1(u) and g(u)=Ψ−1(u), u ∈ [0, 1] can be found. Thus, the membership function μ(z) can be represented by means of a complex-valued function f(u) + ig(u), where i is an imaginary unit. Then the pair (f, g) is a quaternion. The quaternion-valued, square integrable functions form a tool for constructing the new class of mortality models.

Fuzzy Modeling of Survival Function from Interval or Censored Observations

In medicine, biology, or actuarial theory the so called survival function is often used, i.e., a probability P(X ＞ x) that the time X from the beginning of an initial event to a final one will not exceed x. The observed data are mostly right-censored what causes that the observed variable is T = min(X, Z) where Z is the so called censoring variable. The celebrated and widely used nonparametric estimator of survival function is the Kaplan-Meier estimator which suffers from a disadvantage that it is stepwise and therefore it may happen that in considerably distinct points the values of the estimator may be equal.Rossa and Zielinski proposed a local smoothing of the Kaplan-Meier estimator based on an approximation by means of the piecewise Weibull survival function. They have shown that Mean Square Error and Mean Absolute Deviation of the smoothed estimator have been significantly smaller. Moreover, the Weibull approximation method has appeared to be a quite simple algorithm based on logarithmic transformations of the data and by applying the standard estimating procedure of the simple regression model y=ax+b. However, the censoring variables introduce uncertainty, which can be treated as the source of fuzziness. Thus, the estimation problem can be transferred into the fuzzy analysis. Another estimator leading to fuzzy model is the semi-parametric model proposed by Rossa. Parametric part of the estimator contains formulae based on estimates of the Weibull parameters, whereas the censored observations yield uncertainty.The proposed approaches have a general character. As it has been pointed out in Rossa and Zielinski, the Kaplan-Meier survival function can be approximated with a prescribed level of accuracy by a piecewise Weibull survival function. Using double logarithms of that Weibull pieces we get the piecewise linear intervals. All the intervals can have the same slop parameters or can have changing points separating different slops. Thus, the basic tool in the approach is the fuzzy linear regression. For simplicity, it will be assumed the fuzzy regression coefficients have symmetric triangular membership functions.

The Nelson–Aalen and Kaplan–Meier Estimators Under a Sequential Sampling Scheme

This article deals with some statistical properties of the modified Kaplan–Meier and Nelson–Aalen estimators defined for a right-censored sample under a certain sequential sampling scheme. Both types of estimators were proposed by Rossa in the article “Estimation of Survival Distributions Under Right-Censoring When Sample Size Is Random” (Sequent. Anal. 2008, 27(2), 174–184).