The main focus of this chapter is the estimation of the distribution function and probability (density) function of duration and loss variables. The methods used depend on whether the data are for individual or grouped observations, and whether the observations are complete or incomplete.

For complete individual observations, the relative frequency distribution of the sample observations defines a discrete distribution called the empirical distribution. Moments and df of the true distribution can be estimated using the empirical distribution. Smoothing refinements can be applied to the empirical df to improve its performance. We also discuss kernel-based estimation methods for the estimation of the df and pdf.

When the sample observations are incomplete, with left truncation and/or right censoring, the Kaplan–Meier (product-limit) estimator and the Nelson–Aalen estimator can be used to estimate the survival function. These estimators compute the conditional survival probabilities using observations arranged in increasing order. They make use of the data set-up discussed in the last chapter, in particular the risk set at each observed data point. We also discuss the estimation of their variance, the Greenwood formula, and interval estimation.

For grouped data, smoothing techniques are used to estimate the moments, the quantiles, and the df. The Kaplan–Meier and Nelson–Aalen estimators can also be applied to grouped incomplete data.

Learning objectives

1. Empirical distribution

2. Moments and df of the empirical distribution

3. Kernel estimates of df and pdf

4. Kaplan–Meier (product-limit) estimator and Nelson–Aalen estimator

5. Greenwood formula

6. Estimation based on grouped observations