relife.nonparametric.NelsonAalen

relife.nonparametric.NelsonAalen

class relife.nonparametric.NelsonAalen[source]

Bases: object

Nelson-Aalen Estimator.

Compute the non-parametric Nelson-Aalen estimator of the cumulative hazard function from lifetime data.

Notes

For a given time instant \(t\) and \(n\) total observations, this estimator is defined as:

\[\hat{H}(t) = \sum_{i: t_i \leq t} \frac{d_i}{n_i}\]

where \(d_i\) is the number of failures until \(t_i\) and \(n_i\) is the number of assets at risk just prior to \(t_i\).

The variance estimation is obtained by:

\[\widehat{Var}[\hat{H}(t)] = \sum_{i: t_i \leq t} \frac{d_i}{n_i^2}\]

Note that the alternative survivor function estimate:

\[\tilde{S}(t) = \exp{(-\hat{H}(t))}\]

is sometimes suggested for the continuous-time case.

References

1

Lawless, J. F. (2011). Statistical models and methods for lifetime data. John Wiley & Sons.

Methods

fit

Fit the Nelson-Aalen estimator to lifetime data.

plot

Plot the Nelson-Aalen estimator of the cumulative hazard function.
fit(time: numpy.ndarray, event: Optional[numpy.ndarray] = None, entry: Optional[numpy.ndarray] = None) → relife.nonparametric.NelsonAalen[source]

Fit the Nelson-Aalen estimator to lifetime data.

Parameters

  • time (1D array) – Array of time-to-event or durations.

  • event (1D array, optional) –

    Array of event types coded as follows:

    • 0 if observation ends before the event has occurred (right censoring)

    • 1 if the event has occured

    • 2 if observation starts after the event has occurred (left censoring)

    by default the event has occured for each asset.

  • entry (1D array, optional) – Array of delayed entry times (left truncation), by default None.

Returns

The fitted Nelson-Aalen estimator as the current object.

Return type

NelsonAalen

plot(alpha_ci: float = 0.05, **kwargs: numpy.ndarray) → None[source]

Plot the Nelson-Aalen estimator of the cumulative hazard function.

Parameters

  • alpha_ci (float, optional) – \(\alpha\)-value to define the \(100(1-\alpha)\%\) confidence interval, by default 0.05 corresponding to the 95% confidence interval. If set to None or if the model has not been fitted, no confidence interval is plotted.

  • **kwargs – Extra arguments to specify the plot properties (see matplotlib.pyplot.plot documentation).

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