By Jain A.K., Strawderman R.L.

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**Extra info for Flexible hazard regression modeling for medical cost data**

**Example text**

F has a unique singularity at 0 with f (0− ) = 1, and f (0+ ) = −1, 2. f : [−1, 1] \ {0} → [−1, 1], √ 3. f is C 1 on [−1, 1] \ {0}, and f ′ (x) > 2 for x = 0, These conditions are enough to prove that almost all points in [−1, 1] have dense orbits under f . It is also clear that f exhibits exponential sensitivity. By pulling the information back to the original return map, it is possible to prove that the attracting set of the model flow is a generalized non-trivial hyperbolic attractor (also known as a singular hyperbolic attractor).

Even when the number of boxes is small (4 or 16) the improvement is significant. Also note that it is possible to obtain an arbitrarily good approximation but the number of boxes to improve the approximation grows very fast. Another approach to reduce the wrapping eﬀect is based on the mean value form {f (x) : x ∈ x} ⊂ f (x) + f ′ (x)(x − x), (2) where x ∈ x. Instead of computing f (x) in interval arithmetic we choose a single point x ∈ x, find its image under f , compute the Jacobian matrix of f over x, and finally use the formula (2) to find an enclosure.

In this section we assume that f : Rm → Rm is a diﬀerentiable map. In order to study the existence of zeros of f within the interval vector x one evaluates interval Newton operator x), N(x) = x ˆ − f ′ (x)−1 f (ˆ (4) where f ′ (x) is an interval matrix containing the Jacobian matrices f ′ (x) for x ∈ x and xˆ is an arbitrary point from x. Usually, one chooses x ˆ to be the center of x. Computational Methods for Rigorous Analysis of Chaotic Systems 33 The following theorem [21, 20] can be used for proving the existence and uniqueness of zeros: Theorem 1.

### Flexible hazard regression modeling for medical cost data by Jain A.K., Strawderman R.L.

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