# Information Matrix

## Filter Information matrix

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### Smoothing

The late Iain Currie was a long-time advocate of smoothing certain parameters in mortality models. In an earlier blog he showed how smoothing parameters in the Lee-Carter model could improve the quality of the forecast. As Iain himself wrote, "this idea is not new" and traced its origins to Delwarde, Denuit & Eilers (2007).

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: P-splines

### Fathoming the changes to the Lee-Carter model

Ancient Greek philosophers had a paradox called "The Ship of Theseus"; if pieces of a ship are replaced over time as they wear out until every one of the original components is gone, is it still the same ship? At this point you could be forgiven for thinking (a) that this couldn't possibly be further removed from mortality modelling, and (b) that I had consumed something a lot more potent than tea at breakfast.

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: Lee-Carter, Filter information matrix by tag: P-splines, Filter information matrix by tag: ARIMA

### Back to the future with Whittaker smoothing

**Written by:**Iain Currie

**Tags:**Filter information matrix by tag: Whittaker smoothing, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function

### Forecasting with penalty functions - Part III

This is the last of my three blogs on forecasting with penalties. I discussed the 1-d case in the first blog and the 2-d case in the second. Here we discuss some of the properties of 2-d forecasting. Some readers may find some of my remarks surprising, even paradoxical.

**Written by:**Iain Currie

**Tags:**Filter information matrix by tag: forecasting, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function, Filter information matrix by tag: mortality crossover

### Forecasting with penalty functions - Part II

Our first blog in this series of three looked at forecasting log mortality with penalties in one dimension, i.e. forecasting with data for a single age. We now look at the same problem, but in two dimensions. Figure 1 shows our data. We see an irregular surface sitting on top of the age-year plane. Just as in the 1-d case, we see an underlying smooth surface, and it is this surface that we wish both to estimate and to forecast.

**Written by:**Iain Currie

**Tags:**Filter information matrix by tag: forecasting, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function, Filter information matrix by tag: mortality crossover

### Forecasting with penalty functions - Part I

There is much to say on the topic of penalty forecasting, so this is the first of three blogs. In this blog we will describe penalty forecasting in one dimension; this will establish the basic ideas. In the second blog we will discuss the case of most interest to actuaries: two-dimensional forecasting. In the final blog we will discuss some of the properties of penalty forecasting in two dimensions.

**Written by:**Iain Currie

**Tags:**Filter information matrix by tag: forecasting, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function

### Effective dimension

Actuaries often need to smooth mortality rates. Gompertz (1825) smoothed mortality rates by age and his famous law was a landmark in this area. Figure 1 shows the Gompertz model fitted to CMI assured lives data for ages 20–90 in the year 2002. The Gompertz Law usually breaks down below about age 40 and a more general smooth curve would be appropriate. However, a more general smooth curve would obviously require more parameters than the two for the simple Gompertz model.

**Written by:**Iain Currie

**Tags:**Filter information matrix by tag: effective dimension, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines

### Stabilising projections

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: P-splines, Filter information matrix by tag: time series, Filter information matrix by tag: mortality projections, Filter information matrix by tag: CMI

### Features of the survival curve

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: Lee-Carter, Filter information matrix by tag: P-splines, Filter information matrix by tag: ARIMA