Statistics with R

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# Statistics with R
## Statistical Modelling with R for Actuarial Students

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***CS2B – Risk Modelling and Survival Analysis ***

* The emphasis is placed on being able to apply statistical methods to actuarial problems using real data sets and the open-source software environment R.

* Time Series. Probability Distributions, Survival Analysis

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Suppose the transition probabilities are a function of the age of a person. The transition
probability of a person aged x moving:

* from Healthy to Sick state is 0.007x
* from Healthy to Death state is 0.001x
* from Sick to Death state is 0.002(100-x)
* from Sick to Healthy is 0.006(100-x).

#### Exercises

Assuming 100 is the maximum age. Calculate the probability of:

1.  Healthy person aged 30 will be in Sick state after 4 years.

2.  Sick person aged 25 will be in Death state after 7 years.

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The transition matrix can be characterized as follows:
$$
`\begin{pmatrix}
1-H2S(x)-H2D(x) & H2S(x) & H2D(x) \\
S2H(x) & 1-S2H(x)-S2D(x) & S2D(x) \\
0 & 0 & 1 \\
\end{pmatrix}`
$$

N.B. This matrix varies for differing values of `$x$`, and hence the process is not a Markov Process.

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### Functions

Transition Matrix is a function of age (x)

```r
H2S<-function(x){ 0.007*x}
H2D<-function(x){ 0.001*x}
S2H<-function(x){ 0.006*(100-x)}
S2D<-function(x){ 0.002*(100-x)}
```

```r
transmat<-function(x){
  M<-matrix(0,nrow=3,ncol=3)
  M[1,1] <- 1 - H2S(x) - H2D(x)
  M[1,2] <- H2S(x)
  M[1,3] <- H2D(x)
  M[2,1] <- S2H(x)
  M[2,2] <- 1 - S2H(x) - S2D(x)
  M[2,3] <- S2D(x) 
  M[3,1] <- 0
  M[3,2] <- 0
  M[3,3] <- 1
  M
}
```

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Transition Matrix at 30 Years

```r
transmat(30)
```

```
##      [,1] [,2] [,3]
## [1,] 0.76 0.21 0.03
## [2,] 0.42 0.44 0.14
## [3,] 0.00 0.00 1.00
```

Transition Matrix at 25 Years

```r
transmat(25)
```

```
##      [,1]  [,2]  [,3]
## [1,] 0.80 0.175 0.025
## [2,] 0.45 0.400 0.150
## [3,] 0.00 0.000 1.000
```

---

#### Exercise 1

Healthy person aged 30 will be in Sick state after 4 years.

```r
n <- 30
B <- c(1,0,0)
```

```r
i<- 1:4
n +i-1
```

```
## [1] 30 31 32 33
```

```r
for (i in 1:4){
  B=B%*%transmat(n+i-1)
  }
```

---

```r
B
```

```
##       [,1]  [,2]  [,3]
## [1,] 0.545 0.261 0.194
```
Hence the probability of Healthy person aged 30 will be in Sick state after 4 years is 0.2608.

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#### Exercise 2.

Sick person aged 25 will be in Death state after 7 years.

```r
n <- 25
C <- c(0,1,0)
for (i in 1:7){
  C=C%*%transmat(n+i-1)
  }
```

```r
C
```

```
##       [,1]  [,2]  [,3]
## [1,] 0.395 0.172 0.433
```

Hence the probability of sick person aged 25 will be in Death state after 7 years is 0.4333.

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