COMPRENSIÓN DEL SUICIDIO DE RAFAEL RANGEL DES- DES-DE LA PERSPECTIVA DES-DEL PSICOANÁLISIS

Response distributions

This chapter introduces the common statistical distributions used in insurance data analysis and generalized linear modeling. This sets the stage for the development and understanding of the generalized linear model. In statisti-cal analysis, an outcome such as the size of a claim is regarded as at least partially determined by chance. This setup is formalized by the introduction of random variables.

2.1 Discrete and continuous random variables

A random variable y is a real number determined by chance. This chance mechanism may arise naturally, from random sampling or may be a useful hypothetical construct. The set of values that y can take on is called the sample space, denoted Ω.

(i) Discrete random variables. In this case Ω is a finite or countable set of real numbers, for example the non-negative integers{0, 1, 2, . . .}.

Associated with the random variable is a probability function f (y) indicating for each y in Ω (written y∈ Ω) the probability the random variable takes on the value y. The function f (y) is only non-negative on values in Ω: f (y) > 0 if y∈ Ω and 0 otherwise. Further



y∈Ω

f (y) = 1 .

The expected value and variance of a discrete random variable y are defined as:

μ = E(y)≡

y∈Ω

y f (y) , Var(y)≡ E{(y−μ)²} =

y∈Ω

(y−μ)²f (y) .

If y is the number of events, usually within a fixed period of time and/or in a fixed area, then it is a count random variable. Counts are by their

20

2.2 Bernoulli 21 nature non-negative integers. Examples are given in Section 1.2. Dis-tributions widely used to model counts include the binomial, Poisson and the negative binomial. These are discussed below.

(ii) Continuous random variables. In this case Ω is an interval of the real line. Probabilities are specified using a probability density function f (y) with f (y) ≥ 0 for y ∈ Ω and 0 otherwise. Areas under f(y)

Hence the probability of y taking on any given value is zero and

 _∞

−∞f (y)dy = 1 .

Analogous to the discrete case, the mean and variance of a continuous random variable y are defined as

μ = E(y)≡

 _∞

−∞yf (y)dy , Var(y)≡

 _∞

−∞(y− μ)²f (y)dy . Probability functions are a mathematical tool to describe uncertain situations or where outcomes are subject to chance. To simplify discussion the nota-tion f (y) is used for the probability funcnota-tion, for both discrete and continuous random variables. No notational distinction is made between probability den-sity functions, which belong to continuous random variables, and probability (mass) functions, which belong to discrete random variables. It is understood that when integrating f (y) for a discrete random variable y, the appropriate operation is summation.

2.2 Bernoulli

The Bernoulli distribution admits only two possible outcomes, usually coded as 0 or 1 and hence Ω ={0, 1}. The event y = 1 is often called a “success,”

the other, y = 0, a “failure.” Further f (1) = π and f (0) = 1− π where 0 < π < 1. Insurance examples are a claim or no claim on a policy in a given year, or a person dying or surviving over a given year. The event of interest (claim, death) is generally coded as 1, and π is the probability of the event occurring.

Elementary calculations show that the mean and variance of a Bernoulli random variable are π and π(1− π) respectively. The variance is a maximum when π = 0.5. The probability function is

f (y) = π^y(1− π)^1−y , y = 0, 1 . (2.1)

22 Response distributions

Vehicle insurance. This data set consists of 67 856 policies enacted in a given year, of which 4624 had a claim. If the probability π of the occurrence of a claim for each policy is assumed constant, then a rough estimate of π is 4624/67 856 = 0.068.

In practice each policy is not exposed for the full year. Some policies come into force partly into the year while others are canceled before the year’s end.

Define 0 < t ≤ 1 as the amount of exposure during the year, with t = 1 indicating full exposure. For these data the average amount of exposure is 0.469. This suggests a modified model for the claim probability

f (y) = (tπ)^y(1− tπ)^1−y, y = 0, 1 . (2.2) A reasonable estimate of π is the average number of claims weighted by expo-sure. This leads to an estimate of π of 0.145, considerably higher than the previous estimate. Exposure adjustments are often appropriate in insurance studies and a more detailed treatment is given in Section 7.4.

2.3 Binomial

The binomial distribution generalizes the Bernoulli distribution and is used to model counts such as the total number of policies making a claim. If there are n independent Bernoulli random variables, each with success probability π, then the total number of successes has the binomial distribution denoted y∼ B(n, π). The probability function is given by

f (y) =

n y



π^y(1− π)^n−y, y = 0, 1, . . . , n . (2.3) The distribution depends on one unknown parameter π since n is known. Some typical shapes are displayed in Figure 2.1. Elementary calculations show the mean and variance are E(y) = nπ and Var(y) = nπ(1− π), respectively. A Bernoulli random variable is a special case of the binomial random variable with n = 1, and hence y ∼ B(1, π). The binomial is practically and his-torically important and leads directly to the Poisson distribution as discussed below.

A binomial random variable is often transformed into a proportion by divid-ing by n. The resultdivid-ing random variable y/n is called the binomial proportion and has the probability function (2.3) shifted onto 0, 1/n, 2/n, . . . , 1.

Grouped binary data. Binary data is often grouped, in which case the num-ber of occurrences of the event in each subgroup is analyzed. For example, policies may be grouped according to geographical area and the number of policies in each area on which there is a claim, recorded. If the Bernoulli trials within each area are independent with constant probability π, then the

2.4 Poisson 23

number of claims arising from each area is binomial of the form (2.3), with varying numbers of policies n and varying probabilities π. Of interest might be the relationship between π and area characteristics such as socioeconomic indicators.

2.4 Poisson

Suppose with the binomial distribution n becomes large while π becomes small but in such a way that the mean μ = nπ stays constant. In the limit this yields the probability function

f (y) = e^−μμ^y

y! , y = 0, 1, 2, . . . , (2.4) which is the Poisson distribution, denoted y ∼ P(μ). The probability function depends on the single parameter μ and has E(y) = μ and Var(y) = μ. Thus the variance equals the mean. Examples of (2.4) for different values of μ are shown in Figure 2.2.

Number of children. The frequency distribution of the number of children of 141 pregnant women, described on page 15, is given in the “Observed”

column in Table 2.1. The mean number of children in the sample is 0.603, which is taken as the estimate of μ. The observed frequencies are compared with expected frequencies from the P(0.603) distribution:

141× f(y) = 141 ×e^−0.6030.603^y

y! , y = 0, 1, 2, . . . .

24 Response distributions

Table 2.1. Observed and expected number of children per mother Children Obser v ed P(0.603) Expected

0 89 0.547 77.2

The fitted Poisson probabilities and expected frequencies are given in Table 2.1, and the observed and expected frequencies graphed in Figure 2.3. There is good agreement of the data with the Poisson model. In the sample there are more women with no children, and fewer with one child, compared to the Poisson model predictions.