Grouped Data

Study guide: Exam FAM

Grouped Data record each observation only by the interval it fell in: the data are the boundaries $c_0 < c_1 < \cdots < c_k$ and the number of observations $n_j$ in each interval $(c_{j-1}, c_j]$ . Individual data, by contrast, record the exact value of each observation. Either kind is complete when no observation is truncated or censored, and maximum likelihood handles grouped data by giving each interval the probability that the model assigns to it.

$L(\theta) = \prod_{j=1}^{k}\big[F(c_j \mid \theta) - F(c_{j-1} \mid \theta)\big]^{n_j}$

$L(\theta) = \prod_{i=1}^{n} f(x_i \mid \theta)$

$f_n(x) = \frac{n_j}{n\,(c_j - c_{j-1})}, \quad c_{j-1} \le x < c_j$

$F_n(x) = F_n(c_{j-1}) + (x - c_{j-1})\,f_n(x)$

Symbols: Following Klugman, Panjer & Willmot (Loss Models, chapter 11), $n = \sum n_j$ is the sample size and $F(x \mid \theta)$ is the model's distribution function. An open last interval has $c_k = \infty$ and $F(\infty \mid \theta) = 1$ , so it contributes $S(c_{k-1} \mid \theta)^{n_k}$ . The second block is the likelihood for individual data, in which each exact value contributes its density.
What grouping costs: Grouped data say nothing about where in its interval each value fell, so the grouped-data estimate has a larger variance than the individual-data estimate from the same losses. Grouping is a form of interval censoring, and claims reported only by size band are the usual source.
Closed forms: The grouped likelihood usually has to be maximised numerically. With only two groups, the maximum sets $F(c_1 \mid \theta) = n_1/n$ for any one-parameter model. For the exponential with equally spaced boundaries, every term is a power of $p = e^{-h/\theta}$ , where $h$ is the spacing, and the maximum has a closed form.
Histogram and ogive: The histogram $f_n$ is constant on each interval, with total area 1. The ogive $F_n$ , its integral, joins the empirical distribution function at the boundaries with straight lines, which assumes that the observations are spread evenly within each interval. These are the grouped-data counterparts of the empirical distribution, and neither is defined on an open last interval.

Exponential Fit to Claims Counted by Size Band A claims system reports 100 claims only by size band: 60 at or below $1,000, 25 between $1,000 and $2,000, and 15 above $2,000. Fit an exponential distribution with mean $\theta$ by maximum likelihood.
Answer Write $p = e^{-1{,}000/\theta}$ , so that the three intervals have probabilities $1 - p$ , $p - p^2 = p(1 - p)$ and $p^2$ .
 $\begin{align*} L &= (1 - p)^{60}\,\big[p(1 - p)\big]^{25}\,\big(p^2\big)^{15} \\ &= (1 - p)^{85}\,p^{55} \end{align*}$ 
Setting the derivative of $85\ln(1 - p) + 55\ln p$ to 0 gives $p = 55/140$ .
 $\begin{align*} \hat p &= 0.39286 \\ \hat\theta &= \frac{-1{,}000}{\ln 0.39286} \\ &= \frac{1{,}000}{0.93431} \\ &= 1{,}070.31 \end{align*}$ 
The fitted model puts 60.7%, 23.9% and 15.4% of claims in the three bands, close to the observed 60%, 25% and 15%. Had the 100 claim amounts been recorded individually, the estimate would simply be their average, which these counts do not reveal.

Ogive and Histogram from Grouped Claims One hundred claims are grouped as follows: 30 in $(0, 500]$ , 35 in $(500, 1{,}000]$ , 25 in $(1{,}000, 2{,}500]$ and 10 in $(2{,}500, 5{,}000]$ . Find the histogram value $f_{100}(800)$ , the ogive value $F_{100}(800)$ , and the ogive's estimate of the probability that a claim exceeds 1,500.
Answer At the boundaries the empirical distribution function is $F_{100}(500) = 0.30$ , $F_{100}(1{,}000) = 0.65$ and $F_{100}(2{,}500) = 0.90$ .
 $\begin{align*} f_{100}(800) &= \frac{35}{100(500)} \\ &= 0.0007 \\ F_{100}(800) &= 0.30 + 300(0.0007) \\ &= 0.51 \\ F_{100}(1{,}500) &= 0.65 + \frac{500}{1{,}500}(0.25) \\ &= 0.7333 \end{align*}$ 
The estimate of $\Pr(X > 1{,}500)$ is $1 - 0.7333 = 0.2667$ . It rests on the assumption that the 25 claims in the third band are spread evenly across it, which is the assumption the ogive always makes.

How it appears on the exam

FAM asks candidates to estimate severity and frequency parameters by maximum likelihood for complete individual data, complete grouped data, and truncated or censored data. A grouped-data question gives boundaries and counts and asks for the estimate of a one-parameter model whose likelihood reduces to a closed form.

Individual data: With exact values, the same objective becomes the standard likelihood $\prod f(x_i \mid \theta)$ , which gives familiar closed forms such as $\hat\theta = \bar{x}$ for the exponential.

Mixed records: A question can combine individual values, grouped counts and censored claims in one likelihood. Each record contributes its own term, so the parts are written separately and multiplied.

Exam traps

Density instead of probability: A grouped observation contributes the probability of its interval, $F(c_j) - F(c_{j-1})$ . Using the density at the interval's midpoint treats the data as individual and gives a different estimate.

The open last interval: An interval with no upper bound contributes the survival function $S(c_{k-1})$ . Leaving it out of the likelihood discards the largest claims and pulls the estimate of a mean downward.

Dividing by the width: The histogram divides each count by $n$ times the interval's width. With unequal widths, using $n_j/n$ as the height makes wide intervals look far more likely than they are.

Ogive between boundaries: The ogive equals the empirical distribution function only at the boundaries. Between them it is an interpolation, so it is not the observed proportion of claims below $x$ .