Goodness-of-Fit Procedures for Compound Distributions with an Application to Insurance

Abstract

Goodness-of-fit procedures are introduced for testing the validity of compound models. New tests that utilize the Laplace transform as well as classical tests based on the distribution function are investigated. A major area of application of compound laws is in insurance, to model total claims resulting from specific claim frequencies and individual claim sizes. Monte Carlo simulations are used to compare the different test procedures under a variety of specifications for these two components of total claims. A detailed application to an insurance dataset is presented.

Appendix: Consistency and Limit Null Distribution

In this section, we discuss the consistency and limiting distribution of the LT-based test statistics under the null hypothesis $H_0$. We focus our attention on the test criterion $S_{n,w}$ defined in (28), but note that similar results may be obtained for the test statistic $T_{n,w}$. We begin with the consistency of the test based on $S_{n,w}$ under the following assumptions:

• (A.1) The estimator satisfies ${{\widehat{\vartheta }}}\rightarrow {\widetilde{\vartheta }}$, a.s., as $n\rightarrow \infty$, for some ${\widetilde{\vartheta }} \in \Theta$, with ${\widetilde{\vartheta }}\equiv \vartheta _0$ when the null hypothesis $H_0$ is true, with $\vartheta _0$ being the true value.

• (A.2) The LT $L^X_0(\cdot ;\vartheta )$ is continuous in $\vartheta$.

• (A.3) The weight function satisfies,

  • (i) $w(t)>0, \forall t>0$, except for a set of measure zero,

  • (ii) $\int _0^\infty w(t) \mathrm{{d}} t<\infty$.

Theorem 1

Let $L^X(t)$ denote the LT of X. Then if assumptions (A.1) to (A.3) are satisfied,

$$\begin{aligned} \frac{S_{n,w}}{n} \rightarrow \int _0^\infty \left( L^{X}(t)-L^{X}_0(t;{\widetilde{\vartheta }})\right) ^2 w(t) \mathrm{{d}} t, \end{aligned}$$

(45)

a.s., as $n\rightarrow \infty$.

Proof

: Clearly the strong consistency of the empirical Laplace transform and the continuity of $L_0(\cdot ;\vartheta )$ imply that

$$\begin{aligned} \mathrm{{SE}}_{n} \rightarrow L^X(t)-L^X_0(t;\widetilde{\vartheta }), \end{aligned}$$

a.s., as $n\rightarrow \infty$. Then since SE$^2_n(t)\le 4$, the result follows by Lebesgue’s dominated convergence theorem. $\square$

The right-hand side of (45) is positive unless $L^X(t)=L^X_0(t;\widetilde{\vartheta })$, for all $t>0$. However, by the uniqueness of the LT, the last identity holds true only under the null hypothesis $H_0$, in which case ${\widetilde{\vartheta }} \equiv \vartheta _0$, thus implying the strong consistency of the test that rejects $H_0$ for large values of $S_{n,w}$.

We continue with the limit distribution of the test statistic $S_{n,w}$ under the null hypothesis $H_0$. For simplicity, we assume that $\vartheta$ is a scalar parameter. To this end assume that

• (A.4) The estimator ${{\widehat{\vartheta }}}:={{\widehat{\vartheta }}}_n$ satisfies the Bahadur representation

  $$\begin{aligned} {{\widehat{\vartheta }}}_n-\vartheta _0=\frac{1}{n}\sum _{j=1}^n \ell (X_j;\vartheta _0)+o_P(1) \end{aligned}$$

  where $\ell (\cdot ;\cdot )$ are such that ${\mathbb {E}}(\ell (X;\vartheta _0))=0$ and ${\mathbb {E}}(\ell ^2(X;\vartheta _0))<\infty$.

• (A.5) The LT $L_0^X(t;\vartheta )$ is twice differentiable with respect to $\vartheta$ with a continuous second derivative in the neighborhood of the true value $\vartheta _0$.

• (A.6) The weight function is such that

$$\begin{aligned} \int _{0}^\infty \left( \frac{\partial L_0^X(t;\vartheta _0)}{\partial \vartheta }\right) ^2 w(t) \mathrm{{d}}t<\infty , \end{aligned}$$

and

$$\begin{aligned} \int _{0}^\infty \left( \frac{\partial ^2 L_0^X(t;\vartheta )}{\partial \vartheta ^2}\right) ^2_{\vartheta =\vartheta ^*} w(t) \mathrm{{d}}t<\infty , \end{aligned}$$

for all $\vartheta ^*$ in a neighborhood of $\vartheta _0$.

Theorem 2

Under assumptions (A.1) to (A.6) we have under $H_0$,

$$\begin{aligned} Z_n(t)=\sqrt{n} (L^X_n(t)-L_0^X(t;{{\widehat{\vartheta }}}_n)){\mathop {\longrightarrow }\limits ^{{{{\mathcal {L}}}}}}Z(t), \end{aligned}$$

as $n\rightarrow \infty$, where Z(t) is the zero-mean Gaussian process with covariance kernel $K(s,t;\vartheta _0)={\mathbb {E}}(Y(t;\vartheta _0)Y(s;\vartheta _0))$ with

$$\begin{aligned} Y(t;\vartheta )=e^{-tX}-L_0^X(t;\vartheta )-\frac{\partial L_0^X(t;\vartheta )}{\partial \vartheta }\ell (X;\vartheta ). \end{aligned}$$

The covariance kernel is specified by

$$\begin{aligned} K(s,t;\vartheta )= & {} L_0^X(t+s,\vartheta )-L_0^X(t;\vartheta )L_0^X(s; \vartheta ) \\- & {} \frac{\partial L_0^X(s;\vartheta )}{\partial \vartheta } {\mathbb {E}}\left( e^{-tX} \ell (X;\vartheta ) \right) -\frac{\partial L_0^X(t;\vartheta )}{\partial \vartheta } {\mathbb {E}}\left( e^{-sX},\ell (X;\vartheta ) \right) \\+ & {} \frac{\partial L_0^X(s;\vartheta )}{\partial \vartheta } \frac{\partial L_0^X(t;\vartheta )}{\partial \vartheta } {\mathbb {E}}(\ell ^2(X;\vartheta )). \end{aligned}$$

Proof

: Along the proof we will write $Z^{(1)}_n\approx Z^{(2)}_n$ if the two random processes $(Z^{(k)}_n(t), k=1,2)$, satisfy $Z^{(1)}_n(t)-Z^{(2)}_n(t)=\varepsilon _n(t)$, and the remainder $\varepsilon _n(t)$ is such that it has no effect on the limit null distribution of the test statistic $S_{n,w}$.

With this understanding using assumption (A.5) and the second part of (A.6), a two-term Taylor expansion yields

$$\begin{aligned} Z_n\approx Z^*_n, \end{aligned}$$

where

$$\begin{aligned} Z^*_n(t)=\sqrt{n} \left( L^X_n(t)-L_0^X(t;\vartheta _0)\right) -\sqrt{n}\left( {{\widehat{\vartheta }}}_n-\vartheta _0\right) \frac{\partial L_0^X(t;\vartheta _0)}{\partial \vartheta }. \end{aligned}$$

(46)

In turn, using assumption (A.4) and the first part of (A.6) in (46) leads to

$$\begin{aligned} Z^*_n\approx Z^{**}_n, \end{aligned}$$

where

$$\begin{aligned} Z^{**}_n(t)=\sqrt{n} \left( L^X_n(t)-L_0^X(t;\vartheta _0)\right) -\frac{\partial L_0^X(t;\vartheta _0)}{\partial \vartheta } \frac{1}{\sqrt{n}}\sum _{j=1}^n \ell (X_j;\vartheta _0). \end{aligned}$$

(47)

The result now follows by applying the Central Limit Theorem in Hilbert spaces, (see e.g. van der Vaart and Wellner [38], p. 50) on the process $Z^{**}_n(t)$ given in (47). $\square$

Now the limit distribution of the test statistic follows from Theorem 2 and the Continuous Mapping theorem. Specifically we have

$$\begin{aligned} S_{n,w}=\int _0^\infty Z^2_n(t) w(t) \mathrm{{d}}t {\mathop {\longrightarrow }\limits ^{{{{\mathcal {L}}}}}}\int _0^\infty Z^2(t) w(t) \mathrm{{d}}t:=Z_w \end{aligned}$$

where Z(t) is the process defined in Theorem 2. The distribution of $Z_w$ is the same as that of $\sum _{j=1}^\infty \lambda _j N^2_j$, where $\lambda _1,\lambda _2, . . .$, are the eigenvalues corresponding to the integral operator

$$\begin{aligned} Ag(s)=\int _0^\infty K(s,t) g(t) w(t) \mathrm{{d}}t, \end{aligned}$$

i.e. the solutions of the equation $Ag(s)=\lambda g(s)$, and where $N_j, \ j\ge 1$, are iid standard normal variates.

Remark 71

The assumptions (A.1)–(A.3) made in order to prove consistency, as well as those pertaining to the limit null distribution, (A.4)–(A.6), are standard in the context of testing goodness-of-fit based on the empirical LT; see for instance Henze [18], Henze and Klar [20], and Henze and Meintanis [21].