Moment-Based Approximation with Mixed Erlang Distributions by Hélène Cossette, David Landriault, Etienne Marceau, and Khouzeima Moutanabbir ABSTRACT Moment-based approximations have been extensively analyzed over the years (see, e.g., Osogami and Harchol-Balter 2006 and references therein). A number of specific phase-type (and non phase-type) distributions have been considered to tackle the moment-matching problem (see, for instance, Johnson and Taaffe 1989). Motivated by the development of more flexible moment- based approximation methods, we develop and examine the use of finite mixture of Erlangs with a common rate parameter for the moment-matching problem. This is primarily motivated by Tijms (1994) who shows that this class of distributions can approximate any continuous positive distribution to an arbitrary level of accu- racy, as well as the tractability of this class of distributions for various problems of interest in quantitative risk management. We consider separately situations where the rate parameter is either known or unknown. For the former case, a direct connection with a discrete moment-matching problem is established. A parallel to the s-convex stochastic order (e.g., Denuit et al. 1998) is also drawn. Numerical examples are considered throughout. KEYWORDS Risk theory, mixed Erlang distributions, moment-matching, distribution fitting, phase-type approximation 166 CASUALTY ACTUARIAL SOCIETY VOLUME 10/ISSUE 1 Moment-Based Approximation with Mixed Erlang Distributions 1. Introduction moment-based approximation of Whitt (1982) when CV > 1 using a Coxian distribution. Alternatively, Mixed Erlang distributions are known to yield ana- Johnson and Taaffe (1989) considered a mixture of lytic solutions to many risk management problems of Erlangs with a common shape (order) parameter as interest. This is primarily due to the tractable features their moment-based approximation. of this distributional class. Among others, the class Most predominantly, there exists a substantial body of mixed Erlang distributions is closed under various of literature on the three-moment approximation operations such as convolutions and Esscher transfor- within the phase-type class of distributions (e.g., Telek mations (e.g., Willmot and Woo 2007 and Willmot and and Heindl 2002, Bobbio et al. 2005, and references Lin 2011). As such, risk aggregation and ruin problems therein). Matching the first three moments is often can more easily be tackled under mixed Erlang assump- viewed as effective to provide a reasonable approxi- tions (e.g., Cheung and Woo 2016, Cossette et al. 2012, mation to the underlying system (e.g., Osogami and and Landriault and Willmot 2009). Also, Tijms (1994) Harchol-Balter 2006 and references therein). How- showed that the class of mixed Erlang distributions is ever, as illustrated in this paper and many others, three dense in the set of all continuous and positive dis- moments does not always suffice, triggering the devel- tributions. Therefore, we consider a moment-based opment of more flexible moment-based approxima- approximation method which capitalizes on the afore- tions. Among others, we mention the work of Johnson mentioned properties of the mixed Erlang distribution. and Taaffe (1989) on mixed Erlang distributions of More precisely, we propose to approximate a distri- common order. Also, Dufresne (2007) proposes two bution with known moments by a moment-matching approximation techniques based on Jacobi polynomial mixed Erlang distribution. Moment-based approxi- expansions and the logbeta distribution to fit combina- mations have been extensively developed in various tions of exponential distributions. This paper is com- research areas, including performance evaluation, plementary to the aforementioned ones by considering queueing theory, and risk theory, to name a few. the family of finite mixture of Erlangs with common Osogami and Harchol-Balter (2006) identify the rate parameter to approximate a distribution on R+, as following four criteria to evaluate moment-matching theoretically justified in the continuous case by Tijms algorithms: (1) the number of moments matched; (1994, Theorem 3.9.1). The reader is also referred to (2) the computational efficiency of the algorithm; Lee and Lin (2010) where fitting of the same class of (3) the generality of the solution; and (4) the mini- distributions is considered using the EM algorithm mality of the number of parameters (phases). It also (which relies on the knowledge of the approximated seems desirable for the approximation to be in itself distribution rather than only its moments). a distribution. This is not mentioned in Osogami It is worth pointing out that other non-phase type and Harchol-Balter (2006) for the obvious reason approximation methods have been widely used in that they consider phase-type distributions as their actuarial science. A good survey paper on this topic moment-based approximation class. There exists an is Chaubey et al. (1998). One of these approximation extensive literature on the approximation of distribu- classes are refinements to the normal approxima- tions by a specific subset of phase-type distributions tion such as the normal power and the Cornish Fisher using moment-based techniques. For instance, Whitt approximations (e.g., Ramsay 1991, Daykin et al. (1982) proposed a mixture of two exponential dis- 1994, and Lee and Lin 1992). These approximations tributions or a generalized Erlang distribution as are based on the first few moments. However, the a moment-based approximation when either the coef- resulting approximation is often not a proper distribu- ficient of variation (CV) is greater than or less than 1, tion. Other moment-based distributional approxima- respectively. Also, both Altiok (1985) and Vanden tions are the translated gamma distribution (e.g., Seal Bosch et al. (2000) proposed an alternative to the 1977), translated inverse Gaussian distribution (e.g., VOLUME 10/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 167 Variance Advancing the Science of Risk Chaubey et al. 1998) and the generalized Pareto dis- As stated in Courtois and Denuit (2007), there tribution (e.g., Venter 1983). It should be noted that exists a non-negative random variable (rv) with dis- all these approximation methods are designed to fit a tribution function (df) F and first m moments lm if specific number of moments, and thus lack the flex- and only if the following two conditions are satisfied: ibility to match an arbitrary number of moments. • det Pk > 0, for k = 1, . , (m - 1)/2; The rest of the paper is constructed as follows. In • det Qk > 0, for k = 1, . , m/2; Section 2, a brief review on admissible moments, mixed Erlang distributions and the approximation where x holds for the integer part of x. In what fol- method of Johnson and Taaffe (1989) is provided. lows, we silently assume the moment set lm is from + Section 3 is devoted to our class of finite mixture of a probability distribution on R . Erlangs with common rate parameter. Theoretical and 2.2. Mixed Erlang distribution practical considerations related to the approximation method are drawn. Various examples are considered We now review some known properties of mixed to examine the quality of the resulting approximation. Erlang distributions with common rate parameter. A In Section 4, we consider applications of our moment- more elaborate review of this class of distributions based approximations of Section 3 when the underly- can be found in Willmot and Woo (2007), Lee and ing distribution is of mixed Erlang form with known Lin (2010), and Willmot and Lin (2011). rate parameter. A parallel is drawn with a discrete Let W be a mixed Erlang rv with common rate moment-matching problem and certain stochastic parameter b > 0 and df orderings, notably the s-convex stochastic order (e.g., FxW ()=ζ∑ k Hx();,k β ,(1) Denuit et al. 1998). An application of Cossette et al. kA∈ l (2002) will be examined in more detail. l where Al = {1, 2, . , l}, and {zk}k=1 is the probability mass function (pmf) of a discrete rv K with support 2. Background Al for a given l ∈ {1, 2, . .} {∞}. The Erlang df H is defined as 2.1. Admissible moments k−1 i x ()βx Karlin and Studden (1966) provide the necessary Hx();,kHβ≡1;−β ()xk,1=−e−β ∑ , and sufficient conditions for a set of (raw) moments i=0 i! lm = (µ1, . , µm) to be from a probability distribution x ≥ 0, (2) defined on R+. To state this result, define the matrices Pk and Qk (k ≥ 1) as where the parameters k and b of the Erlang df are known as the shape and rate parameters, respectively. 1 µµ1 k An alternative and useful representation of the mixed Erlang rv W is WC= K where {C } are iid expo- µµ12 µk+1 ∑ k=1 k k k≥1 Pk = ; nential rv’s with mean 1/b, independent of K, i.e., the rv W follows a compound distribution. µµkk+12 µ k Remark 1. As in, e.g., Willmot and Woo (2007), we consider the class of mixed Erlang dfs (1) rather than µµ12 µk+1 the more general class of combinations of Erlangs µµ23 µk+2 where some z ’s are possibly negative. For the latter Q =.k k class, additional constraints on {z }l exist to ensure k k=1 that the right-hand side of (1) is a non-decreasing µµkk++12 µ21k+ function in x. This presents additional challenges in 168 CASUALTY ACTUARIAL SOCIETY VOLUME 10/ISSUE 1 Moment-Based Approximation with Mixed Erlang Distributions the subsequent moment-matching application, chal- for more details). In general, these moment-based lenges which do not arise in the mixed Erlang case. approximations propose to work with a specific sub- class of all finite and infinite mixed Erlang dis- It is well known that the j-th moment of W is given j -j ∞ j-1 tributions. Among them, we recall the method of by E[W ] = b Σk=1z {∏ (k + i)}.