3. LOCAL ASYMPTOTIC NORMALITY

Let us now assume that we are given a Lévy process with a skewed Lévy measure e^θxg(x)ν(dx) and we aim to estimate the skewness parameter θ. First of all we are interested how the multiplicative term in the Lévy measure changes the distribution of the underlying Lévy process. We obtain the following lemma. Though we shall only look at Lévy processes with unbounded support, the same calculations hold for processes with bounded support, especially for subordinators.

LEMMA 1. Denote by p_t(x) the density of the Lévy process w.r.t. m, given by the triplet (μ,σ²,g(x)ν(dx)) and by p_t(x,θ) the density corresponding to the process with the skewed Lévy measure e^θxg(x)ν(dx). Furthermore, assume that for all θ ∈ U(θ₀) ⊂ Θ, where U(θ₀) is a neighborhood of the true parameter θ₀,

Then we obtain

and the corresponding drift is μ = μ + ∫h(x)(e^θx − 1)g(x)ν(dx).

Remark. (1) For processes with finite variation, i.e., ∫(|x| ∧ 1)g(x)ν(dx) < ∞, e.g., all compound Poisson processes and subordinators, we may take h(x) = 0 and obtain μ = μ.

(2) We consider densities p_t w.r.t. to some measure m. However, of most practical interest, except for compound Poisson processes, is when m equals the Lebesgue measure, as is the case in our examples. Even though the density p_t might not be known explicitly, conditions on the Lévy measure may ensure its existence; e.g., as was shown in Tucker (1962), infiniteness of the Lévy measure together with a Lebesgue density of the Lévy measure already ensures the existence of a Lebesgue density p_t. However, the relation between the existence of a density p_t and the existence of a density of the corresponding Lévy measure and its mass near zero is much more complex. A detailed outline is given in Sato (1999).

Proof. Under the assumption (1) the denominator in (2) is well defined. We assume that the characteristic function corresponding to p_t is

Hence we can calculate the characteristic function of the skewed distribution

which yields the desired result. █

Of course this result is not new. The principle of multiplying with an exponential factor is well established in different disciplines of stochastics but is named differently. In statistics the family of distributions or processes derived by varying the skewness parameter is called the natural exponential family (cf. Janssen, 1992; Küchler and Sørensen, 1997). In finance the the distribution obtained by adding the skewness is called the Esscher transform (cf. Shiryaev, 1999). This concept is also an easy way of explicitly calculating equivalent martingale measures.

Using (2) we can prove LAN for the skewness parameter. Though we start with a general Lévy process with unknown density, we obtain a fully explicit result of the minimal asymptotic variance, depending only on the quantities of the Lévy triplet, for both sampling schemes. Furthermore, our minimal asymptotic variance for the sampling scheme with Δ → 0 turns out to be the same as in the continuously observed model.

THEOREM 2. Assume condition (1) for θ ∈ U(θ₀) ⊂ Θ, a neighborhood of θ₀; then we obtain LAN

Proof. Because we have i.i.d. increments as a result of the structure of the Lévy processes, we can use the results for i.i.d. random variables.

First, we have to calculate the derivatives. We may interchange integration and differentiation if θ is in the interior of Θ and obtain

(i) For the sampling scheme with fixed distance of observations Δ, we are in the framework of i.i.d. random variables with densities; hence conditions under which LAN in θ₀ holds are well known (see, e.g., Witting, 1985, p. 179; Le Cam and Yang, 1990, Ch. 6.3). The conditions are that the density is positive m a.s. and continuously differentiable w.r.t. the unknown parameter θ in a neighborhood U(θ₀). Then we obtain LAN with maximal rate of convergence

and

which is the inverse of the minimal asymptotic variance, provided Γ(θ) is finite for all θ ∈ U(θ₀) and continuous in θ₀. This result applies as for θ ∈ U(θ₀), p_t(x,θ) is continuously differentiable in θ and p_t(x,θ) > 0 [m] . We obtain LAN with

and

because Γ(θ) is obviously finite for θ ∈ U(θ₀) and continuous in θ₀. For the last equation we use the well-known moment representation, by the derivatives of the characteristic function,

.

(ii) For the sampling scheme with Δ → 0 we need a few more conditions to ensure the LAN property, which results basically from conditions needed to perform the CLT and LLN for triangular schemes (see Woerner, 2001; Gnedenko and Kolmogorov, 1968). The conditions are as follows. Assume that there exists a neighborhood U(θ₀) of θ₀ such that the density p_t(x,θ) w.r.t. m of the Lévy process is two times continuously differentiable w.r.t. the unknown parameter θ and regular (i.e.,

) for all θ ∈ U(θ₀). Denote by θ_n = θ_n(x) a measurable function from

. Furthermore, assume that for all ε > 0 as n → ∞ and Δ → 0, such that nΔ → ∞,

Then we obtain LAN with

for the sampling scheme with Δ → 0 as n → ∞.

Finding dominating functions to ensure these conditions is straightforward because of the special simple structure that θ only enters in the exponential term and Δ only as a polynomial factor, by applying the moment representation to the integrals w.r.t. p_Δ(x,θ)m(dx). We look at the details of condition (i); the others can be checked analogously.

as n → ∞, Δ → 0, and nΔ → ∞. █

Example 1 (Normal inverse Gaussian process)

The normal inverse Gaussian process is characterized by the Lévy triplet

where K₁ denotes the modified Bessel function of third order with index 1 and

. Because as |x| → 0

the process is of unbounded variation and possesses infinitely many jumps. Hence it also has a density w.r.t. the Lebesgue measure,

As pointed out in Barndorff-Nielsen (1998) this process is used both for modeling turbulence, in particular when the Reynolds number is high, and in finance. This is due to some special properties of the normal inverse Gaussian process, such as possible asymmetry modeled by the skewness parameter β, unbounded variation, and semiheavy tails; namely, as |x| → ∞

The parameter β may be estimated according to Theorem 2. Here assumption (1) is satisfied. Hence for the sampling scheme with Δ fixed and n → ∞ we have

and

which is indeed the same result as using the density and the results for i.i.d. random variables.

Example 2 (Gamma process)

The gamma process is characterized by the Lévy triplet

with α,β,x > 0. Hence the process is a subordinator and only possesses nonnegative jumps. The density w.r.t. the Lebesgue measure can be calculated, and we can see that the name reflects the fact that the increments are distributed according to a gamma function,

Analogously to Example 1 β can be estimated, and we obtain for the sampling scheme with Δ fixed and

and

which is again the same result as using the density and the i.i.d. results.

Example 3 (Hyperbolic Lévy motion)

The hyperbolic Lévy motion which was introduced by Barndorff-Nielsen (1977) for modeling mass-size distributions of aeolin sand deposits, has also been applied to some other areas of interest, e.g., turbulence data (cf. Barndorff-Nielsen, 1996) and to financial data (cf. Eberlein and Keller, 1995; Keller, 1997; Rydberg, 1997; Prause, 1999; Raible, 2000).

The hyperbolic Lévy motion may be characterized by the Lévy triplet

where J₁ denotes the Bessel function of the first order with index 1 and Y₁ the Bessel function of the second order with index 1. Furthermore, we have α,δ > 0 and 0 ≤ |β| < α. Keller (1997) has established that the density of the Lévy measure behaves like x⁻² at the origin; hence the process is of unbounded variation. Though the process possesses a density, we cannot calculate it analytically. Only the distribution for t = 1 can be written down explicitly; it is the hyperbolic distribution

where K₁ denotes the modified Bessel function of third order with index 1. We can estimate β analogously to Example 1 and obtain for the sampling scheme with Δ fixed and

and

where

using Keller (1997).

Example 4 (CGMY process)

The CGMY process, named after Carr, Geman, Madan, and Yor, or in physical literature called truncated stable or truncated Lévy flight, is given by the Lévy triplet

where C > 0, G ≥ 0, M ≥ 0, and Y < 2. This class of processes is a flexible model for index dynamics and also for the dynamics of individual stocks, because by varying the parameters it allows all features of finite and infinite activity, bounded and unbounded variation, and also skewness to be modeled directly by the characteristic function (cf. Carr et al., 2002). The variance gamma process (cf. Madan and Senata, 1990) is a special case of the CGMY process. In general the density of the process is not known explicitly, but as in the previous examples we can infer the parameters M or G in the special one-sided case, when either G = ∞ or M = ∞, only by the knowledge of the Lévy measure. When G = ∞ we obtain (0,0,C(exp{−M|x|}/|x|^1+Y)1_x>0) and for the sampling scheme with Δ fixed and

and

4. EFFICIENT ESTIMATORS

With this explicit result for the minimal asymptotic variance we can now try to find a sequence of estimators that is efficient. Because the continuous model is a benchmark for the discretely observed model, we look at the continuous likelihood function to get an idea of what discrete estimation functions may look like. In the continuous model the likelihood function is given by

where Y_t denotes the process under the measure p_t(x)m(dx). Changing to the process X_t under the measure e^θxp_t(x)m(dx)/(∫e^θxp_t(x)m(dx)) and discretizing the time, i.e., t = nΔ, leads to the log-likelihood function

which indeed provides an appropriate estimating function.

We obtain that only the knowledge of the last observation is important for the estimation procedure. In other words, the last observation contains all necessary information. This is however not surprising, because from the theory of exponential families it is well known that Y_t or X_nΔ, respectively, is sufficient for {P_t^θ,θ ∈ Θ}.

THEOREM 3 (Optimal estimators).

Proof.

where

.

(ii) This is analogous to using the CLT for triangular schemes as in Theorem 2 for the convergence in distribution and calculating the explicit form of

as in (i). █

With Theorem 3 equation (3) now provides an easily computable estimating function that indeed leads to efficient estimators. It is especially simple because it only involves the observations and the first moment. Depending on the form of the first moment the equation can sometimes even be solved analytically.

Example 5 (Gamma process)

For the gamma process equation (3) is

Hence we obtain

for the sequence of efficient estimators as n → ∞ for the sampling scheme with fixed Δ.

Equation (4) can also be viewed as the simplest form of a martingale estimating function, having the general form

where Y_i = X_iΔ − X_(i−1)Δ i.i.d. distributed according to p_Δ(.,θ). We can now show that our sequence of estimators is not only optimal in the sense of local asymptotic statistics but also in the sense of Godambe and Heyde (1987), which is the classical optimality concept for martingale estimating functions.

Let us first give a short review on the concept of martingale estimating functions and the definition of optimality by Godambe and Heyde (1987).

The basic problem is that we want to draw inference for discretely observed stochastic processes when the likelihood function is unknown. Because in general the maximum likelihood estimator performs quite well, the idea is to approximate the unknown score function to obtain an approximate maximum likelihood estimator. Using an approximation, the problem might occur that we perhaps do not have mean zero and hence eventually will get biased estimates, especially when the distance of observations Δ is bounded away from zero. A solution is to approximate the score function by a zero mean martingale w.r.t. the filtration generated by the observations. This implies that we obtain consistent and asymptotically normal estimators.

The optimality concept of Godambe and Heyde (1987) and Heyde (1988) formalizes the heuristics that the optimal element in a given class of martingale estimating functions is the one whose L²-distance to the true score function is minimal, or in the partial order of nonnegative matrices the distance to the score function is minimal.

DEFINITION (O_A-optimality). Let

where G_n denotes the compensator of

and 〈G,G^T〉 the quadratic characteristic of G_n. Then

is called O_A-optimal in

if and only if

As a result of the special structure of i.i.d. increments, all considerations simplify greatly for our model compared to general stochastic processes. By straightforward calculations we obtain f*(x) = x for the optimal G*. Hence we also have optimality in the sense of Godambe and Heyde (1987) for the estimators in Theorem 3.