Generalized Post-Widder inversion formula with application to statistics

Transcription

1 arxiv:5.9298v [math.st] 3 ov 25 Generalized Post-Widder inversion formula with application to statistics Denis Belomestny, Hilmar Mai 2, John Schoenmakers 3 August 24, 28 Abstract In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the comple domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean miture models. We construct a nonparametric estimator for the miing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical eample. Keywords: Laplace transform, inversion formula, Post-Widder formula, variancemean mitures, density estimation. Introduction Let p be a probability density on R +, then the integral L(z) := e z p() d, Re z >, () eists and is called the Laplace transform of p. The Laplace transform is a popular tool for solving differential equations and convolution integral equations. Its inversion is of importance in many problems from e.g. physics, engineering and finance (c.f. [2] and [6] for various eamples). In general, the compleity of the inversion problem for L depends on the information available about the Laplace transform. If the Laplace transform is eplicitly given on its Duisburg-Essen University, Essen and IITP RAS, Moscow, Russia denis.belomestny@uni-due.de 2 CREST ESAE ParisTech, Paris, hilmar.mai@ensae.fr, (corresponding author) 3 Weierstrass Institute, Berlin, schoenma@wias-berlin.de

2 half-plane of convergence, the density p can be reconstructed using the so-called Bromvich contour integral (see, e.g. [7]) p() = 2πi c+i c i e z L(z) dz, >. In the real case, i.e. in the situation where the Laplace transform of p is known on the real ais only, the inversion of L is a well-known ill-posed problem (see for eample [4], [5] and references therein). One popular solution for this case is given by the well known Post-Widder formula which reads as follows (cf. [7]): ( ) p() = lim! ( ) + L () ( In some situations, the Laplace transform L can only be computed on some curve l in C, which is different from R + or {Re(z) = c} for some c >. In this paper we generalize the Post-Widder formula to the case of rather general curves l and derive the convergence rates of the resulting estimator. As an application of our results we consider the problem of estimating the miing density in a variance mean miture model (see e.g. [] and [3]). After constructing the estimator we derive bounds for its root mean square error (RMSE) and demonstrate its performance in a short numerical eample. An advantage of using the generalized Post-Widder formula here is that the resulting estimator can be evaluated without any numerical integration. The paper is organized as follows. In Section 2 we introduce the generalized Post- Widder inversion formula and discuss its convergence behavior. Section 3 is devoted to the statistical inference for variance mean mitures together with some numerical results. Finally, the proofs of our results are given in Section 5 to 7. 2 Generalized Post-Widder Laplace inversion In this section we will introduce a generalized Post-Widder inversion formula that etends the classical result by Post and Widder [7] to the situation when the Laplace transform of a continuous density on [, ] is given on a curve in the comple plane. Subsequently, we prove a convergence result and derive the rates of convergence for the resulting inverse Laplace transform. 2. Inversion formula and its kernel representation Let p be a continuous probability density on [, ) and let its Laplace transform L(z) be given on a curve: l := {z = y + ic(y) : y R + }, (2) such that c is piecewise smooth with c(y) = o(y) as y. In this setting the generalized Post-Widder formula can be described as follows. 2 ).

3 Definition 2. (Generalized Post-Widder formula). For any fied >, we introduce the generalized Post-Widder formula by p () := ( )! ( g ( )) + ( L () g ( )), (3) where L () denotes the th-derivative of the Laplace transform L and g(y) := y + ic(y). For fied >, we define the generalized Post-Widder kernel via K (t, ) := ( ( + ic )) + t e (+ic( ))t, t >. (4)! Our first result deals with the convergence of p to p as. Such a convergence follows from the properties of the generalized Post-Widder kernel K and a representation formula for p in terms of K and p. The latter representation is given by the following proposition. Proposition 2.2. It holds p () = p(t)k (t, ) dt. The following result states that K (t, ) converges to the delta function δ(t ) on (, ) for any fied >. Proposition 2.3. The following statements hold. (i) For r =,, 2, it holds t r ( + r/) ( + /) K (t, )dt = ( + i(/)c(/)) r (5) = + r ( c(/) r ir + O / + r c(/) ) 2,, >. / (6) Hence, in particular we have K (t, )dt = for all. (7) (ii) Let > be fied. For any δ (, ), there eists a natural number δ t r K (t, ) dt Ce (ln(+δ) δ)/8, > δ. { t δ, t } such that for r =,, 2, and some constant C not depending on and δ. 3

4 2.2 Convergence analysis By combining Proposition 2.2 and Proposition 2.3, the point-wise convergence of p to p follows for as stated in the following corollary. Corollary 2.4. For any fied and any continuous density p on [, ), we have lim p () = p(). (8) We may now sharpen the statement (8) under additional smoothness assumptions on the density p. In the following propositions we give eplicit convergence rates for p as. It turns out that the rates crucially depend on the growth behavior of the function c(y) as y. We henceforth assume that [ ] c 2 (y) γ := lim sup <. (9) y y The notation f(, ) = O (r(, )) for fied R and means in the sequel the usual O-notation where the actual order coefficient may depend on. We start with a local Lipschitz condition on p. Proposition 2.5. Let p be a locally Lipschitz continuous density on [, ) with Laplace transform () given on the curve (2). It then holds p () = p() + R (), where for and each >. R () = O ( /2 ) When the density p is differentiable, the rates of Proposition 2.5 can be improved as the following result shows. Proposition 2.6. Let p be a differentiable density on [, ) with Laplace transform () given on the curve (2). We then have for γ <, where Re [p ()] = p() + R (), R () = o( /2 ) for and each >. Further we have that p () = p() + O ( /2 ) for < γ <, and p () = p() + o( /2 ) for γ =. 4

5 We conclude this section by considering the Laplace inversion problem for a differentiable density p with locally Lipschitz derivative. It turns out that we can achieve the error term R of the order in this case. Proposition 2.7. Let p be a smooth density on [, ) such that its derivative p is locally Lipschitz, and its Laplace transform () is given on the curve (2). Then for γ <, where Re [p ()] = p() + R (), R () = O( ) for and each >. Moreover we have that p () = p() + O ( /2 ) for < γ <, and p () = p() + O ( ) for γ =. In the net section we discuss some applications of the generalized Post-Widder formula (3). 3 Application to statistical inference for variancemean mitures The problem of inverting a Laplace transform that is given on a curve l in the comple domain appears naturally in the contet of statistical inference for variance-mean miture models. In this section we apply our generalized Post-Widder Laplace inversion formula to estimate the miture density in a variance mean miture model. We start the construction of the estimator from the empirical characteristic function that can be written as the Laplace transform of the miture density evaluated on a certain curve in the comple plain. By inverting this Laplace transform we obtain a nonparametric estimator for the miing density p. Then we derive bounds for the RMSE and conclude by a numerical eample. 3. Variance-mean miture models A normal variance-mean miture model is defined as ( ) sµ q() := σ s ν σ p(s) ds, s where µ R, σ R +, ν is the density of a standard normal distribution and p is a miing density on R +. Variance-mean miture models play an important role in both theory and practice of statistics. In particular, such mitures appear as limit distributions in asymptotic theory for dependent random variables and they are useful for modeling data stemming from heavy-tailed and skewed distributions, see, e.g. [] and [3]. 5

6 As can be easily seen, the variance-mean miture distribution q coincides with the distribution of the random variable σ ξ X + µ ξ, where X is standard normal and ξ is a nonnegative random variable with the density p, which is independent of X. The class of variance-mean miture models is rather large. For eample, the class of the normal variance miture distributions (µ = ) can be described as follows: q is the density of a normal variance miture if and only if F[q]( u) is a completely monotone function in u. 3.2 Estimating the miing density Here we consider the problem of statistical inference for the miing density p based on a sample X,..., X n from the distribution q. The Fourier transform of the density q is given by Φ(u) := F[q](u) = e sψ(u) p(s) ds = L[p](ψ(u)) () with ψ(u) := iuµ + u 2 σ 2 /2 and from our data we can directly estimate the Fourier transform of q, e.g. by means of the so-called empirical Fourier transform: Φ n (u) := n n e iux k. () Then we end up with the problem of reconstructing the density p from its empirical Laplace transform observed on the curve k= l := {z = Re ψ(u) + i Im ψ(u) : u R + }, where we have Re[ψ(u)] = u 2 σ 2 /2 and Im[ψ(u)] = uµ. ote that l = {z = y + c(y) : y R + } with c(y) = µ 2y/σ. If σ than the function c is smooth and satisfies c(y) = o(y) as y. Moreover it holds [ ] c 2 (y) γ = lim sup = 2µ 2 /σ 2. (2) y y Hence if p is a differentiable density on [, ) such that p is locally Lipschitz, we can apply Proposition 2.7 to get the following asymptotic bound { O (/), µ =, p () p() = O (/ (3) ), µ, for p defined in (3) with g(y) = y iµ 2y/σ. Due to (), we have L(z) = Φ(ξ(z)), where ξ is the inverse of ψ on l. Without loss of generality we may assume that σ =, then ξ(z) = 2z µ 2 + iµ using the principal branch of the square root. So, for l we have that ξ (l) (z) = ( ) l (2 (l ))! ( ) 2z µ 2 2 l, 2 l (l )! 6

7 and by Faa di Bruno s formula it follows that for z l, L () (z) = = k,...,k, k +2k k = k +k k =k! k!... k!(!) k... (!) k Φ (k) (ξ(z)) ( ξ (l) (z) ) k l Φ (k) (ξ(z)) ( ) k ( 2z µ 2) 2 k F,k. (4) k= The coefficients F,k can be epressed as follows F,k := = k,...,k, k +2k k = k +k k =k! k!... k! k,...,k, k +2k ( k+)k k+ = k +k k k+ =k, = B,k (,..., (2 ( k))! 2 k ( k)! l= ( ) kl (2 (l ))! 2 l (l )!l!! k! k k+! ), k+ l= l= ( (2(l ))! ) kl 2 l (l )! l! where B,k stand for the partial Bell polynomials. In view of () and (4), we now introduce L () n (z) := Φ (k) n (ξ(z)) ( ) k ( 2z µ 2) 2 k F,k k= as an unbiased estimator for L () (z) at every z l. We so arrive at an empirical estimate for the miing density p: p n, () := ( )! = ( )! (g(/)) + L () n (g(/)) n (g(/)) + n j= e iξ(g(/))x j (ix j ) k ( ) k ( 2g(/) µ 2) 2 k F,k, (5) k= which obviously satisfies E [p n, ()] = p (). The coefficients F,k can be computed by evaluating the partial Bell polynomials B,k that are available in most computational algebra packages. Hence, we obtain an eplicit estimator for p that circumvents the use of numerical integration procedures as needed in other Laplace inversion techniques. 7

8 3.3 Convergence of the estimator Let us now analyze the variance of p n,. Theorem 3.. For some constant C >, depending on >, it holds that [ where β 2k := E X 2k]. Var [p n, ()] C n k β 2k, (6) k= Based on the estimate (6), we can derive upper bounds of the root mean square error (RMSE) for the density estimator p n,. Theorem 3.2. Fi some > and suppose that R () = p () p() = O ( ρ ). We then have the following bounds for the RMSE of p n,. (i) If β 2k A k k k for some A > and all natural k >, then ( ) RMSE(p n, ) = O ln ρ, n provided = where the constant C comes from the bound (6). ln n ln(ac) 2ρ ln ln n, (7) ln(ac) (ii) In the case β 2k A k k bk, k for some b >, it holds that which is achieved by choosing = RMSE(p n, ) lnρ ln n ln ρ n, (2ρ + ln n) (b ) ln ((2ρ + ln n) / (b )) 2ρ b. (8) Remark 3.3. Because of the inequality Var [p n, ()] Var [Re[p n, ()]], the results of Theorem 3.2 remain valid with p n, replaced by Re[p n, ]. 3.4 umerical eample Let the miing density p be the eponential density, i.e. p() = ep( ), then E [ X 2k] = 2k Γ( + k)γ(k + /2) A k k 2k π 8

9 p 4,2 p 4,3 ep(-) p 5 4,2 p 5 4,3 ep(-) Figure : Approimations p n, (real parts) for sample sizes n = (above), n = 5 (below) and different values of of the true eponential density p() = ep( ) in the normal mean variance miture model with parameters µ =. and σ =. 9

10 for some A >. Combining Proposition 2.7 with Theorem 3.2, we obtain RMSE(Re[p n, ]) ln ln(n) ln(n), n. In Figure one can see the result of numerical estimation of the underlying eponential density p() = ep( ) based on different numbers of terms in (5) and different sample sizes n. As can be observed, the estimation error increases as +. This effect can be eplained by noting that g(/) as + and so the variance increases for small (see the proof of Theorem 3.). 4 Acknowledgment The first author was supported by the Deutsche Forschungsgemeinschaft through the SFB 823 Statistical modelling of nonlinear dynamic processes and by a Russian Scientific Foundation grant (project 4-5-5). The second author received founding by the LABEX Louis Bachelier Finance et Croissance Durable in Paris. 5 Proofs In this section we gather the proofs of our results from Section Proof of Proposition 2.2 We start by proving the integral representation of p in terms of p and the generalized Post-Widder kernel K. We have by definition L(z) = differentiating -times results in L () (z) = e uz p(u)du, Re z > ( u) e uz p(u)du, yielding finally ( ( )) + p () = ( )! + ic ( u) e u( +ic( )) p(u) du = ( ( )) +! + ic (t) e t(+ic( )) p(t) dt ( ( )) + = + ic t e t(+ic( )) p(t) dt! = p(t)k (t, ) dt.

11 5.2 Proof of Proposition 2.3 (i): For r =,, 2,... we have t r K (t, )dt = =! ( + ic! ( + ic ( ( )) + t +r e t(+ic( )) dt )) r (+ic( )) z +r e z dz. ote that on the set { z : arc(z) = R e iθ, π/2 < θ < π/2 } it holds that z +r e z = R +r e R cos θ for R. Thus, by the Cauchy integral theorem, (+ic( )) z +r e z dz = t +r e t dt = ( + r)! from which (5) follows. Thus (5) holds for any integer integer r. The asymptotic epression (6) for r = and r = 2 can be seen from taking the logarithm of (5): ( + r/) ( + /) r ln ( + i(/)c(/)) r = ln ( + l/) r ln ( + i(/)c(/)) l= = r irc(/) / e ( + O + c(/) ) 2. / (ii): By Stirling s formula it follows that ( ( + i ) ( c )) + + K (t, ) = t e t e i(t)c( 2π ) ( + O(/)) and hence ( ( ) ) + = + ic 2π t e t e e i(t)c( ) ( + O(/)) ln K (t, ) = i(t)c ( ) ( + ( + ) ln + i ( )) c + 2 ln + ln t t + + O(/) 2π ( ( ) ) ( ) j j = ( + ) ic + j 2 ln 2π j= ( ) + ln t t + i(t)c + O(/) = 2 ln ( ) ) 2 (c 2π + O + O(/) ( ) ( ) + ln t t + + ic ( t) + ic. (9)

12 In particular, for t we have K (t, ) = ep 2 ln 2π + O(/) + ln } t {{ t + } ( ( ) ) 2 + O c < }{{} as. (2) Let us fi > and δ > arbitrarily. W.l.o.g. we may assume that δ <. Because c( ) for, there eist a number δ such that for any > δ and any t > with t δ, i.e. and so for r =,, 2, > δ, ( ( ) ) 2 ln t t + + O c < (ln t t + ), 2 [ K (t, ) ep 2 ln 2π + O(/) + ] (ln t t + ), 2 { t δ, t } t r K (t, ) dt C 2π { t δ, t } t r ( te t+) /2 dt. (2) ote that for t δ we have te t+ = e ln t t+ ln t t+ + e ln(+δ) δ 2 2, hence for r =,, 2, > δ, { t δ, t } t ( r te t+) /2 e (ln(+δ) δ)/4 e (ln(+δ) δ)/4 { t δ, t } { t δ, t } t r e (ln t t+)/4 dt t r e (ln t t+)/4 dt e (ln(+δ) δ)/4 t r+ e ( t+)/4 dt C 2 e (ln(+δ) δ)/4, where ln ( + δ) δ < and C 2 >. It net follows from (2) that for some constant C > t r K (t, ) dt Ce (ln(+δ) δ)/8 (22) for, > δ. { t δ, t } 5.3 Proof of Proposition 2.5 In order to derive the convergence rates in (8), we proceed with the following lemma. 2

13 Lemma 5.. For r =, 2,.. [ ( ( t r (r+)/2 Γ ((r + ) /2) K (t, ) dt C6 ep O c 2 r/2 Proof. Let us fi >. For δ > with < δ < we have by (2), +δ [ ( ( ) ) ] 2 t r K (t, ) dt C 2π ep O c where +δ (t ) r ep [ (ln t t + )] dt = δ Hence by (23) and Theorem 2.3-(ii) +δ t r K (t, ) dt C 6 (r+)/2 ) 2 )]. (t ) r ep [ (ln t t + )] dt, (23) u r ep [ (ln ( + u) u)] du 2 Γ ((r + ) /2) r/2 = 6(r+)/2 2 [ ( ( ep O c 2 Γ ((r + ) /2) (r+)/2. ) 2 )]. Similarly, δ [ ( ( ) )] t r K (t, ) dt = C 2π ep O c 2 2 δ ( t) r ep [ (ln t t + )] dt, where δ δ = ( t) r dt ep [ (ln t t + )] u r du ep [ (ln ( u) + u)] δ u r dt ep [ u 2 /6 ]. By applying Theorem 2.3-(ii) once again the statement of the lemma is proved. 3

14 Let us now fi >. By the Lipschitz assumption on p we thus have for fied < δ <, p () p() p(t) p() K (t, ) dt K t K (t, ) dt + K e (ln(+δ) δ)/8 due to assumption (9). t <δ [ )] K /2 ep O (c 2 ( )2 K 2 /2 5.4 Proof of Proposition 2.6 Let us fi >. By differentiability of p we may find for any ε > a δ ε with < δ ε < such that p(t) =: p() + (t ) (p () + E t ), with Et < ε for all t > with t < δ ε.due to Proposition 2.2 we then have p () p() = (p(t) p()) K (t, )dt = (p(t) p()) K (t, )dt t δ +p () (t )K (t, )dt + (t )Et K (t, )dt =: ( ) + ( ) 2 + ( ) 3. t <δ t <δ Since p is bounded we have by Theorem (2.3)-(ii) that ( ) D e (ln(+δε) δε)/8 (24) for some D > and > δ ε (cf. the proof of Theorem 2.3)-(ii). By Theorem (2.3) we have (t )K (t, )dt = (t )K (t, )dt (t )K (t, )dt t <δ ε t δ ε, t = ( ic(/) / + O + c(/) ) 2 + f,δε (25) / with f,δε D 2 e (ln(+δε) δε)/8 for > δ ε. et, by Lemma 5. and assumption (9) we have [ )] ε ( ) 3 ε (t K (t, ) dt D 3 ep O (c 2 ( )2 (26) t <δ /2 ε D 4 /2 4

15 From (24) (26) we gather that ( ) p () = p() + p () ic(/) / ow, since + O ( + c(/) / )2 + o( /2 ). (27) c(/) / = O ( /2), if < γ <, and c(/) / = o( /2 ) if γ =, (28) the statements follow by taking the real part of (27). 5.5 Proof of Proposition 2.7 Let us fi >. By the Lipschitz assumption on p we may find a δ with < δ < such that p(t) =: p() + (t )p () + 2 (t )2 R t, with Rt < R for some constant R > and for all t > with t < δ. Due to Proposition 2.2 we thus have p () p() = (p(t) p()) K (t, )dt = (p(t) p()) K (t, )dt + t <δ ( (t )p () + 2 (t )2 R t t δ ) K (t, )dt =: ( ) + ( ) 2. (29) Since p is bounded we have by Theorem 2.3-(ii) again that ( ) D e (ln(+δ) δ)/8 for some D > and > δ (cf. the proof of Theorem (2.3)-(ii)). ow let us consider ( ) 2. From Theorem 2.3 it follows similar to the proof of Proposition 2.6 that t <δ (t )K (t, )dt = ic(/) / + O ( + c(/) ) 2 + f,δ (3) / with f,δ D 2 e (ln(+δ) δ)/8, and by Lemma 5. we have that (t ) 2 Rt K (t, )dt R t 2 K (t, ) dt t <δ D [ )] 3 ep O (c 2 ( )2 (3) We thus get by (29), (3), (3), and assumption (9), ( ) ( p () = p() + p () ic(/) + O / + c(/) ) 2 + O( ) / from which the statements follow by taking the real part and taking (28) into account. 5

16 6 Proof of Proposition 3. For a generic constant C >, depending on and changing in this proof from line to line, we may write Var [p n, ()] C 2+2 n E (ix 2 ) k ( ) k ( 2 2g(/) µ 2) 2 k F,k k= C n E (ix ) k ( ) k ( 2 2g(/) µ 2) 2 k F,k C n C n k= k= k= [ E X 2k ] 2g(/) µ 2 k 2 F,k 2 [ k 2 F,kE 2 X 2k]. It is not difficult to see that for l =,..., k +, (2 (l ))! 2 l (l )! Cl l! and so from the definition of the Bell polynomials it follows that F,k C k B,k (!,..., ( k + )!) ( ) ( ) = C k ( k)! k k C k. 7 Proof of Proposition 3.2 (i): Without loss of generality we may assume that A >. Since for k, k k k, we get from (6), Var [p n, ()] C A k A (AC). (32) n A n k= By substituting according to (7) into (32) we obtain Var [p n, ()] A (AC) A n = A A ln 2ρ n, while for the squared bias we have ( ) R 2 () = O ( 2ρ ) = O ln 2ρ, n 6

17 hence (i) follows. (ii): In this case (6) yields Var [p n, ()] C n k= A k k (b )k C n (b ) (33) for another constant C >. From (8) it follows straightforwardly that ( ln (b )+2ρ 2ρ + ln n = ln ((2ρ + ln n) / (b )) ln (2ρ + ln n) (b ) ln ((2ρ + ln n) / (b )) 2ρ ) b 2ρ + ln n = = ln ln ((2ρ + ln n) / (b )) [ (2ρ + ln n) / (b ) ln ((2ρ + ln n) / (b )) 2ρ + ln n ln ((2ρ + ln n) / (b )) ln (2ρ + ln n) / (b ) ( )] ln ((2ρ + ln n) / (b )) 2ρ 2ρ + ln n ( ) ln ln n ln ((2ρ + ln n) / (b )) 2ρ + O ln n ( ) ln ln n + O. ln n ln ln ((2ρ + ln n) / (b )) = ln n (2ρ + ln n) ln ((2ρ + ln n) / (b )) On the other hand, from (8), So, for n, hence We thus obtain by (33), while which gives (ii). ln C (2ρ + ln n) ln C (b ) ln ((2ρ + ln n) / (b )) ln ( C (b )+2ρ) ( ) ln ln n ln n + O ln n ln ln ((2ρ + ln n) / (b )) (2ρ + ln n) ln ((2ρ + ln n) / (b )) }{{} + ln C (b ) ln ln ((2ρ + ln n) / (b )), }{{} C (b )+2ρ 2n Var(p n, ()) 2 2ρ ln2ρ ln n ln 2ρ n ( ln 2ρ ) R 2 () = O ( 2ρ ln n ) = O ln 2ρ n 7

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