Parameter Estimation for Non-Stationary Fisher-Snedecor Diffusion


The problem of parameter estimation for the non-stationary ergodic diffusion with Fisher-Snedecor invariant distribution, to be called Fisher-Snedecor diffusion, is considered. We propose generalized method of moments (GMM) estimator of unknown parameter, based on continuous-time observations, and prove its consistency and asymptotic normality. The explicit form of the asymptotic covariance matrix in asymptotic normality framework is calculated according to the new iterative technique based on evolutionary equations for the point-wise covariations. The results are illustrated in a simulation study covering various starting distributions and parameter values.

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N.N. Leonenko was supported in particular by Cardiff Incoming Visiting Fellowship Scheme and International Collaboration Seedcorn Fund, Cardiff Data Innovation Research Institute Seed Corn Funding, Australian Research Council’s Discovery Projects funding scheme (project number DP160101366), and by projects MTM2012-32674 and MTM2015-71839-P (co-funded with Federal funds), of the DGI, MINECO, Spain.

N. Šuvak and I. Papić were supported by the scientific project UNIOS-ZUP 2018-31 funded by the J. J. Strossmayer University of Osijek.

At last, authors would like to thank Danijel Grahovac (Department of Mathematics, J.J. Strossmayer University of Osijek) for many useful discussions regarding the tail-index estimation.

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Appendix A: Spectral Representation of Transition Density of Fisher-Snedecor Diffusion

For deriving the closed-form results in the framework of statistical analysis, e.g. calculation of asymptotic covariances in explicit form, the explicit expression for diffusion transition density

$$ p(x, t) = p(x; x_{0}, t) = \frac{d}{dx} P(X_{t} \leq x \mid X_{0} = x_{0}), \quad x > 0, \quad t \geq 0, $$

can be extremely useful.

For canonical FSD such representation is given in terms of the spectrum of the corresponding infinitesimal generator and it is thoroughly studied in Avram et al. (2013b). In the general case of the FSD Eq. 2.9 satisfying the SDE Eq. 1.1 and having the invariant density Eq. 2.8, infinitesimal generator is defined as follows:

$$ (\mathcal{G}f)(x) = \frac{2\theta}{\beta - 2} x \left( x + \frac{\varrho}{\alpha} \right) f^{\prime \prime}(x) - \theta \left( x - \frac{\varrho}{\beta - 2} \right)f^{\prime}(x), \quad x > 0. $$

The domain of the operator \(\mathcal {G}\) is the space of functions

$$ \begin{array}{@{}rcl@{}} D(\mathcal{G}) &=& \left\{ f \in L^{2}((0, \infty), \mathfrak{p}(x)) \cap C^{2}((0, \infty )): \mathcal{G}f\right.\\ &\in&\left. L^{2}((0, \infty ), \mathfrak{p}(x)), \lim\limits_{x \to 0} \frac{f^{\prime}(x)}{\mathfrak{s}(x)} = \lim\limits_{x \to \infty} \frac{f^{\prime}(x)}{\mathfrak{s}(x)} = 0 \right\}, \end{array} $$

where \(\mathfrak {s}(x)\) is the scale density given by Eq. 2.4 with κ = ϱ/(β − 2).

As for the canonical FSD, the spectrum of the operator \((-\mathcal {G})\) consists of two disjoint parts: the discrete spectrum and the essential spectrum (see Avram et al. (2013b), Subsection 4.3). The discrete spectrum of the operator \((-\mathcal {G})\) is the finite set \(\sigma _{d}(-\mathcal {G}) = \left \{\lambda _{n}, n = 0, \ldots , \left \lfloor \beta /4 \right \rfloor \right \}\), where the eigenvalues λn are given by

$$ \lambda_{n} = \frac{\theta}{\beta - 2} n (\beta - 2n), \quad \theta > 0, \quad \beta > 2, \quad n = 0, \ldots, \left\lfloor \frac{\beta}{4} \right\rfloor, $$

and the corresponding eigenfunctions are orthogonal Fisher-Snedecor polynomials given by the Rodrigues formula

$$ P_{n}(x) = K_{n} \widetilde{P}_{n}(x) = K_{n} x^{1-\frac{\alpha}{2}} \left( x + \frac{\varrho}{\alpha}\right)^{\frac{\alpha+\beta}{2}} \frac{d^{n}}{dx^{n}} \left\{ x^{\frac{\alpha}{2} + n - 1} \left( x + \frac{\varrho}{\alpha} \right)^{n - \frac{\alpha+\beta}{2}} \right\}, $$

where \(\widetilde {P}_{n}(x)\) are non-normalized polynomials and the normalization constant Kn can be expresses explicitly. The essential spectrum of the operator \((-\mathcal {G})\) is \(\sigma _{ess}(-\mathcal {G}) = [{\Lambda }, \infty )\), where

$$ {\Lambda} = \frac{\theta \beta^{2}}{8 (\beta-2)}, \quad \theta > 0, \quad \beta > 2. $$

Moreover, operator \((-\mathcal {G})\) has the absolutely continuous spectrum of multiplicity one in \(({\Lambda }, \infty )\), i.e. \(\sigma _{ac}(-\mathcal {G}) \subseteq ({\Lambda }, \infty ) \subset \sigma _{ess}(-\mathcal {G})\), whose elements could be parameterized by

$$ \lambda = {\Lambda} + \frac{2\theta k^{2}}{\beta-2} = \frac{2\theta}{\beta-2} \left( \frac{\beta^{2}}{16} + k^{2} \right), \quad \theta > 0, \quad \beta > 2, \quad k > 0, $$

and where Λ is the cutoff between the absolutely continuous spectrum and the discrete spectrum. According to Borodin and Salminen (2002), in the general case for spectral representation of the transition density two linearly independent solutions of the SL equation, one of which is strictly increasing while the other one is strictly decreasing, are crucial. Such solutions in the case of the FSD are

$$ f_{1}(x) = f_{1}(x, -\lambda) \phantom{a} = {{~}_{2}F_{1}} \left( z_{+,\lambda}, z_{-,\lambda}; \frac{\alpha}{2}; -\frac{\alpha}{\varrho} x \right), $$
$$ f_{4}(x) = f_{4}(x, -\lambda) = \left( \frac{\alpha}{\varrho} x \right)^{-z_{+,\lambda}} {{~}_{2}F_{1}} \left( z_{+,\lambda}, u_{+,\lambda}; 1 + 2 {\Delta}_{\lambda}; -\frac{\varrho}{\alpha x} \right), $$

where λ > Λ is the spectral parameter,

$$ {\Delta}_{\lambda} = \sqrt{\frac{\beta^{2}}{16} - \frac{\lambda (\beta-2)}{2\theta}},\quad z_{\pm, \lambda} = -\frac{\beta}{4} \pm {\Delta}_{\lambda}, \quad u_{\pm, \lambda}=1-{\alpha\over 2}+z_{\pm, \lambda} $$

and 2F1(a, b; c; ⋅) is the Gauss hypergeometric function (see e.g. Nikiforov and Uvarov (1988) or Luke (1969)). Due to the procedure of the analytic continuation of the function 2F1(a, b; c; ⋅), solutions f1(x,−λ) and f4(x,−λ) are well defined on the whole state space of the FSD. Spectral representation of transition density p(x; x0, t) is given in Theorem A.1. The proof can be conducted analogously as in the canonical case for which we refer to Avram et al. (2013b), Theorem 4.1.

Theorem A.1

Spectral representation of the transition density of the FSD with the PDF Eq. 2.8 with parametersα > 2, \(\alpha \notin \{2(m+1), m \in \mathbb {N}\}\), β > 2, ϱ > 0 and 𝜃 > 0 is of the form

$$ p(x; x_{0}, t) = p_{d}(x; x_{0}, t) + p_{c}(x; x_{0}, t) . $$

The discrete part of the spectral representation

$$ p_{d}(x; x_{0}, t) = \mathfrak{p}(x) \sum\limits_{n=0}^{\left\lfloor \frac{\beta}{4} \right\rfloor}e^{-\lambda_{n} t} P_{n}(x_{0}) P_{n}(x) $$

is given in terms of the eigenvaluesλngiven by Eq. A.2and the normalized Fisher-Snedecor polynomialsPn(⋅) given by Eq. A.3. The continuous part of the spectral representation

$$ p_{c}(x; x_{0}, t) = \mathfrak{p}(x) \frac{1}{\pi} \int\limits_{\Lambda}^{\infty} e^{-\lambda t} k(\lambda) \times $$
$$ \times \left| \frac{ B^{\frac{1}{2}}\left( \frac{\alpha}{2}, \frac{\beta}{2} \right) {\Gamma}\left( -\frac{\beta}{4} + ik(\lambda) \right) {\Gamma}\left( \frac{\alpha}{2} + \frac{\beta}{4} + ik(\lambda) \right)}{\Gamma\left( \frac{\alpha}{2} \right) {\Gamma}\left( 1 + 2ik(\lambda) \right)} \right|^{2} f_{1}(x_{0}, -\lambda) f_{1}(x, -\lambda) d\lambda $$

is given in terms of the elementsλof the absolutely continuous spectrum of the operator\((-\mathcal {G})\)given by Eq. A.4, solutionf1(⋅,−λ) of the Sturm-Liouville equation\((\mathcal {G}f)(x) = -\lambda f(x)\)forλ > Λ given by Eq. A.5and parameterk(λ) = −iΔλ, where Δλis given in Eq. A.7.

Furthermore, the explicit expression for the corresponding two-dimensional density is given by following expression:

$$ p(x, y, t) = \frac{\partial^{2}}{\partial x \partial y} P(X_{s + t} \!\leq\! x, X_{s} \!\leq\! y) = \mathfrak{p}(y) p(x; y, t) = \mathfrak{p}(y) \left( p_{d}(x; y, t) + p_{c}(x; y, t) \right), $$

where pd(x; y, t) is given by Eq. A.9 and pc(x; y, t) is given by Eq. A.10. Representation Eq. A.11 of two-dimensional density of the FSD can be used in calculation of explicit form of expectation \(E[{X_{s}^{m}} {X_{t}^{n}}]\), \(s, t \in (0, \infty )\), which is very useful for calculating the explicit expressions of asymptotic covariances of parameter estimator in asymptotic normality framework (see Avram et al. (2011)).

Appendix B: Important Results on Non-Stationary Fisher-Snedecor Diffusion

B.1 Coupling, Ergodicity, and β-Mixing

This section collects the results on ergodic behavior of the FSD. A traditional tool for proving the ergodicity of a Markov process X is the coupling construction. A coupling for a pair of processes U and V is any two-component process Z = (Z(1), Z(2)) such that Z(1) has the same distribution as U and Z(2) has the same distribution as V. According to this terminology, for a Markov process X and every pair of probability distributions \(\mu , \nu \in \mathcal {P}\), where \(\mathcal {P}\) is the family of probability distributions on the Borel σ-algebra on the diffusion state space \(\mathbb {X}\), we consider two versions X(μ) and X(ν) of the process X with the initial distributions μ and ν, respectively. Any two-component process Z = (Z(1), Z(2)) which is a coupling for X(μ) and X(ν) is called (μ, ν)-coupling for the process X. According to Kulik and Leonenko (2013), the Markov process X admits an exponential ϕ-coupling if there exists an invariant measure π for this process and positive constants C and c such that, for every \(\mu \in \mathcal {P}\), there exists a (μ, π)-coupling Z = (Z(1), Z(2)) such that

$$ E\left[\phi({Z^{1}_{t}})+\phi({Z_{t}^{2}})\right]1\!\! \mathrm{I}_{{Z_{t}^{1}}\not={Z_{t}^{2}}}\leq C e^{-c t}{\int}_{\mathbb{X}}\phi d\mu, \quad t \geq 0. $$

In Kulik (2011) an exponential ϕ-coupling is introduced, and it was demonstrated that it is a convenient tool for studying convergence rates of Lp-semigroups, generated by a Markov process, and spectral properties of respective generators. In Kulik and Leonenko (2013) it is shown that this notion is also efficient for proving LLN and CLT for the FSD in non-stationary setting.

Next we provide the definition of the well-known β-mixing coefficient, also known as complete regularity or Kolmogorov’s coefficient. Generally, β-mixing coefficient of the process X is defined as

$$ \beta^{\mu}(t)=\sup\limits_{s\geq 0}E_{\mu}\sup\limits_{B\in \mathcal{F}^{X}_{\geq t+s}}|P_{\mu}(B|\mathcal{F}^{X}_{s})-P_{\mu}(B)|,\quad \mu\in \mathcal{P}, \quad t \geq 0, $$

where \(\mathcal {F}^X_{\geq r}\) for a given r ≥ 0 denotes the σ-algebra generated by the process X at times vr.

The state-dependent β-mixing coefficient is defined by

$$ \beta_{x}(t)=\sup\limits_{s\geq 0}E_{x}\sup\limits_{B\in \mathcal{F}^{X}_{\geq t+s}}|P_{x}(B|\mathcal{F}^{X}_{s})-P_{x}(B)|,\quad x\in \mathbb{X}, \quad t \geq 0, $$

where the initial distribution of X is the degenerate distribution μ = δx.

The stationary β-mixing coefficient is defined by

$$ \beta(t)=\sup\limits_{s\geq 0}E_{\pi}\sup\limits_{B\in \mathcal{F}^{X}_{\geq t+s}}|P_{\pi}(B|\mathcal{F}^{X}_{s})-P_{\pi}(B)|,\quad x\in \mathbb{X}, \quad t \geq 0, $$

where π denotes the (unique) invariant distribution of the process X. For more information about various types of mixing coefficients see e.g. Bradley (2005).

Finally, results concerning the ϕ-coupling and β-mixing for the non-stationary FSD are stated in the Theorem B.1. For the proof we refer to Kulik and Leonenko (2013), Theorem 3.1.

Theorem Appendix B.1

Let the functionϕbe defined as\(\phi = \phi _{\lozenge } + \phi _{\blacklozenge }\), whereϕ ≥ 1, \(\phi _{\lozenge }, \phi _{\blacklozenge } \in C^2(0,\infty )\), \(\phi _{\lozenge } = 0\)on\([2, \infty )\), \(\phi _{\blacklozenge } = 0\)on (0, 1], \(\phi _{\lozenge }(x) = x^{-\gamma }\) for xsmall enough and\(\phi _{\blacklozenge }(x) = x^{\delta }\)forxlarge enough with non-negativeγandδsatisfying\(\gamma < \displaystyle \frac {\alpha }{2}-1\)and\(\delta < \displaystyle \frac {\beta }{2}\). Then the following statements hold true.

  1. 1.

    FSD admits an exponentialϕ-coupling.

  2. 2.

    Finite-dimensional distributions of the FSD admit the following convergence rate in the weighted total variation norm with the weightϕ: for anym ≥ 1 and 0 ≤ t1 < … < tm

    $$ \|\mu_{t+t_{1}, \dots, t+t_{m}}-\pi_{t_{1}, \dots, t_{m}}\|_{\phi,var}\leq m C e^{-ct}{\int}_{\mathbb{X}}\phi d\mu,\quad \mu\in \mathcal{P}, \quad t\geq 0. $$

    Here\(\mu _{t_{1}, \ldots , t_{m}}\), 0 ≤ t1 < … < tm, m ≥ 1, denotes finite-dimensional distributions of the respective diffusion with the initial distributionμ, while\(\pi _{t_{1}, \ldots , t_{m}}\)denotes the corresponding finite-dimensional invariant distribution. ConstantsCandcare the same as in the bound (B.1) in the definition of an exponentialϕ-coupling.

  3. 3.

    FSD admits the following bound for theβ-mixing coefficient:

    $$ \beta^{\mu}(t)\leq C^{\prime}e^{-ct}{\int}_{\mathbb{X}}\phi d\mu, \quad \mu\in \mathcal{P}, \quad t \geq 0. $$

    Here the constantcis the same as in the bound (B.1), and\(C^{\prime }\)is a positive constant which can be given explicitly (see Kulik and Leonenko (2013), relation (5.15)).

Furthermore, from Eq. B.6 and Corollary 3.1 from Kulik and Leonenko (2013), the following bounds for the β-mixing coefficients can be obtained:

  • bound for the state-dependent β-mixing coefficient:

    $$ \beta_{x}(t)\leq C^{\prime}e^{-ct}\phi(x), \quad x\in \mathbb{X}, \quad t \geq 0 $$
  • bound for the stationary β-mixing coefficient:

    $$ \beta(t)\leq C^{\prime\prime}e^{-ct}, \quad t\geq 0, \quad C^{\prime\prime}:=C^{\prime}{\int}_{\mathbb{X}}\phi d\pi<+\infty. $$

B.2 Limit Theorems for Additive Functionals for Random Samples from the Fisher-Snedecor Diffusion

Here we state the LLN and CLT for additive functionals of the FSD X, separately for the discrete-time and the continuous-time observations. For the proofs we refer to the recent paper (Kulik and Leonenko 2013), Theorems 3.3 and 3.4. For clarity of the exposition, we introduce the notation \(X^{st} = \left (X^{st}_t, t \in (-\infty , \infty ) \right )\) for the stationary version of the FSD X, by which we understand the strictly stationary process such that for every m ≥ 1 and t1 < … < tm the distribution of the random vector \(X^{st}_{t_1}, \ldots , X^{st}_{t_m}\) is \(\pi _{0, t_2-t_1,\dots , t_m-t_1}\) (time-shift invariance of the finite-dimensional distributions). Heuristically, Xst is a solution of the SDE (1.1) defined on the whole time axis and starting at \((-\infty )\) from the invariant distribution π.

Theorem Appendix B.2

(Discrete-time case)

Let, for somer, k ≥ 1, a vector-valued function

$$ f =(f_{1}, \ldots, f_{k}) \colon \mathbb{X}^{r} \to \mathbb{R}^{k} $$

be such that for anyi = 1,…,k for some γi, δi such that γi < (α/2) − 1 and δi < β/2

$$ |f_{i}(x)| \leq C \sum\limits_{j=1}^{r}\left( x^{-\gamma_{i}}_{j}+x^{\delta_{i}}_{j}\right), \quad x = (x_{1}, \ldots, x_{r}) $$

with some constantC. Then the following statements hold true.

  1. 1.

    Law of large numbers

    For arbitrary initial distributionμ of X and arbitrary t1,…,tr ≥ 0,

    $$ {\frac{1}{n}} \sum\limits_{l=1}^{n}f\left( X_{t_{1}+l}, \ldots, X_{t_{r}+l}\right) \overset{P}{\to} a_{f}, $$

    with the asymptotic mean vector

    $$ a_{f} = Ef\left( X_{t_{1}}^{st}, \ldots, X_{t_{r}}^{st}\right). $$

    If, in addition, the initial distribution is such that for some positiveε

    $$ {\int}_{\mathbb{X}}\left( x^{-\gamma_{i}-\varepsilon}+x^{\delta_{i}+\varepsilon}\right)\mu(dx)<\infty,\quad i = 1, \ldots, k, $$

    then Eq. B.10holds true in the mean sense.

  2. 2.

    Central limit theorem

    Assume in addition that there existsε > 0 such that

    $$ E\left\|f\left( X_{t_{1}}^{st}, \dots, X_{t_{r}}^{st}\right)\right\|^{2+\varepsilon}<\infty. $$


    $$ {\frac{1}{\sqrt n}} \sum\limits_{l=1}^{n}\left( f\left( X_{t_{1}+l}, \dots, X_{t_{r}+l}\right)-a_{f}\right) \Rightarrow \mathcal{N}(0, \mathbf{\Sigma}), $$

    where the components of the asymptotic covariance matrix Σ are given as follows:

    $$ (\mathbf{\Sigma})_{i,j}= \sum\limits_{l=-\infty}^{\infty} \text{Cov} \left( f_{i}\left( X_{t_{1}+l}^{st}, \dots, X_{t_{r}+l}^{st}\right), f_{j}\left( X_{t_{1}}^{st}, \ldots, X_{t_{r}}^{st}\right))\right), \quad i,j =1, \dots, k. $$

Theorem Appendix B.3

(Continuous-time case)

Let the components of a vector-valued function\(f \colon \mathbb {X}^r \to \mathbb {R}^k\)satisfy (B.9) withγi, δisatisfyingγi < α/2 andδi < β/2 for everyi = 1,…,k. Then the following statements hold true.

  1. 1.

    Law of large numbers

    For arbitrary initial distributionμofX

    $$ {\frac{1}{T}}{{\int}_{0}^{T}}f\left( X_{t_{1}+s}, \dots, X_{t_{r}+s}\right) ds \overset{P}{\to} a_{f}. $$

    If, in addition, the initial distribution is such that for some positiveε

    $$ {\int}_{\mathbb{X}}\left( x^{-(\gamma_{i}-1)\vee 0-\varepsilon}+x^{\delta_{i}+\varepsilon}\right)\mu(dx)<\infty, \quad i = 1, \ldots, k, $$

    then Eq. B.14holds true in the mean sense.

  2. 2.

    Central limit theorem Assume in addition that

    $$ \gamma_{i} < {\frac{\alpha}{4}}+{\frac{1}{2}}, \quad \delta_{i} < {\frac{\beta}{4}}, \quad i = 1, \ldots, k. $$

    Then for arbitrary initial distributionμofX

    $$ {\frac{1}{\sqrt T}}{{\int}_{0}^{T}}\left( f\left( X_{t_{1}+s}, \dots, X_{t_{r}+s}\right)-a_{f} \right) ds \Rightarrow \mathcal{N}(0, \mathbf{\Sigma}), $$

    where the components of the asymptotic covariance matrix Σ are given as follows:

    $$ (\mathbf{\Sigma})_{i,j}={\int}_{-\infty}^{\infty} \text{Cov} \left( f_{i}\left( X_{t_{1}+s}^{st}, \dots, X_{t_{r}+s}^{st}\right), f_{j}\left( X_{t_{1}}^{st}, \ldots, X_{t_{r}}^{st}\right))\right) ds, \quad i, j = 1, \ldots, k. $$

Statements of Theorems B.2 and B.3 clearly show that the technique for calculation of asymptotic covariances (Σ)i, j in discrete and continuous-time setting rely on properties of the stationary FSD Xst. Therefore, we refer to Appendix A, where we give a short overview of the most important probabilistic properties of the stationary FSD in the canonical case.

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Kulik, A.M., Leonenko, N.N., Papić, I. et al. Parameter Estimation for Non-Stationary Fisher-Snedecor Diffusion. Methodol Comput Appl Probab 22, 1023–1061 (2020).

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  • Fisher-Snedecor diffusion
  • Generalized method of moments (GMM)
  • P-consistency
  • Asymptotic normality
  • Iterative technique for the calculation of the asymptotic covariance matrix

Mathematics Subject Classification (2010)

  • 33C05
  • 33C47
  • 35P10
  • 60G10
  • 60J60
  • 62M05
  • 62M15