Abstract
The problem of parameter estimation for the nonstationary ergodic diffusion with FisherSnedecor invariant distribution, to be called FisherSnedecor diffusion, is considered. We propose generalized method of moments (GMM) estimator of unknown parameter, based on continuoustime 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 pointwise covariations. The results are illustrated in a simulation study covering various starting distributions and parameter values.
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Acknowledgments
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 MTM201232674 and MTM201571839P (cofunded with Federal funds), of the DGI, MINECO, Spain.
N. Šuvak and I. Papić were supported by the scientific project UNIOSZUP 201831 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 tailindex estimation.
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Appendices
Appendix A: Spectral Representation of Transition Density of FisherSnedecor Diffusion
For deriving the closedform results in the framework of statistical analysis, e.g. calculation of asymptotic covariances in explicit form, the explicit expression for diffusion transition density
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:
The domain of the operator \(\mathcal {G}\) is the space of functions
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 λ_{n} are given by
and the corresponding eigenfunctions are orthogonal FisherSnedecor polynomials given by the Rodrigues formula
where \(\widetilde {P}_{n}(x)\) are nonnormalized polynomials and the normalization constant K_{n} can be expresses explicitly. The essential spectrum of the operator \((\mathcal {G})\) is \(\sigma _{ess}(\mathcal {G}) = [{\Lambda }, \infty )\), where
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
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
where λ > Λ is the spectral parameter,
and _{2}F_{1}(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 _{2}F_{1}(a, b; c; ⋅), solutions f_{1}(x,−λ) and f_{4}(x,−λ) are well defined on the whole state space of the FSD. Spectral representation of transition density p(x; x_{0}, 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
The discrete part of the spectral representation
is given in terms of the eigenvaluesλ_{n}given by Eq. A.2and the normalized FisherSnedecor polynomialsP_{n}(⋅) given by Eq. A.3. The continuous part of the spectral representation
is given in terms of the elementsλof the absolutely continuous spectrum of the operator\((\mathcal {G})\)given by Eq. A.4, solutionf_{1}(⋅,−λ) of the SturmLiouville equation\((\mathcal {G}f)(x) = \lambda f(x)\)forλ > Λ given by Eq. A.5and parameterk(λ) = −iΔ_{λ}, where Δ_{λ}is given in Eq. A.7.
Furthermore, the explicit expression for the corresponding twodimensional density is given by following expression:
where p_{d}(x; y, t) is given by Eq. A.9 and p_{c}(x; y, t) is given by Eq. A.10. Representation Eq. A.11 of twodimensional 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 NonStationary FisherSnedecor Diffusion
B.1 Coupling, Ergodicity, and βMixing
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 twocomponent 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 σalgebra 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 twocomponent 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 ϕcoupling 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
In Kulik (2011) an exponential ϕcoupling is introduced, and it was demonstrated that it is a convenient tool for studying convergence rates of L_{p}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 nonstationary setting.
Next we provide the definition of the wellknown βmixing coefficient, also known as complete regularity or Kolmogorov’s coefficient. Generally, βmixing coefficient of the process X is defined as
where \(\mathcal {F}^X_{\geq r}\) for a given r ≥ 0 denotes the σalgebra generated by the process X at times v ≥ r.
The statedependent βmixing coefficient is defined by
where the initial distribution of X is the degenerate distribution μ = δ_{x}.
The stationary βmixing coefficient is defined by
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 ϕcoupling and βmixing for the nonstationary 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 nonnegativeγandδsatisfying\(\gamma < \displaystyle \frac {\alpha }{2}1\)and\(\delta < \displaystyle \frac {\beta }{2}\). Then the following statements hold true.

1.
FSD admits an exponentialϕcoupling.

2.
Finitedimensional distributions of the FSD admit the following convergence rate in the weighted total variation norm with the weightϕ: for anym ≥ 1 and 0 ≤ t_{1} < … < t_{m}
$$ \\mu_{t+t_{1}, \dots, t+t_{m}}\pi_{t_{1}, \dots, t_{m}}\_{\phi,var}\leq m C e^{ct}{\int}_{\mathbb{X}}\phi d\mu,\quad \mu\in \mathcal{P}, \quad t\geq 0. $$(B.5)Here\(\mu _{t_{1}, \ldots , t_{m}}\), 0 ≤ t_{1} < … < t_{m}, m ≥ 1, denotes finitedimensional distributions of the respective diffusion with the initial distributionμ, while\(\pi _{t_{1}, \ldots , t_{m}}\)denotes the corresponding finitedimensional invariant distribution. ConstantsCandcare the same as in the bound (B.1) in the definition of an exponentialϕcoupling.

3.
FSD admits the following bound for theβmixing coefficient:
$$ \beta^{\mu}(t)\leq C^{\prime}e^{ct}{\int}_{\mathbb{X}}\phi d\mu, \quad \mu\in \mathcal{P}, \quad t \geq 0. $$(B.6)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 βmixing coefficients can be obtained:

bound for the statedependent βmixing coefficient:
$$ \beta_{x}(t)\leq C^{\prime}e^{ct}\phi(x), \quad x\in \mathbb{X}, \quad t \geq 0 $$(B.7) 
bound for the stationary βmixing 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.8)
B.2 Limit Theorems for Additive Functionals for Random Samples from the FisherSnedecor Diffusion
Here we state the LLN and CLT for additive functionals of the FSD X, separately for the discretetime and the continuoustime 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 t_{1} < … < t_{m} the distribution of the random vector \(X^{st}_{t_1}, \ldots , X^{st}_{t_m}\) is \(\pi _{0, t_2t_1,\dots , t_mt_1}\) (timeshift invariance of the finitedimensional distributions). Heuristically, X^{st} 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
(Discretetime case)
Let, for somer, k ≥ 1, a vectorvalued function
be such that for anyi = 1,…,k for some γ_{i}, δ_{i} such that γ_{i} < (α/2) − 1 and δ_{i} < β/2
with some constantC. Then the following statements hold true.

1.
Law of large numbers
For arbitrary initial distributionμ of X and arbitrary t_{1},…,t_{r} ≥ 0,
$$ {\frac{1}{n}} \sum\limits_{l=1}^{n}f\left( X_{t_{1}+l}, \ldots, X_{t_{r}+l}\right) \overset{P}{\to} a_{f}, $$(B.10)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
$$ {\int}_{\mathbb{X}}\left( x^{\gamma_{i}\varepsilon}+x^{\delta_{i}+\varepsilon}\right)\mu(dx)<\infty,\quad i = 1, \ldots, k, $$(B.11)then Eq. B.10holds true in the mean sense.

2.
Central limit theorem
Assume in addition that there existsε > 0 such that
$$ E\left\f\left( X_{t_{1}}^{st}, \dots, X_{t_{r}}^{st}\right)\right\^{2+\varepsilon}<\infty. $$(B.12)Then
$$ {\frac{1}{\sqrt n}} \sum\limits_{l=1}^{n}\left( f\left( X_{t_{1}+l}, \dots, X_{t_{r}+l}\right)a_{f}\right) \Rightarrow \mathcal{N}(0, \mathbf{\Sigma}), $$(B.13)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
(Continuoustime case)
Let the components of a vectorvalued function\(f \colon \mathbb {X}^r \to \mathbb {R}^k\)satisfy (B.9) withγ_{i}, δ_{i}satisfyingγ_{i} < α/2 andδ_{i} < β/2 for everyi = 1,…,k. Then the following statements hold true.

1.
Law of large numbers
For arbitrary initial distributionμofX
$$ {\frac{1}{T}}{{\int}_{0}^{T}}f\left( X_{t_{1}+s}, \dots, X_{t_{r}+s}\right) ds \overset{P}{\to} a_{f}. $$(B.14)If, in addition, the initial distribution is such that for some positiveε
$$ {\int}_{\mathbb{X}}\left( x^{(\gamma_{i}1)\vee 0\varepsilon}+x^{\delta_{i}+\varepsilon}\right)\mu(dx)<\infty, \quad i = 1, \ldots, k, $$(B.15)then Eq. B.14holds true in the mean sense.

2.
Central limit theorem Assume in addition that
$$ \gamma_{i} < {\frac{\alpha}{4}}+{\frac{1}{2}}, \quad \delta_{i} < {\frac{\beta}{4}}, \quad i = 1, \ldots, k. $$(B.16)Then for arbitrary initial distributionμofX
$$ {\frac{1}{\sqrt T}}{{\int}_{0}^{T}}\left( f\left( X_{t_{1}+s}, \dots, X_{t_{r}+s}\right)a_{f} \right) ds \Rightarrow \mathcal{N}(0, \mathbf{\Sigma}), $$(B.17)where the components of the asymptotic covariance matrix Σ are given as follows:
$$ (\mathbf{\Sigma})_{i,j}={\int}_{\infty}^{\infty} \text{Cov} \left( f_{i}\left( X_{t_{1}+s}^{st}, \dots, X_{t_{r}+s}^{st}\right), f_{j}\left( X_{t_{1}}^{st}, \ldots, X_{t_{r}}^{st}\right))\right) ds, \quad i, j = 1, \ldots, k. $$(B.18)
Statements of Theorems B.2 and B.3 clearly show that the technique for calculation of asymptotic covariances (Σ)_{i, j} in discrete and continuoustime setting rely on properties of the stationary FSD X^{st}. 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 NonStationary FisherSnedecor Diffusion. Methodol Comput Appl Probab 22, 1023–1061 (2020). https://doi.org/10.1007/s1100901909755z
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Keywords
 FisherSnedecor diffusion
 Generalized method of moments (GMM)
 Pconsistency
 Asymptotic normality
 Iterative technique for the calculation of the asymptotic covariance matrix
Mathematics Subject Classification (2010)
 33C05
 33C47
 35P10
 60G10
 60J60
 62M05
 62M15