First we study estimation of the drift parameter in the fractional Ornstein-Uhlenbeck process whose marginal distribution is Student -distribution. We obtain Spearman’s correlation based estimator, quantile estimator and Brownian excursion based estimator of the drift parameter. Then we study method of moments estimator and quantile estimator in fractional inverse Gaussian and fractional gamma Ornstein-Uhlenbeck processes.
ArayaH.BahamondeN.TorresS., & ViensF. (2019). Donsker type theorem for fractional Poisson process. Statist. Probab. Letters, 150, 1-8.
2.
Barndorff-NeilsenO.E., & ShephardN. (1998). Processes of normal inverse Gaussian type. Finance and Stochastics, 2, 41-68.
3.
Barndorff-NielsenO.E., & ShephardN. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society, Series B, 63, 167-241.
4.
BishwalJ.P.N. (2006). Rates of weak convergence of the approximate minimum contrast estimators for the discretely observed Ornstein-Uhlenbeck process. Statistics and Probability Letters, 76(13), 1397-1409.
5.
BishwalJ.P.N. (2008). Parameter Estimation in Stochastic Differential Equations, Lecture Notes in Mathematics, 1923, Springer-Verlag.
6.
BishwalJ.P.N. (2010). Uniform rate of weak convergence for the minimum contrast estimator in the Ornstein-Uhlenbeck process. Methodology and Computing in Applied Probability, 12(3), 323-334.
7.
BishwalJ.P.N. (2011a). Sufficiency and Rao-Blackwellization of Vasicek model. Theory of Stochastic Processes, 17(1), 12-15.
8.
BishwalJ.P.N. (2011b). Maximum quasi-likelihood estimation in fractional Levy stochastic volatility model. Journal of Mathematical Finance, 1(3), 12-15.
9.
BishwalJ.P.N. (2011c). Berry-Esseen inequalities for the discretely observed Ornstein-Uhlenbeck-Gamma process. Markov Processes and Related Fields, 17(1), 119-150.
10.
BishwalJ.P.N. (2011d). Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling. Fractional Calculus and Applied Analysis, 14(3), 375-410.
11.
BishwalJ.P.N. (2011e). Some new estimators of integrated volatility. American Open Journal of Statistics, 1(2), 74-80.
12.
BishwalJ.P.N. (2011f). Financial extremes: A short review. Advances and Applications in Statistics, 25(1), 1-14.
13.
BishwalJ.P.N. (2022a). Quasi-likelihood Estimation in fractional Levy SPDEs from Poisson sampling. European Journal of Mathematical Analysis, 2(2), 1-16.
BishwalJ.P.N. (2023). Parameter estimation for SPDEs driven by cylindrical stable processes. Eur. J. Math. Anal, 3(4), 1-25. doi: 10.28924/ada/ma.3.4.
16.
BlumenthalR.M., & GetoorR.K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech, 10, 493-516.
17.
BrockwellP.J.DavisR.A., & YangY. (2007). Estimation of non-negative Levy driven Ornstein-Uhlenbeck processes. Journal of Applied Probability, 44, 977-989.
18.
BorovkovK., & DecrouezG. (2013). Ornstein-Uhlenbeck processes with heavy distribution tails. Theory of Probability and its Applications, 57(3), 396-418.
19.
DavisR.A., & McCormickW.P. (1989). Estimation for first order autoregressive processes with positive or bounded innovations, Stochastic Process. Appl, 31, 237-250.
20.
DavisR.A.KnightK., & LiuJ. (1992). M-estimation for autoregressions with infinite variance. Stochastic Processes and their Applications, 40, 145-180.
21.
DavisR.A., & DunsmuirW.T.M. (1997). Least absolute deviation estimation for regression with ARMA errors. Journal of Theoretical Probability, 10, 481-497.
22.
DavisR.A., & McCormickW.P. (1989). Estimation for first order autoregressive processes with positive or bounded innovations. Stochastic Process. Appl, 31, 237-250.
23.
EngleR., & RussellJ. (1998). The autoregressive conditional duration model: A new model for irregularly spaced data. Econometrica, 66, 1127.1162.
24.
GaverD.P., & LewisP.A.W. (1980). First order autoregressive gamma sequences and point processes. Adv. Appl. Probab, 12, 727-745.
25.
GlassermanP. (2004). Monte Carlo Methods in Financial Engineering, Springer, New York.
JongbloedG.van der MeulenF.H., & van der VaartA.W. (2005). Nonparametric inference for Levy driven Ornstein-Uhlenbeck processes. Bernoulli, 11(5), 759-791.
30.
KleptsynaM.L., & Le BretonA. (2002). Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Statistical Inference for Stochastic Processes, 5(3), 229-248.
31.
KnightK. (1998). Limit distributions for L1 regression estimators under general conditions. Annals of Statistics, 26, 755-770.
32.
KoenkerR. (2005). Quantile Regression, Cambridge University Press, Cambridge.
33.
KolmogorovA.N. (1940). Wiener skewline and other interesting curves in Hilbert space. Doklady Akad. Nauk, 26, 115-118.
34.
LevyP. (1948). Processus stochastiques et movement Brownien, Paris.
35.
LingS. (2005). Self-weighted least absolute deviation estimation for infinite variance autoregressive models. Journal of Royal Statistical Society, Series B, Statistiacl Methodology, 67, 381-393.
36.
LeonenkoN.N.PapicI.SikorskiiA., & SuvakN. (2020). Approximation of heavy-tailed fractional Pearson diffusions in Skorohod topology. Journal of Mathematical Analysis and Applications, 486, 1-22.
MandelbrotB., & Van NessJ.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review, 10, 422-437.
39.
MasudaH. (2005). Classical method of moments for partially and discretely observed ergodic models. Statistical Inference for Stochastic Processes, 8, 25-50.
40.
MasudaH. (2010). Asymptotic mixed normality in the self-weighted LAD estimation of Levy OU process. Electronic Journal of Statistics, 4, 525-565.
41.
NeilsenB., & ShephardN. (2003). Likelihood analysis of a first order autoregressive model with exponential innovations. J. Time Series Anal, 24(3), 337-344.
42.
NorrosI.ValkeilaE., & VirtamoJ. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion. Bernoulli, 5, 571-587.
43.
PortnoyS., & KoenkerR. (1997). The Gaussian hare and Laplacian tortoise: Computability of squared-error versus absolute-error estimators. Statistical Science, 12, 279-300.
44.
RosinskiJ. (2000). Stochastic series representations of inverse Gaussian and other exponentially tempered stable processes, Research Report 42, Center for Mathematics, Physics and Stochastics, University of Aarhus, Aarhus.
45.
TudorC.A., & ViensF. (2007). Statistical aspects of the fractional stochastic calculus. Annals of Statistics, 35(3), 1183-1212.
46.
Van der VaartA.W. (2000). Asymptotic Statistics, Cambridge University Press, Cambridge.
47.
WolfeA.J. (1982). On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn. Stochastic Process Appl, 12, 301-312.
