# zbMATH — the first resource for mathematics

Conditional least squares estimation in nonstationary nonlinear stochastic regression models. (English) Zbl 1181.62133
Summary: Let $$\{Z_n\}$$ be a real nonstationary stochastic process such that
$E(Z_n\,|\,{\mathcal F}_{n-1})\overset{\text{a.s.}}<\infty\text{ and } E(Z^2_n\,|\,{\mathcal F}_{n-1})\overset{\text{a.s.}}<\infty,$
where $$\{{\mathcal F}_n\}$$ is an increasing sequence of $$\sigma$$-algebras. Assuming that
$E(Z_n\,|\,{\mathcal F}_{n-1})=g_n(\theta_0,\nu_0)=g_n^{(1)}(\theta_0)+g_n^{(2)}(\theta_0,\nu_0),$
$\theta_0\in\mathbb R^p,\quad p<\infty,\;\nu_0\in\mathbb R^q\text{ and }q\leq\infty,$
we study the asymptotic properties of $$\widehat\theta_n:=\text{argmin}_{\theta}\sum^n_{k=1}(Z_k-g_k(\theta,\widehat\nu))^2\lambda^{-1}_k$$, where $$\lambda_k$$ is $${\mathcal F}_{k-1}$$-measurable, $$\widehat\nu=\{\widehat\nu_k\}$$ is a sequence of estimations of $$\nu_0$$, $$g_n(\theta,\widehat\nu)$$ is Lipschitz in $$\theta$$ and $$g_n^{(2)}(\theta_0,\widehat\nu)-g_n^{(2)}(\theta,\widehat\nu)$$ is asymptotically negligible relative to $$g_n^{(1)}(\theta_0)-g_n^{(1)}(\theta)$$.
We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of $$\{\widehat\theta_n\}$$ in the linear model. For that, we prove a strong law of large numbers for a class of submartingales. Again using this strong law, we derive the general conditions leading to the asymptotic distribution of $$\{\widehat\theta_n\}$$. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60F15 Strong limit theorems 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators 62M05 Markov processes: estimation; hidden Markov models 62M09 Non-Markovian processes: estimation 60G46 Martingales and classical analysis 62E20 Asymptotic distribution theory in statistics
Full Text:
##### References:
  Anderson, T. W. and Taylor, J. B. (1979). Strong consistency of least squares estimates in dynamic models. Ann. Statist. 7 484-489. · Zbl 0407.62040 · doi:10.1214/aos/1176344670  Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201  Dion, J.-P. (1974). Estimation of the mean and the initial probabilities of a branching process. J. Appl. Probab. 11 687-694. JSTOR: · Zbl 0311.62054 · doi:10.2307/3212552 · links.jstor.org  Dacunha-Castelle, D. and Duflo, M. (1993). Probabilités et Statistiques. 2. Problèmes à Temps Mobile . Masson, Paris. · Zbl 0535.62004  Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987-1007. JSTOR: · Zbl 0491.62099 · doi:10.2307/1912773 · links.jstor.org  Feigin, P. D. (1977). A note on maximum likelihood estimation for simple branching processes. Austral. J. Statist. 19 152-154. · Zbl 0406.62061 · doi:10.1111/j.1467-842X.1977.tb01282.x  Godambe, V. P. and Heyde, C. C. (1987). Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55 231-244. JSTOR: · Zbl 0671.62007 · doi:10.2307/1403403 · links.jstor.org  Guttorp, P. (1991). Statistical Inference for Branching Processes . Wiley, New York. · Zbl 0778.62077  Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application . Wiley, New York. · Zbl 0462.60045  Hall, P. and Yao, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71 285-317. JSTOR: · Zbl 1136.62368 · doi:10.1111/1468-0262.00396 · links.jstor.org  Harris, T. E. (1948). Branching processes. Ann. Math. Statist. 19 474-494. · Zbl 0041.45603 · doi:10.1214/aoms/1177730146  Hutton, J. E. and Nelson, P. I. (1986). Quasilikelihood estimation for semimartingales. Stochastic Process. Appl. 22 245-257. · Zbl 0616.62113 · doi:10.1016/0304-4149(86)90004-9  Jacob, C. and Peccoud, J. (1998). Estimation of the parameters of a branching process from migrating binomial observations. Adv. in Appl. Probab. 30 948-967. · Zbl 0923.62087 · doi:10.1239/aap/1035228202  Jacob, C. (2008). Conditional least squares estimation in nonlinear and nonstationary stochastic regression models: Asymptotic properties and examples. Technical Report UR341, INRA, Jouy-en-Josas, France.  Lalam, N. and Jacob, C. (2004). Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process. Adv. in Appl. Probab. 36 582-601. · Zbl 1046.62082 · doi:10.1239/aap/1086957586  Jacob, C., Lalam, N. and Yanev, N. (2005). Statistical inference for processes depending on exogenous inputs and application in regenerative processes. Pliska Stud. Math. Bulgar. 17 109-136.  Jagers, P. and Klebaner, F. C. (2003). Random variation and concentration effects in PCR. J. Theoret. Biol. 224 299-304. · doi:10.1016/S0022-5193(03)00166-8  Jennrich, R. I. (1969). Asymptotic properties of nonlinear least squares estimators. Ann. Math. Statist. 40 633-643. · Zbl 0193.47201 · doi:10.1214/aoms/1177697731  Klebaner, F. C. (1984). On population-size dependent branching processes. Adv. in Appl. Probab. 16 30-55. JSTOR: · Zbl 0528.60080 · doi:10.2307/1427223 · links.jstor.org  Lai, T. L., Robbins, H. and Wei, C. Z. (1979). Strong consistency of estimates in multiple regression. II. J. Multivariate Anal. 9 343-361. · Zbl 0416.62051 · doi:10.1016/0047-259X(79)90093-9  Lai, T. L. and Wei, C. Z. (1982). Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist. 10 154-166. · Zbl 0649.62060 · doi:10.1214/aos/1176345697  Lai, T. L. and Wei, C. Z. (1983). Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters, J. Multivariate Anal. 13 1-23. · Zbl 0509.62081 · doi:10.1016/0047-259X(83)90002-7  Lai, T. L. (1994). Asymptotic properties of nonlinear least squares estimates in stochastic regression models. Ann. Statist. 22 1917-1930. · Zbl 0824.62054 · doi:10.1214/aos/1176325764  Lalam, N., Jacob, C. and Jagers, P. (2004). Modelling of the PCR amplification process by a size-dependent branching process and estimation of the efficiency. Adv. in Appl. Probab. 36 602-615. · Zbl 1051.92021 · doi:10.1239/aap/1086957587  Maaouia, F. and Touati, A. (2005). Identification of multitype branching processes. Ann. Statist. 33 2655-2694. · Zbl 1099.62096 · doi:10.1214/009053605000000561  Ngatchou-Wandji, J. (2008). Estimation in a class of nonlinear heteroscedastic time-series models. Electron. J. Stat. 2 40-62. · Zbl 1135.62369 · doi:10.1214/07-EJS157  Peccoud, J. and Jacob, C. (1996). Theoretical uncertainty of measurements using quantitative polymerase chain reaction. Biophysical J. 71 101-108.  Rahimov, I. (1995). Random Sums and Branching Stochastic Processes. Lecture Notes in Statistics 96 195. Springer, New York. · Zbl 0812.60003  Rebolledo, R. (1980). Central limit theorems for local martingales. Z. Wahrsch. Verw. Gebiete 51 269-286. · Zbl 0432.60027 · doi:10.1007/BF00587353  Schnell, S. and Mendoza, C. (1997). Enzymological considerations for a theoretical description of the quantitative competitive polymerase chain reaction. J. Theoret. Biol. 184 433-440.  Skouras, K. (2000). Strong consistency in nonlinear stochastic regression models. Ann. Statist. 28 871-879. · Zbl 1105.62355 · doi:10.1214/aos/1015952002 · euclid:aos/1015952002  Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61 439-447. JSTOR: · Zbl 0292.62050 · links.jstor.org  Wei, C. Z. (1985). Asymptotic properties of least-squares estimates in stochastic regression models. Ann. Statist. 13 1498-1508. · Zbl 0582.62062 · doi:10.1214/aos/1176349751  Wu, C. F. (1981). Asymptotic theory of nonlinear least squares estimation. Ann. Statist. 9 501-513. · Zbl 0475.62050 · doi:10.1214/aos/1176345455  Yanev, N. (2009). Statistical inference for branching processes. In Records and Branching Processes (M. Ahsanullah and G. Yanev, eds.). Nova Science Publishers, New York.  Yao, J. F. (2000). On least squares estimation for stable nonlinear AR processes. Ann. Inst. Statist. Math. 52 316-331. · Zbl 0959.62077 · doi:10.1023/A:1004117906532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.