arXiv subjects

Ngai Hang Chan

Publications and source records attributed to Ngai Hang Chan.

10 recordsLinked to original sources

arXiv2026

Generalised Linear Models Driven by Latent Processes: Asymptotic Theory and Applications

This paper introduces a class of generalised linear models (GLMs) driven by latent processes for modelling count, real-valued, binary, and positive continuous time series. Extending earlier latent-process regression frameworks based on Poisson or one-parameter exponential family assumptions, we allow the conditional distribution of the response to belong to a bi-parameter exponential family, with the latent process entering the conditional mean multiplicatively. This formulation substantially broadens the scope of latent-process GLMs, for instance, it naturally accommodates gamma responses for positive continuous data, enables estimation of an unknown dispersion parameter via method of moments, and avoids restrictive conditions on link functions that arise under existing formulations. We establish the asymptotic normality of the GLM estimators obtained from the GLM likelihood that ignores the latent process, and we derive the correct information matrix for valid inference. In addition, we provide a principled approach to prediction and forecasting in GLMs driven by latent processes, a topic not previously addressed in the literature. We present two real data applications on measles infections in North Rhine-Westphalia (Germany) and paleoclimatic glacial varves, which highlight the practical advantages and enhanced flexibility of the proposed modelling framework.

stat.ME↗

arXiv2021

Nearly Unstable Integer-Valued ARCH Process and Unit Root Testing

This paper introduces a Nearly Unstable INteger-valued AutoRegressive Conditional Heteroskedasticity (NU-INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU-INARCH process endowed with a Skorohod topology weakly converges to a Cox-Ingersoll-Ross diffusion. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to non-stationarity. A unit root test is proposed and its Type-I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID-19 in the United Kingdom.

stat.ME↗

arXiv2013

Marked empirical processes for non-stationary time series

Consider a first-order autoregressive process $X_i=βX_{i-1}+\varepsilon_i,$ where $\varepsilon_i=G(η_i,η_{i-1},\ldots)$ and $η_i,i\in\mathbb{Z}$ are i.i.d. random variables. Motivated by two important issues for the inference of this model, namely, the quantile inference for $H_0: β=1$, and the goodness-of-fit for the unit root model, the notion of the marked empirical process $α_n(x)=\frac{1}{n}\sum_{i=1}^ng(X_i/a_n)I(\varepsilon_i\leq x),x\in\mathbb{R}$ is investigated in this paper. Herein, $g(\cdot)$ is a continuous function on $\mathbb{R}$ and $\{a_n\}$ is a sequence of self-normalizing constants. As the innovation $\{\varepsilon_i\}$ is usually not observable, the residual marked empirical process $\hat α_n(x)=\frac{1}{n}\sum_{i=1}^ng(X_i/a_n)I(\hat{\varepsilon}_iłeq x),x\in\mathbb{R},$ is considered instead, where $\hat{\varepsilon}_i=X_i-\hatβX_{i-1}$ and $\hatβ$ is a consistent estimate of $β.$ In particular, via the martingale decomposition of stationary process and the stochastic integral result of Jakubowski (Ann. Probab. 24 (1996) 2141-2153), the limit distributions of $α_n(x)$ and $\hatα_n(x)$ are established when $\{\varepsilon_i\}$ is a short-memory process. Furthermore, by virtue of the results of Wu (Bernoulli 95 (2003) 809-831) and Ho and Hsing (Ann. Statist. 24 (1996) 992-1024) of empirical process and the integral result of Mikosch and Norvaiša (Bernoulli 6 (2000) 401-434) and Young (Acta Math. 67 (1936) 251-282), the limit distributions of $α_n(x)$ and $\hatα_n(x)$ are also derived when $\{\varepsilon_i\}$ is a long-memory process.

math.ST↗

arXiv2013

Moment bounds and mean squared prediction errors of long-memory time series

A moment bound for the normalized conditional-sum-of-squares (CSS) estimate of a general autoregressive fractionally integrated moving average (ARFIMA) model with an arbitrary unknown memory parameter is derived in this paper. To achieve this goal, a uniform moment bound for the inverse of the normalized objective function is established. An important application of these results is to establish asymptotic expressions for the one-step and multi-step mean squared prediction errors (MSPE) of the CSS predictor. These asymptotic expressions not only explicitly demonstrate how the multi-step MSPE of the CSS predictor manifests with the model complexity and the dependent structure, but also offer means to compare the performance of the CSS predictor with the least squares (LS) predictor for integrated autoregressive models. It turns out that the CSS predictor can gain substantial advantage over the LS predictor when the integration order is high. Numerical findings are also conducted to illustrate the theoretical results.

math.ST↗

arXiv2012

Uniform moment bounds of Fisher's information with applications to time series

In this paper, a uniform (over some parameter space) moment bound for the inverse of Fisher's information matrix is established. This result is then applied to develop moment bounds for the normalized least squares estimate in (nonlinear) stochastic regression models. The usefulness of these results is illustrated using time series models. In particular, an asymptotic expression for the mean squared prediction error of the least squares predictor in autoregressive moving average models is obtained. This asymptotic expression provides a solid theoretical foundation for some model selection criteria.

math.ST↗

arXiv2012

Residual empirical processes for long and short memory time series

This paper studies the residual empirical process of long- and short-memory time series regression models and establishes its uniform expansion under a general framework. The results are applied to the stochastic regression models and unstable autoregressive models. For the long-memory noise, it is shown that the limit distribution of the Kolmogorov-Smirnov test statistic studied in Ho and Hsing [Ann. Statist. 24 (1996) 992-1024] does not hold when the stochastic regression model includes an unknown intercept or when the characteristic polynomial of the unstable autoregressive model has a unit root. To this end, two new statistics are proposed to test for the distribution of the long-memory noises of stochastic regression models and unstable autoregressive models. (With Correction.)

math.ST↗

arXiv2010

On Parameter Estimation of Threshold Autoregressive Models

This paper studies the threshold estimation of a TAR model when the underlying threshold parameter is a random variable. It is shown that the Bayesian estimator is consistent and its limit distribution is expressed in terms of a limit likelihood ratio. Furthermore, convergence of moments of the estimators is also established. The limit distribution can be computed via explicit simulations from which testing and inference for the threshold parameter can be conducted. The obtained results are illustrated with numerical simulations.

math.ST↗

arXiv2009

Integrated functionals of normal and fractional processes

Consider $Z^f_t(u)=\int_0^{tu}f(N_s) ds$, $t>0$, $u\in[0,1]$, where $N=(N_t)_{t\in\mathbb{R}}$ is a normal process and $f$ is a measurable real-valued function satisfying $Ef(N_0)^2<\infty$ and $Ef(N_0)=0$. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191--216] showed that $Z_t^f/t^{1/2}$ converges in distribution to a multiple of standard Brownian motion as $t\to\infty$. If the dependence is sufficiently strong, then $Z_t/(EZ_t(1)^2)^{1/2}$ converges in distribution to a higher order Hermite process as $t\to\infty$ by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53--83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices $H\in(0,1)$, we give their limiting distributions. In this context, we show that the known results are only applicable to $H<3/4$ and $H>3/4$, respectively, whereas our result covers $H=3/4$.

math.PR↗

arXiv2007

Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence

This paper considers the effect of least squares procedures for nearly unstable linear time series with strongly dependent innovations. Under a general framework and appropriate scaling, it is shown that ordinary least squares procedures converge to functionals of fractional Ornstein--Uhlenbeck processes. We use fractional integrated noise as an example to illustrate the important ideas. In this case, the functionals bear only formal analogy to those in the classical framework with uncorrelated innovations, with Wiener processes being replaced by fractional Brownian motions. It is also shown that limit theorems for the functionals involve nonstandard scaling and nonstandard limiting distributions. Results of this paper shed light on the asymptotic behavior of nearly unstable long-memory processes.

math.ST↗

arXiv2007

Fractional constant elasticity of variance model

This paper develops a European option pricing formula for fractional market models. Although there exist option pricing results for a fractional Black-Scholes model, they are established without accounting for stochastic volatility. In this paper, a fractional version of the Constant Elasticity of Variance (CEV) model is developed. European option pricing formula similar to that of the classical CEV model is obtained and a volatility skew pattern is revealed.

math.ST↗