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We propose a new method to construct confidence intervals for quantities that are associated with a stationary time series, which avoids direct estimation of the asymptotic variances. Unlike the existing tuning-parameter-dependent approaches, our method has the attractive convenience of being free of any user-chosen number or smoothing parameter. The interval is constructed on the basis of an asymptotically distribution-free self-normalized statistic, in which the normalizing matrix is computed by using recursive estimates. Under mild conditions, we establish the theoretical validity of our method for a broad class of statistics that are functionals of the empirical distribution of fixed or growing dimension. From a practical point of view, our method is conceptually simple, easy to implement and can be readily used by the practitioner. Monte Carlo simulations are conducted to compare the finite sample performance of the new method with those delivered by the normal approximation and the block bootstrap approach.

In this paper, we investigate the recursive L1 estimator of the conditional mode when the input variable takes values in a pseudo-metric space. The new proposed estimator is constructed under an ergodicity assumption, which provides a robust alternative to the standard mixing processes in various practical settings. The particular interest of this contribution arises from the difficulty in incorporating the mathematical properties of a functional mixing process. In contrast, ergodicity is characterized by the Kolmogorov–Sinai entropy, which measures the dynamics, the sparsity, and the microscopic fluctuations of the functional process. Using an observation sampled from ergodic functional time series (fts), we establish the asymptotic properties of this estimator. In particular, we derive its convergence rate and show Borel–Cantelli (BC) consistency. The general expression for the convergence rate is then specialized to several notable scenarios, including the independence case, the classical kernel method, and the vector-valued case. Finally, numerical experiments on both simulated and real-world datasets demonstrate the superiority of the L1-recursive estimator compared to existing competitors.

This paper considers the Recursive Kernel Estimator (RKE) of the expectile-based conditional shortfall. The estimator is constructed under a functional structure based on the ergodicity assumption. More preciously, we assume that the input-variable is valued in a pseudo-metric space, output-variable is scalar and both are sampled from ergodic functional time series data. We establish the complete convergence rate of the RKE-estimator of the considered functional shortfall model using standard assumptions. We point out that the ergodicity assumption constitutes a relevant alternative structure to the mixing time series dependency. Thus, the results of this paper allows to cover a large class of functional time series for which the mixing assumption is failed to check. Moreover, the obtained results is established in a general way, allowing to particularize this convergence rate for many special situations including the kernel method, the independence case and the multivariate case. Finally, a simulation study is carried out to illustrate the finite sample performance of the RKE-estimator. In order to examine the feasibility of the recursive estimator in practice we consider a real data example based on financial time series data.

Identification of the Wiener--Hammerstein system consisting of a linear subsystem in a cascade with a static nonlinearity $f(\\cdot)$ followed by another linear subsystem with internal noises is considered. On the basis of input and noisy output the impulse responses of the two linear subsystems are estimated by stochastic approximation (SA) algorithms, and the nonlinear function is also estimated by SA algorithms but with kernel functions. The system input is taken to be a sequence of independent and identically distributed (iid) Gaussian random variables $u_{k}\\in$ $\\mathcal{N}(0,\\vartheta^2)$ with $\\vartheta>0$. For convergence of the proposed algorithms, the properties of martingale difference sequences (mds) and $\\alpha$-mixings play an important role. The estimates for coefficients of the linear subsystems as well as for values of the nonlinear function are proved to converge to the true values with probability one. Three numerical examples with nonlinearities possessing different properties are given, justifying the theoretical analysis. [PUBLICATION ABSTRACT]

This article introduces a class of generalized duration models and shows that the autoregressive conditional duration (ACD) models and stochastic conditional duration (SCD) models discussed in the literature are special cases. The martingale estimating functions approach, which provides a convenient framework for deriving optimal inference for nonlinear time series models, is described. It is shown that when the first two conditional moments are functions of the same parameter, and information about higher order conditional moments of the observed duration process become available, combined estimating functions are optimal and are more informative than component estimating functions. The combined estimating functions approach is illustrated on three classes of generalized duration models, viz., multiplicative random coefficient ACD models, random coefficient models with ACD errors, and log-SCD models. Recursive estimation of model parameters based on combined estimating functions provides a mechanism for fast estimation in the general case, and is illustrated using simulated data sets.

In the non-Gaussian clutter modeled as independent and identically distributed spherically invariant random vectors, three estimators of sample covariance matrix （SCM）, normalized sample covariance matrix （NSCM） and the corresponding recursive estimator （NSCM-RE） are analyzed. Based on the uniform theorem, three corresponding adaptive normalized matched filters （ANMF） are evaluated from the standpoint of constant false alarm rate （CFAR） property. The theoretical results demonstrate that the SCM-ANMF is only CFAR to the normalized clutter covariance matrix （NCCM）; the NSCM-ANMF is only CFAR to the clutter power level; and the NSCM-RE-ANMF with finite number of iterations is still not CFAR to the NCCM. To ensure CFAR property of ANMF, an adaptive estimator （AE） is devised. Moreover, with AE as the initialization matrix for the iterations, the AE-RE is proposed. With finite number of iterations, the corresponding AE-REANMF guarantees CFAR property to both of the NCCM and the clutter power level. Finally, the performance assessment conducted by Monte Carlo simulation confirms the effectiveness of the proposed detectors.

In this work we propose a new filtering approach for linear discrete time non-Gaussian systems that generalizes a previous result concerning quadratic filtering [A. De Santis, A. Germani, and M. Raimondi, IEEE Trans. Automat. Control, 40 (1995) pp. 1274-1278]. A recursive $\\nu $th-order polynomial estimate of finite memory $\\Delta $ is achieved by defining a suitable extended state which allows one to solve the filtering problem via the classical Kalman linear scheme. The resulting estimate will be the mean square optimal one among those estimators that take into account $\\nu $-polynomials of the last $\\Delta $ observations. Numerical simulations show the effectiveness of the proposed method.

For i = 1,2,..., let Xiand Yibe Rd-valued (d ≥ 1 integer) and R-valued, respectively, random variables, and let {(Xi, Yi)}, i ≥ 1, be a strictly stationary and α-mixing stochastic process. Set m(x) = E(Y1∣ X1= x), x ∈ Rd, and let m̂n(x) be a certain recursive kernel estimate of m(x). Under suitable regularity conditions and as n → ∞, it is shown that m̂n(x), properly normalized, is asymptotically normal with mean 0 and a specified variance. This result is established, first under almost sure boundedness of the Yi's, and then by replacing boundedness by continuity of certain truncated moments. It is also shown that, for distinct points x1,...,xNin Rd(N ≥ 2 integer), the joint distribution of the random vector, (m̂n(x1),...,m̂n(xN)), properly normalized, is asymptotically N-dimensional normal with mean vector 0 and a specified covariance function.

This paper considers the parameter estimation problem of nonlinear models, which are related to the impulse or step response functions of linear time-invariant (LTI) dynamical systems, based on the response data. In terms of the nonlinear characteristic of the models, the nonlinear dynamical optimization scheme is adopted for obtaining the system parameter estimates. By constructing a gradient criterion function, a gradient recursion algorithm is derived. In order to overcome the difficulty of determining the step-size in the gradient recursion algorithm, a trying method and a numerical approach are proposed to achieve the step-size. On this basis, a stochastic gradient estimation method is presented by using a recursive step-size. Furthermore, a multi-innovation stochastic gradient method is deduced for enhancing the estimation accuracy by using the dynamical window data. Finally, a dynamical length stochastic gradient estimation technique is offered to obtain more accurate parameter estimates by using dynamical length measured data from the step response. The examples are provided to examine the algorithm performance and the simulation results indicate that the presented approaches are effective.

Radiotherapy is increasingly used to treat oligometastatic patients. We sought to identify prognostic criteria in oligometastatic patients undergoing definitive hypofractionated image-guided radiotherapy (HIGRT). Exclusively extracranial oligometastatic patients treated with HIGRT were pooled. Characteristics including age, sex, primary tumor type, interval to metastatic diagnosis, number of treated metastases and organs, metastatic site, prior systemic therapy for primary tumor treatment, prior definitive metastasis-directed therapy, and systemic therapy for metastasis associated with overall survival (OS), progression-free survival (PFS), and treated metastasis control (TMC) were assessed by the Cox proportional hazards method. Recursive partitioning analysis (RPA) identified prognostic risk strata for OS and PFS based on pretreatment factors. 361 patients were included. Primary tumors included non-small cell lung (17%), colorectal (19%), and breast cancer (16%). Three-year OS was 56%, PFS was 24%, and TMC was 72%. On multivariate analysis, primary tumor, interval to metastases, treated metastases number, and mediastinal/hilar lymph node, liver, or adrenal metastases were associated with OS. Primary tumor site, involved organ number, liver metastasis, and prior primary disease chemotherapy were associated with PFS. OS RPA identified five classes: class 1: all breast, kidney, or prostate cancer patients (BKP) (3-year OS 75%, 95% CI 66-85%); class 2: patients without BKP with disease-free interval of 75+ months (3-year OS 85%, 95% CI 67-100%); class 3: patients without BKP, shorter disease-free interval, ≤ two metastases, and age < 62 (3-year OS 55%, 95% CI 48-64%); class 4: patients without BKP, shorter disease-free interval, ≥ three metastases, and age < 62 (3-year OS 38%, 95% CI 24-60%); class 5: all others (3-year OS 13%, 95% CI 5-35%). Higher biologically effective dose (BED) (p < 0.01) was associated with OS. We identified clinical factors defining oligometastatic patients with favorable outcomes, who we hypothesize are most likely to benefit from metastasis-directed therapy.