Journal of Statistical Planning and Inference最新文献

In this paper, we address the problem of kernel estimation of the Expected Shortfall (ES) risk measure for financial losses that satisfy the $\alpha$-mixing conditions. First, we introduce a new non-parametric estimator for the ES measure using a kernel estimation. Given that the ES measure is the sum of the Value-at-Risk and the mean-excess function, we provide an estimation of the ES as a sum of the estimators of these two components. Our new estimator has a closed-form expression that depends on the choice of the kernel smoothing function, and we derive these expressions in the case of Gaussian, Uniform, and Epanechnikov kernel functions. We study the asymptotic properties of this new estimator and compare it to the Scaillet estimator. Capitalizing on the properties of these two estimators, we combine them to create a new estimator for the ES which reduces the bias and lowers the mean square error. The combined estimator shows better stability with respect to the choice of the kernel smoothing parameter. Our findings are illustrated through some numerical examples that help us to assess the small sample properties of the different estimators considered in this paper.

Step-stress is a special type of accelerated life-testing procedure that allows the experimenter to test the units of interest under various stress conditions changed (usually increased) at different intermediate time points. In this paper, we study the problem of testing hypothesis for the scale parameter of a simple step-stress model with exponential lifetimes and under Type-II censoring. We consider several modifications of the log-likelihood ratio statistic and eliminate the distributional dependence on the unknown lifetime parameters by exploiting the scale invariant properties of the normalized failure spacings. The presented results and the ratio statistic are further generalized to the multilevel step-stress case under the log-link assumption. We compare the power performance of the proposed tests via Monte Carlo simulations. As an illustration, the described procedures are applied to a real data example from the literature.

Space-filling designs that possess high separation distance are useful for computer experiments. We propose a novel method to construct high-dimensional high-separation distance designs. The construction involves taking the Kronecker product of sub-Hadamard matrices and rotation. In addition to possessing better separation distance than most existing types of space-filling designs, our newly proposed designs enjoy orthogonality and projection uniformity and are more flexible in the numbers of runs and factors than that from most algebraic constructions. From numerical results, such designs are excellent in Gaussian process emulation of high-dimensional computer experiments. An R package on design construction is available online.

Ordinary differential equations (ODEs) are widely used to model complex dynamics that arise in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally challenging. In this paper, we propose a two-stage nonparametric approach to address this problem. We first extract the de-noised data and their higher order derivatives using boundary kernel method, and then feed them into a sparsely connected deep neural network with rectified linear unit (ReLU) activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and the complexity of the ODE structure. We have shown that our method is consistent if the ODE possesses a general modular structure with each modular component involving only a few input variables, and the network architecture is properly chosen. Theoretical properties are corroborated by an extensive simulation study that also demonstrates the effectiveness of the proposed method in finite samples. Finally, we use our method to simultaneously characterize the growth rate of COVID-19 cases from the 50 states of the United States.

In this paper, we suggest an empirical likelihood-based test for the autoregressive coefficient of an integer-valued AR(1) model, i.e., INAR(1). We derive the limit distributions of the resulting test statistic under both null and alternative hypotheses. It turns out that regardless of whether the INAR process is stable or unstable, the statistic is always chi-squared distributed asymptotically under the null hypothesis, and as a result, it can offer unified inferences for the autoregressive coefficient. The performance of its finite sample is also demonstrated using simulations and an empirical example.

Fractionally integrated autoregressive moving average (FIARMA) processes have been widely and successfully used to model and predict univariate time series exhibiting long range dependence. Vector and functional extensions of these processes have also been considered more recently. Here we study these processes by relying on a spectral domain approach in the case where the processes are valued in a separable Hilbert space $H_0$. In this framework, the usual univariate long memory parameter $d$ is replaced by a long memory operator $D$ acting on $H_0$, leading to a class of $H_0$-valued FIARMA($D,p,q$) processes, where $p$ and $q$ are the degrees of the AR and MA polynomials. When $D$ is a normal operator, we provide a necessary and sufficient condition for the $D$-fractional integration of an $H_0$-valued ARMA($p,q$) process to be well defined. Then, we derive the best predictor for a class of causal FIARMA processes and study how this best predictor can be consistently estimated from a finite sample of the process. To this end, we provide a general result on quadratic functionals of the periodogram, which incidentally yields a result of independent interest. Namely, for any ergodic stationary process valued in $H_0$ with a finite second moment, the empirical autocovariance operator converges, in trace-norm, to the true autocovariance operator almost surely at each lag.

This paper considers the joint estimation of the parameters of a first-order fractional autoregressive model. A one-step procedure is considered in order to obtain an asymptotically-efficient estimator with an initial guess estimator with convergence speed lower than $\sqrt{n}$ and singular asymptotic joint distribution. This estimator is computed faster than the maximum likelihood estimator or the Whittle estimator and therefore allows for faster inference on large samples. The paper also illustrates the performance of this method on finite-size samples via Monte Carlo simulations.

High-dimensional additive quantile regression model via penalization provides a powerful tool for analyzing complex data in many contemporary applications. Despite the fast developments, how to combine the strengths of additive quantile regression with total variation penalty with theoretical guarantees still remains unexplored. In this paper, we propose a new methodology for sparse additive quantile regression model over bounded variation function classes via the empirical norm penalty and the total variation penalty for local adaptivity. Theoretically, we prove that the proposed method achieves the optimal convergence rate under mild assumptions. Moreover, an alternating direction method of multipliers (ADMM) based algorithm is developed. Both simulation results and real data analysis confirm the effectiveness of our method.

We investigate the consistency and the rate of convergence of the adaptive Lasso estimator for the parameters of linear AR(p) time series with a white noise which is a strictly stationary and ergodic martingale difference. Roughly speaking, we prove that $\left(i\right)$ If the white noise has a finite second moment, then the adaptive Lasso estimator is almost sure consistent $\left(ii\right)$ If the white noise has a finite fourth moment, then the error estimate converges to zero with the same rate as the regularizing parameters of the adaptive Lasso estimator. Such theoretical findings are applied to estimate the parameters of INAR(p) time series and to estimate the fertility function of Hawkes processes. The results are validated by some numerical simulations, which show that the adaptive Lasso estimator allows for a better balancing between bias and variance with respect to the Conditional Least Square estimator and the classical Lasso estimator.