Different estimators of the spectral matrix: an empirical comparison testing a new shrinkage estimator

ABSTRACT

In this paper we propose a new non parametric estimator of the spectral matrix of a multivariate stationary stochastic process, with the main goal to locally improve the deficiencies of the smoothed periodogram in terms of mean square error of the estimates. Our estimator is based on a convex linear combination of the frequency averaged periodogram and an estimate of the true mean spectral matrix across frequencies. In a wide simulation study we show that our estimator turns out to be able to markedly improve the frequency averaged periodogram especially at central frequencies.     

Spectral analysis with replicated time series

A doubly stochastic process {x(b,t);b?B,t?Z} is considered, with (B,β,P_β) being a probability space so that for each b, {X(b,t);t ? Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = E_B[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b₁,…,b_r from (B,β,P_β). For each b₁? B, the processes X(b₁,t) are observed at the same times t=1,…,N. Thus, r time series (x(b₁,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.     

Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling

This paper addresses the problem of unbiased estimation of P[X > Y] = θ for two independent exponentially distributed random variables X and Y. We present (unique) unbiased estimator of θ based on a single pair of order statistics obtained from two independent random samples from the two populations. We also indicate how this estimator can be utilized to obtain unbiased estimators of θ when only a few selected order statistics are available from the two random samples as well as when the samples are selected by an alternative procedure known as ranked set sampling. It is proved that for ranked set samples of size two, the proposed estimator is uniformly better than the conventional non-parametric unbiased estimator and further, a modified ranked set sampling procedure provides an unbiased estimator even better than the proposed estimator.     

The periodogram regression:correction and comments

Ratios of periodogram and spectral density at different harmonic frequencies are independent if the frequency zero is approached slowly enough. This is an asymptotically relevant condition for the periodogram regression to work with fractionally integrated series. In finite samples, however, this theoretical condition should be ignored as we illustrate experimentally.     

Nonparametric beta kernel estimator for long and short memory time series

In this article we introduce a nonparametric estimator of the spectral density by smoothing the periodogram using beta kernel density. The estimator is proved to be bounded for short memory data and diverges at the origin for long memory data. The convergence in probability of the relative error and Monte Carlo simulations show that the proposed estimator automatically adapts to the long- and the short-range dependency of the process. A cross-validation procedure is studied in order to select the nuisance parameter of the estimator. Illustrations on historical as well as most recent returns and absolute returns of the S&P500 index show the performance of the beta kernel estimator. The Canadian Journal of Statistics 48: 582–595; 2020 © 2020 Statistical Society of Canada     

Spectral Density Based Goodness-of-Fit Tests for Time Series Models

A new goodness-of-fit test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quantification of how well a parametric spectral density model fits the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justified theoretically. The finite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.     

The Estimation and Testing of the Cointegration Order Based on the Frequency Domain

ABSTRACT

This article proposes a method to estimate the degree of cointegration in bivariate series and suggests a test statistic for testing noncointegration based on the determinant of the spectral density matrix for the frequencies close to zero. In the study, series are assumed to be I(d), 0 < d ? 1, with parameter d supposed to be known. In this context, the order of integration of the error series is I(d ? b), b ∈ [0, d]. Besides, the determinant of the spectral density matrix for the dth difference series is a power function of b. The proposed estimator for b is obtained here performing a regression of logged determinant on a set of logged Fourier frequencies. Under the null hypothesis of noncointegration, the expressions for the bias and variance of the estimator were derived and its consistency property was also obtained. The asymptotic normality of the estimator, under Gaussian and non-Gaussian innovations, was also established. A Monte Carlo study was performed and showed that the suggested test possesses correct size and good power for moderate sample sizes, when compared with other proposals in the literature. An advantage of the method proposed here, over the standard methods, is that it allows to know the order of integration of the error series without estimating a regression equation. An application was conducted to exemplify the method in a real context.     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.     

α-stable laws for noncoding regions in DNA sequences

In this work, we analyze the long-range dependence parameter for a nucleotide sequence in several different transformations. The long-range dependence parameter is estimated by the approximated maximum likelihood method, by a novel estimator based on the spectral envelope theory, by a regression method based on the periodogram function, and also by the detrended fluctuation analysis method. We study the length distribution of coding and noncoding regions for all Homo sapiens chromosomes available from the European Bioinformatics Institute. The parameter of the tail rate decay is estimated by the Hill estimator ?α. We show that the tail rate decay is greater than 2 for coding regions, while for almost all noncoding regions it is less than 2.     

Spatial dependence estimation using FFT of biased covariances

One of the main problems in geostatistics is fitting a valid variogram or covariogram model in order to describe the underlying dependence structure in the data. The dependence between observations can be also modeled in the spectral domain, but the traditional methods based on the periodogram as an estimator of the spectral density may present some problems for the spatial case. In this work, we propose an estimation method for the covariogram parameters based on the fast Fourier transform (FFT) of biased covariances. The performance of this estimator for finite samples is compared through a simulation study with other classical methods stated in spatial domain, such as weighted least squares and maximum likelihood, as well as with other spectral estimators. Additionally, an example of application to real data is given.     

Inferential analysis for the reliability parameter based on the three-parameter Lindley distribution

ABSTRACT

In this article, we consider the estimation of R = P(Y < X), when Y and X are two independent three-parameter Lindley (LI) random variables. On the basis of two independent samples, the modified maximum likelihood estimator along its asymptotic behavior and conditional likelihood-based estimator are used to estimate R. We also propose sample-based estimate of R and the associated credible interval based on importance sampling procedure. A real life data set involving the times to breakdown of an insulating fluid is presented and analyzed for illustrative purposes.     

A bias-corrected histogram estimator for line transect sampling

The classical histogram method has already been applied in line transect sampling to estimate the parameter f(0), which in turns is used to estimate the population abundance D or the population size N. It is well know that the bias convergence rate for histogram estimator of f(0) is o(h²) as h → 0, under the shoulder condition assumption. If the shoulder condition is not true, then the bias convergence rate is only o(h). This paper proposed two new estimators for f(0), which can be considered as modifications of the classical histogram estimator. The first estimator is derived when the shoulder condition is assumed to be valid and it reduces the bias convergence rate from o(h²) to o(h³). The other one is constructed without using the shoulder condition assumption and it reduces the bias convergence rate from o(h) to o(h²). The asymptotic properties of the proposed estimators are derived and formulas for bin width are also given. The finite properties based on a real data set and an extensive simulation study demonstrated the potential practical use of the proposed estimators.     

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern ⁻¹). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.     

New difference-based estimator of parameters in semiparametric regression models

The paper introduces a new difference-based Liu estimator β?_Ldiff=([Xtilde]′[Xtilde]+I)^?1([Xtilde]′[ytilde]+η β?_diff) of the regression parameters β in the semiparametric regression model, y=Xβ+f+?. Difference-based estimator, β?_diff=([Xtilde]′[Xtilde])^?1[Xtilde]′[ytilde] and difference-based Liu estimator are analysed and compared with respect to mean-squared error (mse) criterion. Finally, the performance of the new estimator is evaluated for a real data set. Monte Carlo simulation is given to show the improvement in the scalar mse of the estimator.     

On the predictive performance of the almost unbiased Liu estimator

ABSTRACT

Regression models are usually used in forecasting (predicting) unknown values of the response variable y. This article considers the predictive performance of the almost unbiased Liu estimator compared to the ordinary least-squares estimator, principal component regression estimator, and Liu estimator. Finally, we present a numerical example to explain the theoretical results and we obtain a region where the almost unbiased Liu estimator is uniformly superior to the ordinary least-squares estimator, principal component regression estimator, and Liu estimator.     

Assessing goodness-of-fit of categorical regression models based on case-control data

Demonstrated equivalence between a categorical regression model based on case‐control data and an I‐sample semiparametric selection bias model leads to a new goodness‐of‐fit test. The proposed test statistic is an extension of an existing Kolmogorov–Smirnov‐type statistic and is the weighted average of the absolute differences between two estimated distribution functions in each response category. The paper establishes an optimal property for the maximum semiparametric likelihood estimator of the parameters in the I‐sample semiparametric selection bias model. It also presents a bootstrap procedure, some simulation results and an analysis of two real datasets.     

Approximate optimal allocation in repeated sampling from a finite population

Samples of size n are drawn from a finite population on each of two occasions. On the first occasion a variate x is measured, and on the second a variate y. In estimating the population mean of y, the variance of the best linear unbiased combination of means for matched and unmatched samples is itself minimized, with respect to the sampling design on the second occasion, by a certain degree of matching. This optimal allocation depends on the population correlation coefficient, which previous authors have assumed known. We estimate the correlation from an initial matched sample, then an approximately optimal allocation is completed and an estimator formed which, under a bivariate normal superpopulation model, has model expected mean square error equal, apart from an error of order n^-2, to the minimum enjoyed by any linear, unbiased estimator.     

Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models

We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n ? k tends to infinity fast enough in relation to the thresholding parameter.     

On Some Aspects of Unbiased Estimation of Parameters in Quasi-Binomial Distributions

In this article, an attempt has been made to settle the question of existence of unbiased estimator of the key parameter p of the quasi-binomial distributions of Type I (QBD I) and of Type II (QBD II), with/without any knowledge of the other parameter φ appearing in the expressions for probability functions of the QBD's. This is studied with reference to a single observation, a random sample of finite size m as also with samples drawn by suitably defined sequential sampling rules.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.