Bias reduction of the maximum-likelihood estimator for a conditional Gaussian MA(1) model

In this paper, we consider an estimation for the unknown parameters of a conditional Gaussian MA(1) model. In the majority of cases, a maximum-likelihood estimator is chosen because the estimator is consistent. However, for small sample sizes the error is large, because the estimator has a bias of O(n^{? 1}). Therefore, we provide a bias of O(n^{? 1}) for the maximum-likelihood estimator for the conditional Gaussian MA(1) model. Moreover, we propose new estimators for the unknown parameters of the conditional Gaussian MA(1) model based on the bias of O(n^{? 1}). We investigate the properties of the bias, as well as the asymptotical variance of the maximum-likelihood estimators for the unknown parameters, by performing some simulations. Finally, we demonstrate the validity of the new estimators through this simulation study.     

Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

Double-smoothing for bias reduction in local linear regression

Local linear regression involves fitting a straight line segment over a small region whose midpoint is the target point x, and the local linear estimate at x   is the estimated intercept of that straight line segment, with an asymptotic bias of order h²$h^2$ and variance of order (nh)^-1$(\mathit{nh}{)}^{-1}$ (h is the bandwidth). In this paper, we propose a new estimator, the double-smoothing local linear estimator, which is constructed by integrally combining all fitted values at x   of local lines in its neighborhood with another round of smoothing. The proposed estimator attempts to make use of all information obtained from fitting local lines. Without changing the order of variance, the new estimator can reduce the bias to an order of h⁴$h^4$. The proposed estimator has better performance than local linear regression in situations with considerable bias effects; it also has less variability and more easily overcomes the sparse data problem than local cubic regression. At boundary points, the proposed estimator is comparable to local linear regression. Simulation studies are conducted and an ethanol example is used to compare the new approach with other competitive methods.     

A note on bias reduction in variable-kernel density estimates

It is well known that the inverse-square-root rule of Abramson (1982) for the bandwidth h of a variable-kernel density estimator achieves a reduction in bias from the fixed-bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O(h⁴) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O(h⁴) bias expression holds uniformly on any set where the target density is bounded away from zero.     

Kernel Method Starting with Half-Normal Detection Function for Line Transect Density Estimation

In this article, we introduce the nonparametric kernel method starting with half-normal detection function using line transect sampling. The new method improves bias from O(h ²), as the smoothing parameter h → 0, to O(h ³) and in some cases to O(h ⁴). Properties of the proposed estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the estimator are investigated and compared with the traditional kernel estimator by using simulation technique. A numerical results show that improvements over the traditional kernel estimator often can be realized even when the true detection function is far from the half-normal detection function.     

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.     

Pseudo‐R2 statistics under complex sampling

Model summaries based on the ratio of fitted and null likelihoods have been proposed for generalised linear models, reducing to the familiar R² coefficient of determination in the Gaussian model with identity link. In this note I show how to define the Cox–Snell and Nagelkerke summaries under arbitrary probability sampling designs, giving a design‐consistent estimator of the population model summary. It is also shown that for logistic regression models under case–control sampling the usual Cox–Snell and Nagelkerke R² are not design‐consistent, but are systematically larger than would be obtained with a cross‐sectional or cohort sample from the same population, even in settings where the weighted and unweighted logistic regression estimators are similar or identical. Implementation of the new estimators is straightforward and code is provided in R.     

Simulated deficiencies of robust estimators of a proportionality constant

In linear regression, robust methods are at the beginning of their use in practice. In the small sample case, such robust methods provide a necessary measure of protection against deviations from the assumed error distribution. This paper studies through simulation the deficiencies of bioptimal estimators and compares them with more common methods like Huber's estimator or Tukey's estimator. Polyoptimal estimators are convex combinations of Pitman estimators and are optimally robust for a confrontation containing several shapes. The word confrontation is due to J.W. Tukey. It expresses the situation when compromising two or several error distributions. The paper uses the confrontation containing the Gaussian distribution along with a symmetric heavy-tailed distribution having a tail of order 0(t^-2) as t→ ±∞.     

Maximum likelihood estimator in a multi-phase random regression model

We consider a random regression model with several-fold change-points. The results for one change-point are generalized. The maximum likelihood estimator of the parameters is shown to be consistent, and the asymptotic distribution for the estimators of the coefficients is shown to be Gaussian. The estimators of the change-points converge, with n ^?1 rate, to the vector whose components are the left end points of the maximizing interval with respect to each change-point. The likelihood process is asymptotically equivalent to the sum of independent compound Poisson processes.     

The Effectiveness of Multistage Ranked Set Sampling in Stratifying the Population

The spectral analysis of Gaussian linear time-series processes is usually based on uni-frequential tools because the spectral density functions of degree 2 and higher are identically zero and there is no polyspectrum in this case. In finite samples, such an approach does not allow the resolution of closely adjacent spectral lines, except by using autoregressive models of excessively high order in the method of maximum entropy. In this article, multi-frequential periodograms designed for the analysis of discrete and mixed spectra are defined and studied for their properties in finite samples. For a given vector of frequencies ω, the sum of squares of the corresponding trigonometric regression model fitted to a time series by unweighted least squares defines the multi-frequential periodogram statistic IM(ω). When ω is unknown, it follows from the properties of nonlinear models whose parameters separate (i.e., the frequencies and the cosine and sine coefficients here) that the least-squares estimator of frequencies is obtained by maximizing I ^M(ω). The first-order, second-order and distribution properties of I ^M(ω) are established theoretically in finite samples, and are compared with those of Schuster's uni-frequential periodogram statistic. In the multi-frequential periodogram analysis, the least-squares estimator of frequencies is proved to be theoretically unbiased in finite samples if the number of periodic components of the time series is correctly estimated. Here, this number is estimated at the end of a stepwise procedure based on pseudo-Flikelihood ratio tests. Simulations are used to compare the stepwise procedure involving I ^M(ω) with a stepwise procedure using Schuster's periodogram, to study an approximation of the asymptotic theory for the frequency estimators in finite samples in relation to the proximity and signal-to-noise ratio of the periodic components, and to assess the robustness of I ^M(ω) against autocorrelation in the analysis of mixed spectra. Overall, the results show an improvement of the new method over the classical approach when spectral lines are adjacent. Finally, three examples with real data illustrate specific aspects of the method, and extensions (i.e., unequally spaced observations, trend modeling, replicated time series, periodogram matrices) are outlined.     

RATES OF CONVERGENCE IN SEMI-PARAMETRIC MODELLING OF LONGITUDINAL DATA

We consider the problem of semi-parametric regression modelling when the data consist of a collection of short time series for which measurements within series are correlated. The objective is to estimate a regression function of the form E[Y(t) | x] =x'ß+μ(t), where μ(.) is an arbitrary, smooth function of time t, and x is a vector of explanatory variables which may or may not vary with t. For the non-parametric part of the estimation we use a kernel estimator with fixed bandwidth h. When h is chosen without reference to the data we give exact expressions for the bias and variance of the estimators for β and μ(t) and an asymptotic analysis of the case in which the number of series tends to infinity whilst the number of measurements per series is held fixed. We also report the results of a small-scale simulation study to indicate the extent to which the theoretical results continue to hold when h is chosen by a data-based cross-validation method.     

Deviation of order p for estimators of the variance in first-order stochastic differential equation (SDE)

In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in L ^p norm (p≥; 1) between the variance and this estimator. The method of the proof consists in writing the L ^p norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for m-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov's formula.     

ROBUST RIDGE REGRESSION BASED ON AN M-ESTIMATOR

Consider the linear regression model y =β₀1 +Xβ+ in the usual notation. It is argued that the class of ordinary ridge estimators obtained by shrinking the least squares estimator by the matrix (X¹X + kI)^-1X'X is sensitive to outliers in the ^variable. To overcome this problem, we propose a new class of ridge-type M-estimators, obtained by shrinking an M-estimator (instead of the least squares estimator) by the same matrix. Since the optimal value of the ridge parameter k is unknown, we suggest a procedure for choosing it adaptively. In a reasonably large scale simulation study with a particular M-estimator, we found that if the conditions are such that the M-estimator is more efficient than the least squares estimator then the corresponding ridge-type M-estimator proposed here is better, in terms of a Mean Squared Error criteria, than the ordinary ridge estimator with k chosen suitably. An example illustrates that the estimators proposed here are less sensitive to outliers in the y-variable than ordinary ridge estimators.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

A Note on Asymptotic Behavior of the Nonparametric Density Estimators in Multivariate Mixtures

The asymptotic behavior of the nonparametric density estimator has been given for a multivariate mixture model. It has been observed that the estimator is asymptotically normally distributed with bias of size h ² and variance of size (nh)^?1.     

On shrinking minimax convergence in nonparametric statistics

‘?…?if we are prepared to assume that the unknown density has k derivatives, then?…?the optimal mean integrated squared error is of order n^{?2 k/(2 k+1)}?…?’ The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman &; Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.     

The asymptotic behavior of monotone percentile regression estimates

The least-absolute-deviation estimate of a monotone regression function on an interval has been studied in the literature. If the observation points become dense in the interval, the almost sure rate of convergence has been shown to be O(n^1/4). Applying the techniques used by Brunk (1970, Nonparametric, Techniques in Statistical Inference. Cambridge Univ. Press), the asymptotic distribution of the l₁ estimator at a point is obtained. If the underlying regression function has positive slope at the point, the rate of convergence is seen to be O(n^1/3). Monotone percentile regression estimates are also considered.     

Estimation in Partially Linear Single-Index Models with Missing Covariates

In this article, we consider a partially linear single-index model Y = g(Z ^τθ₀) + X ^τβ₀ + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.     

A plug-in technique in nonparametric regression with dependence

The problem addressed is that of smoothing parameter selection in kernel nonparametric regression in the fixed design regression model with dependent noise. An asymptotic expression of the optimum bandwidth parameter has been obtained in recent studies, where this takes the form h = C ₀ n ^?1/5. This paper proposes to use a plug-in methodology, in order to obtain an optimum estimation of the bandwidth parameter, through preliminary estimation of the unknown value of C ₀.     

Best Quadratic Unbiased Prediction in a General Linear Model with Stochastic Regression Coefficients

In this article, we discuss on how to predict a combined quadratic parametric function of the form β′ H β + hσ² in a general linear model with stochastic regression coefficients denoted by y  =  X β +  e . Firstly, the quadratic predictability of β′ H β + hσ² is investigated to obtain a quadratic unbiased predictor (QUP) via a general method of structuring an unbiased estimator. This QUP is also optimal in some situations and therefore we hope it will be a fine predictor. To show this idea, we apply the Lagrange multipliers method to this problem and finally reach the expected conclusion through permutation matrix techniques.