Amplitude modulated model for analyzing non-stationary speech signals

Recently amplitude modulated (AM) model in presence of additive white noise was used to analyze certain non-stationary speech data. It is observed that the assumption of white noise may not be proper in many cases. In this article, we consider the AM signal model in presence of stationary noise. We consider the least squares estimators and the estimators obtained by maximizing the Periodogram function. The two estimators are asymptotically equivalent. We study the theoretical properties of both estimators and observe their performances through numerical simulations. One speech data is analyzed and it is observed that the performance of the proposed estimators is quite satisfactory.     

Variable Selection for Partially Linear Models with Randomly Censored Data

This article proposes a variable selection procedure for partially linear models with right-censored data via penalized least squares. We apply the SCAD penalty to select significant variables and estimate unknown parameters simultaneously. The sampling properties for the proposed procedure are investigated. The rate of convergence and the asymptotic normality of the proposed estimators are established. Furthermore, the SCAD-penalized estimators of the nonzero coefficients are shown to have the asymptotic oracle property. In addition, an iterative algorithm is proposed to find the solution of the penalized least squares. Simulation studies are conducted to examine the finite sample performance of the proposed method.     

Estimating Percentiles of Time-to-Failure Distribution Obtained from a Linear Degradation Model Using Kernel Density Method

In this article, we propose a nonparametric estimator for percentiles of the time-to-failure distribution obtained from a linear degradation model using the kernel density method. The properties of the proposed kernel estimator are investigated and compared with well-known maximum likelihood and ordinary least squares estimators via a simulation technique. The mean squared error and the length of the bootstrap confidence interval are used as the basis criteria of the comparisons. The simulation study shows that the performance of the kernel estimator is acceptable as a general estimator. When the distribution of the data is assumed to be known, the maximum likelihood and ordinary least squares estimators perform better than the kernel estimator, while the kernel estimator is superior when the assumption of our knowledge of the data distribution is violated. A comparison among different estimators is achieved using a real data set.     

Equivalence of maximum likelihood estimation and iterative two-stage estimation for seemingly unrelated regression models

The seemingly unrelated regression model is viewed in the context of repeated measures analysis. Regression parameters and the variance-covariance matrix of the seemingly unrelated regression model can be estimated by using two-stage Aitken estimation. The first stage is to obtain a consistent estimator of the variance-covariance matrix. The second stage uses this matrix to obtain the generalized least squares estimators of the regression parameters. The maximum likelihood (ML) estimators of the regression parameters can be obtained by performing the two-stage estimation iteratively. The iterative two-stage estimation procedure is shown to be equivalent to the EM algorithm (Dempster, Laird, and Rubin, 1977) proposed by Jennrich and Schluchter (1986) and Laird, Lange, and Stram (1987) for repeated measures data. The equivalence of the iterative two-stage estimator and the ML estimator has been previously demonstrated empirically in a Monte Carlo study by Kmenta and Gilbert (1968). It does not appear to be widely known that the two estimators are equivalent theoretically. This paper demonstrates this equivalence.     

On approximate least squares estimators of parameters of one-dimensional chirp signal

Chirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.     

Least squares variogram fitting by spatial subsampling

Summary. Least squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.     

Robust Regression Using Data Partitioning and M-Estimation

We propose a new robust regression estimator using data partition technique and M estimation (DPM). The data partition technique is designed to define a small fixed number of subsets of the partitioned data set and to produce corresponding ordinary least square (OLS) fits in each subset, contrary to the resampling technique of existing robust estimators such as the least trimmed squares estimator. The proposed estimator shares a common strategy with the median ball algorithm estimator that is obtained from the OLS trial fits only on a fixed number of subsets of the data. We examine performance of the DPM estimator in the eleven challenging data sets and simulation studies. We also compare the DPM with the five commonly used robust estimators using empirical convergence rates relative to the OLS for clean data, robustness through mean squared error and bias, masking and swamping probabilities, the ability of detecting the known outliers, and the regression and affine equivariances.     

A Comparison of Partially Adaptive and Reweighted Least Squares Estimation

The small sample performance of least median of squares, reweighted least squares, least squares, least absolute deviations, and three partially adaptive estimators are compared using Monte Carlo simulations. Two data problems are addressed in the paper: (1) data generated from non-normal error distributions and (2) contaminated data. Breakdown plots are used to investigate the sensitivity of partially adaptive estimators to data contamination relative to RLS. One partially adaptive estimator performs especially well when the errors are skewed, while another partially adaptive estimator and RLS perform particularly well when the errors are extremely leptokur-totic. In comparison with RLS, partially adaptive estimators are only moderately effective in resisting data contamination; however, they outperform least squares and least absolute deviation estimators.     

Topp–Leone normal distribution with application to increasing failure rate data

In this article, we proposed a new three-parameter probability distribution, called Topp–Leone normal, for modelling increasing failure rate data. The distribution is obtained by using Topp–Leone-X family of distributions with normal as a baseline model. The basic properties including moments, quantile function, stochastic ordering and order statistics are derived here. The estimation of unknown parameters is approached by the method of maximum likelihood, least squares, weighted least squares and maximum product spacings. An extensive simulation study is carried out to compare the long-run performance of the estimators. Applicability of the distribution is illustrated by means of three real data analyses over existing distributions.     

Fitting linear regression models to censored data by least squares and maximum likelihood methods

Several approaches have been suggested for fitting linear regression models to censored data. These include Cox's propor­tional hazard models based on quasi-likelihoods. Methods of fitting based on least squares and maximum likelihoods have also been proposed. The methods proposed so far all require special purpose optimization routines. We describe an approach here which requires only a modified standard least squares routine.

We present methods for fitting a linear regression model to censored data by least squares and method of maximum likelihood. In the least squares method, the censored values are replaced by their expectations, and the residual sum of squares is minimized. Several variants are suggested in the ways in which the expect­ation is calculated. A parametric (assuming a normal error model) and two non-parametric approaches are described. We also present a method for solving the maximum likelihood equations in the estimation of the regression parameters in the censored regression situation. It is shown that the solutions can be obtained by a recursive algorithm which needs only a least squares routine for optimization. The suggested procesures gain considerably in computational officiency. The Stanford Heart Transplant data is used to illustrate the various methods.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

An efficient and fast algorithm for estimating the frequencies of 2-D superimposed exponential signals in presence of zero-mean multiplicative and additive noise

In this paper, a computationally efficient algorithm is proposed for estimating the parameters of two-dimensional (2-D) superimposed exponential signals in presence of independently and identically distributed (i.i.d.) zero-mean multiplicative and additive noise. It is observed that the estimator is consistent and works quite well in terms of biases and mean squared errors. Moreover, the algorithm is efficient when multiple 2-D frequencies pairs share a same 1-D frequency component and the estimators attain the same convergence rate with the least squares estimator (LSE) in presence of additive noise. Finally, it is observed that the algorithm can be used to estimate the frequencies of the evanescent component of texture accurately.     

Efficiency of least squares estimators in the presence of spatial autocorrelation

The effect of spatial autocorrelation on inferences made using ordinary least squares estimation is considered. It is found, in some cases, that ordinary least squares estimators provide a reasonable alternative to the estimated generalized least squares estimators recommended in the spatial statistics literature. One of the most serious problems in using ordinary least squares is that the usual variance estimators are severely biased when the errors are correlated. An alternative variance estimator that adjusts for any observed correlation is proposed. The need to take autocorrelation into account in variance estimation negates much of the advantage that ordinary least squares estimation has in terms of computational simplicity     

An efficient algorithm for estimating the parameters of superimposed exponential signals

An efficient computational algorithm is proposed for estimating the parameters of undamped exponential signals, when the parameters are complex valued. Such data arise in several areas of applications including telecommunications, radio location of objects, seismic signal processing and computer assisted medical diagnostics. It is observed that the proposed estimators are consistent and the dispersion matrix of these estimators is asymptotically the same as that of the least squares estimators. Moreover, the asymptotic variances of the proposed estimators attain the Cramer–Rao lower bounds, when the errors are Gaussian.     

COVARIATE-ADJUSTED REGRESSION FOR LONGITUDINAL DATA INCORPORATING CORRELATION BETWEEN REPEATED MEASUREMENTS

We propose an estimation method that incorporates the correlation/covariance structure between repeated measurements in covariate-adjusted regression models for distorted longitudinal data. In this distorted data setting, neither the longitudinal response nor (possibly time-varying) predictors are directly observable. The unobserved response and predictors are assumed to be distorted/contaminated by unknown functions of a common observable confounder. The proposed estimation methodology adjusts for the distortion effects both in estimation of the covariance structure and in the regression parameters using generalized least squares. The finite-sample performance of the proposed estimators is studied numerically by means of simulations. The consistency and convergence rates of the proposed estimators are also established. The proposed method is illustrated with an application to data from a longitudinal study of cognitive and social development in children.     

A GROUP OF SPLIT-PLOT DESIGNS WITH A HETEROSCEDASTIC MODEL

For a group of split-plot designs it is assumed that the error variance is constant for a particular experiment but it varies from experiment to experiment. Assuming error variances to be known, estimators of the treatment parameters are obtained by weighted (generalised) least squares method and the corresponding analysis is given. For unknown error variances, an adjustment of the statistics using estimated weights is proposed for removing much of the resulting bias. The adjustment stems from a theorem due to Meier (1953).     

Change points in nonparametric regresion functions

Estimators of location and size of jumps or discontinuities in a regression function and/or its derivatives are proposed. The estimators are based on the analysis of residuals obtained from the locally weighted least squares regression. The proposed estimators adapt to both fixed and random designs. The asymptotic properties of the estimators are investigated. The method is illustrated through simulation studies.     

ASYMPTOTIC PROPERTIES OF RESTRICTED MAXIMUM LIKELIHOOD (REML) ESTIMATES FOR HIERARCHICAL MIXED LINEAR MODELS

This paper explores the asymptotic distribution of the restricted maximum likelihood estimator of the variance components in a general mixed model. Restricting attention to hierarchical models, central limit theorems are obtained using elementary arguments with only mild conditions on the covariates in the fixed part of the model and without having to assume that the data are either normally or spherically symmetrically distributed. Further, the REML and maximum likelihood estimators are shown to be asymptotically equivalent in this general framework, and the asymptotic distribution of the weighted least squares estimator (based on the REML estimator) of the fixed effect parameters is derived.     

On discrete-domain multidimensional sinusoidal models

We consider three-dimensional sinusoidal frequency model in a random field. Three-dimensional frequency model has wide applications in statistical signal processing. In this article, we mainly consider the usual least squares estimators and the estimators that can be obtained by maximizing the periodogram function. We obtain consistency and asymptotic normality property of both the estimators. It is observed that they are asymptotically equivalent. Finally we generalize the results to the multidimensional case.     

Consistency of the structured total least squares estimator in a multivariate errors-in-variables model

The structured total least squares estimator, defined via a constrained optimization problem, is a generalization of the total least squares estimator when the data matrix and the applied correction satisfy given structural constraints. In the paper, an affine structure with additional assumptions is considered. In particular, Toeplitz and Hankel structured, noise free and unstructured blocks are allowed simultaneously in the augmented data matrix. An equivalent optimization problem is derived that has as decision variables only the estimated parameters. The cost function of the equivalent problem is used to prove consistency of the structured total least squares estimator. The results for the general affine structured multivariate model are illustrated by examples of special models. Modification of the results for block-Hankel/Toeplitz structures is also given. As a by-product of the analysis of the cost function, an iterative algorithm for the computation of the structured total least squares estimator is proposed.