Swagata Nandi

An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring

We consider the problem of estimation of the parameters of the Marshall-Olkin Bivariate Weibull distribution in the presence of random censoring. Since the maximum likelihood estimators of the parameters cannot be expressed in a closed form, we suggest an EM algorithm to compute the same. Extensive simulations are carried out to conclude that the estimators perform efficiently under random censoring.

Asymptotic properties of the least squares estimators of the parameters of the chirp signals

Chirp signals are quite common in different areas of science and engineering. In this paper we consider the asymptotic properties of the least squares estimators of the parameters of the chirp signals. We obtain the consistency property of the least squares estimators and also obtain the asymptotic distribution under the assumptions that the errors are independent and identically distributed. We also consider the generalized chirp signals and obtain the asymptotic properties of the least squares estimators of the unknown parameters. Finally we perform some simulations experiments to see how the asymptotic results behave for small sample and the performances are quite satisfactory.

Surrogate Data — A Qualitative and Quantitative Analysis

The surrogates approach was suggested as a means to distinguish linear from nonlinear stochastic or deterministic processes. The numerical implementation is straightforward, but the statistical interpretation depends strongly on the stochastic process under consideration and the used test statistic. In the first part, we present quantitative investigations of level accuracy under the null hypothesis, power analysis for several violations, properties of phase randomization, and examine the assumption of uniformly distributed phases. In the second part we focus on level accuracy and power characteristics of Amplitude Adjusted Fourier-Transformed (AAFT) and improved AAFT (IAAFT) algorithms. In our study AAFT outperforms IAAFT. The latter method has a similar performance in many setups but it is not stable in general. We will see some examples where it breaks down.

Determination of Discrete Spectrum in a Random Field

We consider a two dimensional frequency model in a random field, which can be used to model textures and also has wide applications in Statistical Signal Processing. First we consider the usual least squares estimators and obtain the consistency and the asymptotic distribution of the least squares estimators. Next we consider an estimator, which can be obtained by maximizing the periodogram function. It is observed that the least squares estimators and the estimators obtained by maximizing the periodogram function are asymptotically equivalent. Some numerical experiments are performed to see how the results work for finite samples. We apply our results on simulated textures to observe how the different estimators perform in estimating the true textures from a noisy data.

Bootstrap and Resampling

Thebootstrap is by now a standard method in modern statistics. Its roots go back to a lot ofresampling ideas that were around in the seventies. The seminal work of Efron synthesized some of the earlierresampling ideas and established a new framework for simulation based statistical analysis. The idea of thebootstrap is to develop a setup to generate more (pseudo) data using the information of the original data. True underlying sample properties are reproduced as closely as possible and unknown model characteristics are replaced by sample estimates.

Estimation of frequencies in presence of heavy tail errors

In this paper, we consider the problem of estimating the sinusoidal frequencies in presence of additive white noise. The additive white noise has mean zero but it may not have finite variance. We propose to use the least-squares estimators or the approximate least-squares estimators to estimate the unknown parameters. It is observed that the least-squares estimators and the approximate least-squares estimators are asymptotically equivalent and both of them provide consistent estimators of the unknown parameters. We obtain the asymptotic distribution of the least-squares estimators under the assumption that the errors are from a symmetric stable distribution. We propose different methods of constructing confidence intervals and compare their performances through Monte Carlo simulations. We also discuss the properties of the estimators if the errors are correlated and finally we discuss some open problems.

Some theoretical properties of phase-randomized multivariate surrogates

We study the multivariate surrogate data method of Prichard and Theiler [D. Prichard and J. Theiler, Generating surrogate data for time series with several simultaneously measured variables, Phys. Rev. Lett. 73 (1994), pp. 951–954.]. This approach is an extension of the univariate surrogate data method introduced in Theiler et al. [J. Theiler, S. Eubank, A. Longtin, B. Galdrikan, and J.D. Farmer, Testing for nonlinearity in time series: The method of surrogate data, Physica D 58 (1992), pp. 77–94.]. Surrogate data is a resampling method for linear time series that is based on phase-randomized Fourier transforms. Theoretical properties of the multivariate method are derived. We show that the conditional distribution of surrogates is asymptotically normal. Furthermore, it is shown that surrogate data is the only method to construct tests that have constant level on the hypothesis of multivariate stationary Gaussian circular processes. We argue that this result approximately holds for non-circular processes under some regularity conditions. This entails that other bootstrap methods are not available for this general null hypothesis. Bootstrap will work only for more restricted hypotheses. Our results generalize results of Chan [K.S. Chan, On the validity of the method of surrogate data, Fields Inst. Commun. 11 (1997), pp. 77–97.] for the univariate setting.

A note on estimating the fundamental frequency of a periodic function

In this note we consider the estimation of the fundamental frequency of a periodic function. It is observed that the simple least-squares estimators can be used quite effectively to estimate the unknown parameters. The asymptotic distribution of the least-squares estimators is provided. Some simulation results are presented and finally we analyze two real life data sets using different methods.

EFFECT OF JUMP DISCONTINUITY FOR PHASE-RANDOMIZED SURROGATE DATA TESTING *

In this paper we discuss two modifications of the surrogate data method based on phase randomization, see [Theiler et al., 1992]. By construction, phase randomized surrogates are circular stationary. In this respect they differ from the original time series. This can cause level inaccuracies of surrogate data tests. We will illustrate this. These inaccuracies are caused by end to end mismatches of the original time series. In this paper we will discuss two approaches to remedy this problem: resampling from subsequences without end to end mismatches and data tapering. Both methods can be understood as attempts to make non-circular data approximately circular. We will show that the first method works quite well for a large range of applications whereas data tapering leads only to improvements in some examples but can be very unstable otherwise.

Noise space decomposition method for two-dimensional sinusoidal model

The estimation of the parameters of the two-dimensional sinusoidal signal model has been addressed. The proposed method is the two-dimensional extension of the one-dimensional noise space decomposition method. It provides consistent estimators of the unknown parameters and they are non-iterative in nature. Two pairing algorithms, which help in identifying the frequency pairs have been proposed. It is observed that the mean squared errors of the proposed estimators are quite close to the asymptotic variance of the least squares estimators. For illustrative purposes two data sets have been analyzed, and it is observed that the proposed model and the method work quite well for analyzing real symmetric textures.