U. V. Naik-Nimbalkar

Filtering and Smoothing via Estimating Functions

Abstract We consider the problem of filtering and smoothing in state-space models, which include nonlinear and non-Gaussian models. We do not make any distributional assumptions about the processes involved. Our approach to these problems is based on the theory of estimating functions. Filter and smoother are obtained as solutions of estimating equations that are optimal in appropriate classes. We illustrate our procedures by simulation studies of a model where the observational variance depends on the state and a binomial logit model with a covariate. In non-Gaussian cases, procedures based on estimating equations often perform considerably better than the existing semiparametric procedures.

Dynamic Copula-Based Markov Time Series

This article examines a test procedure for checking the constancy of serial dependence via copulas for Markov time series data. It also provides a copula-based modeling approach for the dynamic serial dependence. Various parametric families of copulas offering different dependent structures are investigated. A score test is proposed for checking the constancy of a copula parameter. The score test is constructed and its asymptotic null distribution established under a two-stage estimation procedure. The test does not require specification of the probability distribution for the copula parameter. To capture the dynamics of dependence structure over time, autoregressive moving average and exponential type models are proposed. Illustrations are given based on simulated data and historic coffee prices data.

Empirical likelihood ratio test for equality of k medians in censored data

Most of the existing tests for equality of k medians actually assume that under the null hypothesis, the underlying k survival functions are identical. We propose a test which is based on empirical profile likelihood for median in censored data. This test does not require such an assumption. It is shown that under the standard censoring mechanism, the asymptotic distribution of the test statistic is chi-squared with k - 1 degrees of freedom, when the null hypothesis holds.

Accelerated Failure Time Models For Load Sharing Systems

We model the load sharing phenomenon in a k-out-of- m system through the accelerated failure time model. This model leads to multivariate families of distributions for ordered random variables, which are particular cases of the sequential order statistics. For illustrative purpose, we discuss the model, and the estimation problem for a two component parallel system under the setting of a linear failure rate distribution. In this set up, we discuss a test for the hypothesis that the failure times of components are statistically independent against the alternative that they show the load sharing phenomenon. We report simulation studies showing the performance of the estimators, and the test procedure. The test is also applied to two data sets for illustrative purpose.

Testing of two sample proportional intensity assumption for non-homogeneous Poisson processes

Abstract For two independent non-homogeneous Poisson processes with unknown intensities we propose a test for testing the hypothesis that the ratio of the intensities is constant versus it is increasing on (0,t]. The existing test procedures for testing such relative trends are based on conditioning on the number of failures observed in (0,t] from the two processes. Our test is unconditional and is based on the original time truncated data which enables us to have meaningful asymptotics. We obtain the asymptotic null distribution (as t becomes large) of the proposed test statistic and show that the proposed test is consistent against several large classes of alternatives. It was observed by Park and Kim (IEEE. Trans. Rehab. 40 (1), 1992, 107–111) that it is difficult to distinguish between the power-law and log-linear processes for certain parameter values. We show that our test is consistent for such alternatives also.

Modified Expected Shortfall: A New Robust Coherent Risk Measure

The coherent risk measure expected shortfall is a popular alternative to value-at-risk. However, the estimated value may miscommunicate the actual risk, especially when huge losses are present in the return series. This may force the financial institution to keep extra capital to meet the requirement set by the regulators. We propose a new robust coherent risk measure called modified expected shortfall, which quantifies the authentic risk of a portfolio. In comparison with the expected shortfall, the magnitude of the suggested risk measure is found to be lower. We propose non-parametric estimators of the modified expected shortfall and establish their statistical properties such as consistency and asymptotic normality.

Tests for Equality of Intensities of Failures of a Repairable System Under Two Competing Risks

Let a repairable system be subject to failure due to two competing risks. It is assumed that repairs are instantaneous.

Minimum distance estimation for random coefficient autoregressive models

In this paper, we extend the minimum distance method of Beran (1993) to random coefficient autoregressive (RCA) models. After stating the necessary assumptions the asymptotic properties of the minimum distance estimator are derived.

Optimal unbiased statistical estimating functions for Hilbert space valued parameters

Abstract An extended Cramer-Rao type inequality is shown to hold for unbiased statistical estimating functions (USEFs) when the parameter space is a real separable Hilbert space. This enables a definition of an ‘optimality criterion’ for the USEFs. A definition of the score function is given in this set up and it is shown that the USEF based on it is optimal. A bound is also obtained in the presence of nuisance parameters. Finally some examples are given.

ESTIMATION OF THE SCALE PARAMETER OF A POWER LAW PROCESS USING POWER LAW COUNTS

Analogous to Kingmans Poisson Counts, power law counts are defined. Further, these are used to obtain the maximum likelihood estimator of the scale parameter of a power law process. Comparison of this estimator is done with those obtained by using other sampling schemes. Also, cost comparisons are done under the assumption of equal asymptotic variances under different sampling schemes.