Aysen D. Akkaya

Robust estimation in multiple linear regression model with non-Gaussian noise

The traditional least squares estimators used in multiple linear regression model are very sensitive to design anomalies. To rectify the situation we propose a reparametrization of the model. We derive modified maximum likelihood estimators and show that they are robust and considerably more efficient than the least squares estimators besides being insensitive to moderate design anomalies.

Robust estimation and hypothesis testing under short-tailedness and inliers

Estimation and hypothesis testing based on normal samples censored in the middle are developed and shown to be remarkably efficient and robust to symmetric shorttailed distributions and to inliers in a sample. This negates the perception that sample mean and variance are the best robust estimators in such situations (Tiku, 1980; Dunnett, 1982).

Time series AR(1) model for short-tailed distributions

The innovations in AR(1) models in time series have primarily been assumed to have a normal or long-tailed distributions. We consider short-tailed distributions (kurtosis less than 3) and derive modified maximum likelihood (MML) estimators. We show that the MML estimator of φ is considerably more efficient than the commonly used least squares estimator and is also robust. This paper is essentially the first to achieve robustness to inliers and to various forms of short-tailedness in time series analysis.

Estimating parameters of a multiple autoregressive model by the modified maximum likelihood method

We consider a multiple autoregressive model with non-normal error distributions, the latter being more prevalent in practice than the usually assumed normal distribution. Since the maximum likelihood equations have convergence problems (Puthenpura and Sinha, 1986) [11], we work out modified maximum likelihood equations by expressing the maximum likelihood equations in terms of ordered residuals and linearizing intractable nonlinear functions (Tiku and Suresh, 1992) [8]. The solutions, called modified maximum estimators, are explicit functions of sample observations and therefore easy to compute. They are under some very general regularity conditions asymptotically unbiased and efficient (Vaughan and Tiku, 2000) [4]. We show that for small sample sizes, they have negligible bias and are considerably more efficient than the traditional least squares estimators. We show that our estimators are robust to plausible deviations from an assumed distribution and are therefore enormously advantageous as compared to the least squares estimators. We give a real life example.

Autoregressive models with short-tailed symmetric distributions

Symmetric short-tailed distributions do indeed occur in practice but have not received much attention particularly in the context of autoregression. We consider a family of such distributions and derive the modified maximum likelihood estimators of the parameters. We show that the estimators are efficient and robust. We develop hypothesis-testing procedures.

A Comparative Study of Stochastic Models for Seismic Hazard Estimation

Different stochastic models have been developed over the years for the prediction of earthquake occurrences. The Mathematical rigor and the extent of input data requirement increase as the stochastic models used in the description of the spatial and temporal dependence characteristics of earthquake occurrences get more realistic. In this study, a review of the most widely used stochastic models, namely: Poisson, extreme value and Markov are presented briefly, together with the random field model proposed by the authors. The shortcomings of each model are discussed. The seismic hazard predictions obtained from these models are then compared among themselves based on the data recorded along the most active portion of the North Anatolian fault zone.

Modified maximum likelihood estimators using ranked set sampling

The closed-form maximum likelihood estimators (MLEs) of population mean and variance under ranked set sampling (RSS) do not exist since the likelihood equations involve nonlinear functions and have usually no explicit solutions. We derive modified maximum likelihood (MML) estimators for the population mean and variance under RSS and show that they are considerably more efficient than RSS estimators. Furthermore, we suggest two new estimators for the unknown parameters using two modified ranked set sampling methods and show that these methods make the variances of both MML and RSS estimators smaller.

Estimation of earthquake hazard based on extremes of local integral random functions

Abstract A random field model is developed for the occurrence of earthquakes in the space–time domain based on the theory of random functions. The seismic hazard posed by earthquakes is estimated by applying the level crossing theory to local integral random functions. Analysis of the correlation structure of the strain energy random field associated with the proposed model is described by utilizing the concepts of variance function and scale of fluctuation. The earthquake occurrence rates obtained in the study differ from the classical earthquake occurrence rates in the sense that these rates are adjusted for the degree of correlation in space and time. The theory and the results of the study are applied to a real-life problem involving the assessment of seismic hazard for the North Anatolian Fault Zone (NAFZ). The annual seismic hazard obtained from the random field model developed in this study is found to be consistent with the observed seismic activity along the NAFZ.

Statistical Seismology Science Gateway

Seismic hazard assessment and risk analysis are critical for human life since these fields study and model expected earthquakes to produce seismic hazard maps and to determine the risk of damage from potential earthquakes to buildings, dams, etc. The Statistical Seismology Science Gateway, built on the gUSE framework, holds a comprehensive set of tools and models covering seismic data analysis, hazard assessment and risk analysis. It provides an environment in such a way that skilled users can experiment their own approaches by simply accessing the gateway services, while novice users can easily use existing applications. This chapter focuses on the development of the workflows implementing these tools and models as well as the portlets to access them. It also discusses how easily gUSE framework was customised to build this science gateway and how it is operated.

A New Estimation Technique for AR(1) Model with Long-Tailed Symmetric Innovations

In recent years, it is seen in many time series applications that innovations are non-normal. In this situation, it is known that the least squares (LS) estimators are neither efficient nor robust and maximum likelihood (ML) estimators can only be obtained numerically which might be problematic. The estimation problem is considered newly through different distributions by the use of modified maximum likelihood (MML) estimation technique which assumes the shape parameter to be known. This becomes a drawback in machine data processing where the underlying distribution cannot be determined but assumed to be a member of a broad class of distributions. Therefore, in this study, the shape parameter is assumed to be unknown and the MML technique is combined with Huber’s estimation procedure to estimate the model parameters of autoregressive (AR) models of order 1, named as adaptive modified maximum likelihood (AMML) estimation. After the derivation of the AMML estimators, their efficiency and robustness properties are discussed through simulation study and compared with both MML and LS estimators. Besides, two test statistics for significance of the model are suggested. Both criterion and efficiency robustness properties of the test statistics are discussed, and comparisons with the corresponding MML and LS test statistics are given. Finally, the estimation procedure is generalized to AR(q) models.