Himadri Ghosh

Optimal diallel cross designs for estimation of heritability

Diallel crosses as mating designs are used to study the genetic properties of inbred lines in plant breeding experiments. Most of the theory of optimal diallel cross designs is based on standard linear model assumptions where the general combining ability effects are taken as fixed. In many practical situations, this assumption may not be tenable since we are studying only a sample of inbred lines, from a possibly large hypothetical population. A random effects model is proposed that allows us to first estimate the variance components and then obtain the variances of the estimates. We address the issue of optimal designs in this context by considering the A-optimality criteria. We obtain designs that are A-optimal for the estimation of heritability in the sense that the designs minimize the sum of the variances of the estimates of the variance components. The approach leads to certain connections with the optimization problem under the fixed effects model. Some numerical illustrations are given.

Self Exciting Threshold Autoregressive Models for Describing Cyclical Data

We thoroughly study a very important family of nonlinear timeseries models, viz. Self exciting threshold autoregressive (SETAR) types of models. A heartening feature of this family is that it is capable of describing cyclical data. As an illustration, SETAR models are then applied to countrys lac export data during the period 1900-2000, obtained from Annual reports of Shellac Export Promotion Council, Kolkata. It is shown that fitted model, based on minimum Akaike information criterion (AIC), exhibits a threshold behaviour. Finally, attempts are made to obtain optimal predictor for out-of-sample data based on fitted SETAR model, which is found to be quite satisfactory.

An improved fuzzy time-series method of forecasting based on L -- R fuzzy sets and its application

Classical time-series theory assumes values of the response variable to be ‘crisp’ or ‘precise’, which is quite often violated in reality. However, forecasting of such data can be carried out through fuzzy time-series analysis. This article presents an improved method of forecasting based on L–R fuzzy sets as membership functions. As an illustration, the methodology is employed for forecasting Indias total foodgrain production. For the data under consideration, superiority of proposed method over other competing methods is demonstrated in respect of modelling and forecasting on the basis of mean square error and average relative error criteria. Finally, out-of-sample forecasts are also obtained.

Kalman filter-based modelling and forecasting of stochastic volatility with threshold

We propose a parametric nonlinear time-series model, namely the Autoregressive-Stochastic volatility with threshold (AR-SVT) model with mean equation for forecasting level and volatility. Methodology for estimation of parameters of this model is developed by first obtaining recursive Kalman filter time-update equation and then employing the unrestricted quasi-maximum likelihood method. Furthermore, optimal one-step and two-step-ahead out-of-sample forecasts formulae along with forecast error variances are derived analytically by recursive use of conditional expectation and variance. As an illustration, volatile all-India monthly spices export during the period January 2006 to January 2012 is considered. Entire data analysis is carried out using EViews and matrix laboratory (MATLAB) software packages. The AR-SVT model is fitted and interval forecasts for 10 hold-out data points are obtained. Superiority of this model for describing and forecasting over other competing models for volatility, namely AR-Generalized autoregressive conditional heteroscedastic, AR-Exponential GARCH, AR-Threshold GARCH, and AR-Stochastic volatility models is shown for the data under consideration. Finally, for the AR-SVT model, optimal out-of-sample forecasts along with forecasts of one-step-ahead variances are obtained.

Bootstrap study of parameter estimates for nonlinear Richards growth model through genetic algorithm

Richards nonlinear growth model, which is a generalization of the well-known logistic and Gompertz models, generally provides a realistic description of many phenomena. However, this model is very rarely used as it is extremely difficult to fit it by employing nonlinear estimation procedures. To this end, utility of using a very powerful optimization technique of genetic algorithm is advocated. Parametric bootstrap methodology is then used to obtain standard errors of the estimates. Subsequently, bootstrap confidence-intervals are constructed by two methods, viz. the Percentile method, and Bias-corrected and accelerated method. The methodology is illustrated by applying it to Indias total annual foodgrain production time-series data.

Wavelet Frequency Domain Approach for Statistical Modeling of Rainfall Time-Series Data

The powerful methodology of “Wavelet analysis in frequency domain” for analyzing time-series data is studied. As an illustration, Indian monsoon rainfall time-series data from 1879–2006 is considered. The entire data analysis is carried out using SPLUS WAVELET TOOLKIT software package. The discrete wavelet transform (DWT) and multiresolution analysis (MRA) of the data are computed to analyze the behaviour of trend present in the time-series data in terms of different times and scales. By using bootstrap method, size and power of the test for testing significance of trend in the data is computed. It is found that the size of the test for Daubechies wavelet is more than that for Haar wavelet. In respect of both Daubechies and Haar wavelet filters, it is found that the test for presence of trend is unbiased. Also, power of the test for both Daubechies (D4) and Haar wavelets, at level 5 is less than the one at level 6. Further, Haar wavelet at level 6 has generally performed better than Daubechies (D4) wavelet at level 6 in terms of power of the test. Using the former wavelet, a declining trend in the data under consideration is revealed.

Nonlinear Time Series Modeling and Forecasting for Periodic and ARCH Effects

Generalized autoregressive conditional heteroscedastic (GARCH) nonlinear time series model may be employed to describe data sets depicting volatility. This model along with its estimation procedure is thoroughly studied. Lagrange multiplier (LM) test for testing presence of Autoregressive conditional heteroscedastic (ARCH) effects is also discussed. As an illustration, modeling and forecasting of monthly rainfall data of Sub-Himalayan West Bengal meteorological subdivision, India is carried out. As the data exhibits presence of seasonal component, Hylleberg, Engle, Granger and Yoo (1990) [HEGY] seasonal unit root test is applied to the data with a view to make the series stationary through “differencing filter”. Subsequently, GARCH model is employed on the residuals obtained after carrying out Periodic autoregressive (PAR) modeling of the seasonal variation. Further, Mixture periodic ARCH (MPARCH) model, which is an extension of GARCH model, is also applied on zero conditional mean residual series to identify time varying volatility in the data set. The performance of fitted models is examined from the viewpoint of dynamic one-step and two-step ahead forecast error variances along with Mean square prediction error (MSPE), Mean absolute prediction error (MAPE) and Relative mean absolute prediction error (RMAPE). Salient feature of the work done is that, for selected model, best predictor and prediction error variance for carrying out out-of-sample forecasting up to three-steps ahead are derived analytically by recursive use of conditional expectation and conditional variance. The SPSS, SAS and EViews software packages are used for data analysis. By carrying out a comparative study, it is concluded that, for the data under consideration, the PAR model with AR-GARCH errors has performed better than the Seasonal autoregressive integrated moving average (SARIMA) model for modeling as well as forecasting

A Bootstrap Study of Variance Estimation under Heteroscedasticity using Genetic Algorithm

The conventional ordinary least squares (OLS) variance-covariance matrix estimator for a linear regression model under heteroscedastic errors is biased and inconsistent. Accordingly, several estimators have so far been proposed by various researchers. However, none of these perform well under the finite-sample situation. In this paper, the powerful optimization technique of Genetic algorithm (GA) is used to modify these estimators. Properties of these newly developed estimators are thoroughly studied by Monte Carlo method for various sample sizes. It is shown that GA-versions of the estimators are superior to corresponding non-GA versions as there are significant reductions in the Total relative bias as well as Total root mean square error.

Non-Linear Time Series Modeling of Volatile Onion Price Data Using AR(p )-ARCH( q)-In-Mean

Onion prices have been in the headline since 1998 for its tear jerking effect on consumers, farmers and Government alike. Present paper attempts to develop a realistic time-series model to explain the behaviour of monthly onion price data during April, 1996 to October, 2001 collected from National Agricultmal Cooperative Marketing Federation (NAFED), New Delhi. In the first step, attempts are made to apply Seasonal Autoregressive (SAR) model to the detrended data. However, residual analysis reveals that assumption of constant one-period ahead forecast variance does not hold true. Accordingly, a new class of stochastic processes, called Autoregressive Conditional Heterosqedastic (ARCH) process, is studied to model the residual series. To this end, computer programs are written in EViews and package, Version 4.0 and IML in SAS, Version 8e to perform Lagrange-Multiplier test for possible presence of ARCH, to fit the AR(p )- ARCH( q) model, and to carry out residual analysis after fitting ARCH model. It is shown that ARCH model provides a good description of the data under consideration. Finally, the identified model is employed for forecasting purposes.

Gompertz growth model in random environment with time-dependent diffusion

ABSTRACTThe Gompertz nonlinear growth (GNG) model with independently and identically distributed (i.i.d.) errors is often employed for describing growth data. However, the corresponding stochastic differential equation (SDE) variant is more realistic for modeling growth data, as it is capable of taking into account the effect of randomly fluctuating parameters, such as birth and death rates. However, one limitation of this prescription is that the diffusion term is assumed to be time independent. The purpose of this article is to generalize the Gompertz SDE model by taking the diffusion coefficient as time-varying. The resultant model is solved analytically and methodology for estimation of parameters, based on the method of maximum likelihood, is developed. Formulas for optimal predictors and prediction error variances and the linear Gompertz SDE (LGSDE) model and modified Gompertz SDE (MGSDE) model are also derived. Superiority of the proposed MGSDE model is shown over the LGSDE and GNG models for pig g...