Showing papers in "Journal of statistical theory and practice in 2010"

Ratio Estimation of the Mean of a Sensitive Variable in the Presence of Auxiliary Information

Abstract: We propose a ratio estimator for the mean of sensitive variable utilizing information from a nonsensitive auxiliary variable. Expressions for the Bias and MSE of the proposed estimator (correct up to first and second order approximations) are derived. We show that the proposed estimator does better than the ordinary RRT mean estimator that does not utilize the auxiliary information. We also show that there is hardly any difference in the first order and second order approximations for MSE even for small sample sizes. We also generalize the proposed estimator to the case of transformed ratio estimators but these transformations do not result in any significant reduction in MSE. An extensive simulation study is presented to evaluate the performance of the proposed estimator. The procedure is also applied to some financial data (purchase orders (sensitive variable) and gross turn-over (non-sensitive variable)) in 2009 for 5090 companies in Portugal from a survey on Information and Communication Tech...

Finding the optimal pre-set boundaries for pairs trading strategy based on cointegration technique

Abstract: Pairs trading is one of the arbitrage strategies that can be use in trading stocks on the stock market. This paper incorporates pairs trading with the use of cointegration technique to exploit stocks that are temporarily out of equilibrium. In determining which two stocks can be a pair, Banerjee, Dolado, Galbraith and Hendry (1993) and Vidyamurthy (2004) showed that the cointegration technique is more effective than correlation criterion for extracting profit potential in temporary pricing anomalies between two stock prices driven by common underlying factors. By using stationary properties of cointegration errors following an AR(1) process, this paper explores the ways in which the pre-set boundaries chosen to open a trade can influence the minimum total profit over a specified trading horizon. The minimum total profit relates to the pre-set boundaries for opening trades, the higher the profit per trade but the lower the trade nummbers. The number of trades over a specified trading horizon is es...

Estimation of Mean, Ratio and Product Using Auxiliary Information in the Presence of Measurement Errors in Sample Surveys

Abstract: For estimating the mean, ratio and product we have considered the estimation of general parameter R(α)(= μ Y0/μ(α) Y1,μ Y1 ≠ 0) using the information on an auxiliary variable X whose population mean μ X is assumed to be known, in the presence of measurement errors, where μ Y0 and μ Y1 are the population means of the study variables Y 0 and Y 1 respectively; α being a scalar takes values, 1 (for population ratio R = μ Y0/μ Y1) −1 (for population product P = μ Y0μ Y1), and 0 (for population mean μ Y0). A class of estimators for estimating the general parameter R (α) has been defined and asymptotic expressions for its bias and mean squared error (MSE) have been obtained. A comparative study has been made among the proposed estimators and the conventional estimators Numerical illustration is given in support of the present study.

On the Two-Sample Behrens-Fisher Problem for Functional Data

Abstract: In this paper, we propose and study an L 2-norm based test for the two-sample Behrens Fisher problem for functional data, which tests the equality of mean functions of two Gaussian processes with possibly unequal covariance functions. The distributions of the proposed test statistic under the null hypothesis and a sequence of local alternatives are derived. Under certain regularity conditions, the proposed test is shown to be χn-consistent. To well approximate the null distribution, we propose a Welch-type (two-cumulant matched χ2-approximation) method with the unknown parameters estimated using a simple naive method. When the two functional samples have the same covariance functions, some existing testing procedures for functional data can be adaptively applied. However, when this assumption is violated, we illustrate, via a simulation study, that the effect of unequal co-variance functions can be very significant. The methodologies are motivated by and illustrated with a real data example.

Change-Point Problems: Bibliography and Review

Abstract: Five types of change-point problems concerning change in mean, variance, slope, hazard rate, and space-time distribution are briefly reviewed and a list of comprehensive bibliography is provided. Directions for future studies are discussed.

Near-Exact Distributions for Certain Likelihood Ratio Test Statistics

Abstract: In this paper we will show how, using an expansion of a Logbeta distribution as an infinite mixture of Gamma distributions we are able to obtain near-exact distributions for the negative logarithm of the l.r.t. (likelihood ratio test) statistics used in Multivariate Analysis to test the independence of several sets of variables, the equality of several mean vectors, sphericity and the equality of several variance-covariance matrices which will match as many of the exact moments as we wish and for which we will be able to have an a priori upper-bound for the difference between their c.d.f. and the exact c.d.f., These near-exact distributions also display very good performance, with an agreement with the exact distribution which may virtually be taken as far as we wish and which it is not possible to obtain with the usual asymptotic distributions. Furthermore, based on the results presented it will be easy to build near-exact distributions for any l.r.t. statistics which may be built as the product...

Bayesian Estimation of the Parameter of Maxwell Distribution under Different Loss Functions

Abstract: In this paper, Bayes estimators of parameter of Maxwell distribution have been derived by considering non-informative as well as conjugate priors under different scale invariant loss functions, namely, Quadratic Loss Function, Squared-Log Error Loss Function and Modified Linear Exponential Loss Function. The risk functions of these estimators have been studied.

Bayesian Approach on the Generalized Exponential Distribution in the Presence of Outliers

Abstract: In this paper, the maximum likelihood and the Bayes estimators are derived for sample from the Generalized-Exponential distribution in the presence of k outliers. These estimators are obtained using Newton-Raphson method and Lindley's approximation (L-approximation). The proposed Bayes estimators are obtained under symmetric and asymmetric loss functions. These estimators are compared empirically using Monte Carlo simulation, when all the parameters are unknown.

Probabilistic Set-Membership Approach for Robust Regression

Abstract: Interval constraint propagation methods have been shown to be efficient and reliable to solve difficult nonlinear bounded-error estimation problems. However they are considered as unsuitable in a probabilistic context, where the approximation of a probability density function by a set cannot be accepted as reliable. This paper shows how probabilistic estimation problems can be transformed into a set estimation problem by assuming that some rare events will never happen. Since the probability of occurrence of those rare events can be computed, we can give some prior lower bounds for the probability associated to solution set of the corresponding set estimation problem. The approach will be illustrated on a parameter estimation problem.

Evaluating a New Risk Marker's Predictive Contribution in Survival Models

Abstract: Although the area under the receiver operating characteristic (ROC) curve (AUC) is the most popular measure of the performance of prediction models, it has limitations, especially when it is used to evaluate the added discrimination of a new risk marker in an existing risk model. Pencina et al. (2008) proposed two indices, the net reclassification improvement (NRI) and integrated discrimination improvement (IDI), to supplement the improvement in the AUC (IAUC). Their NRI and IDI are based on binary outcomes in case-control settings, which do not involve time-to-event outcome. However, many disease outcomes are time-dependent and the onset time can be censored. Measuring discrimination potential of a prognostic marker without considering time to event can lead to biased estimates. In this paper, we extended the NRI and IDI to time-to-event settings and derived the corresponding sample estimators and asymptotic tests. Simulation studies showed that the time-dependent NRI and IDI have better performance than Pencina’s NRI and IDI for measuring the improved discriminatory power of a new risk marker in prognostic survival models.

Finite Mixture of Exponential Model and its Applications to Renewal and Reliability Theory

Abstract: In the present paper, we study the properties of finite mixture of exponential model in the context of renewal theory. We obtain a generalized mixture of gamma distributions in terms of the confluent hypergeometric function, as the waiting time distribution. We present various applications of the model to reliability.

Expanding the Statistical Flowgraph Model Framework to Use Any Transition Distribution

Abstract: Historically, parametric statistical flowgraph models (SFGMs) have exclusively used distributions with moment generating functions (MGFs). This is a significant limitation because it does not allow the use of some common distributions. This paper extends SFGM methodology by using the mathematical construct of a complex Laplace transform in lieu of MGFs. This extension enables modeling all “smooth” densities in SFGMs. We demonstrate this method using an illustrative and a real data example; both the frequentist and Bayesian approaches are considered. This enhancement of parametric SFGMs notably extends their use and flexibility. R-code is available from the authors.

Two-Level Supersaturated Designs: A Review

Abstract: Supersaturated Designs (SSDs) are fractional factorial designs in which the run size is not enough to estimate the main effects of all the factors in the experiment Two-level SSDs have been studied extensively in the literature. The thrust of research has been on obtaining lower bounds to the value of E(s 2), a measure of departure from orthogonality, and constructing designs that attain these lower bounds. The focus of this paper is to review the literature on two-level SSDs.

Variance estimation in two-stage cluster sampling under imputation for missing data

Abstract: Variance estimation in the presence of imputed data has been widely studied in the literature. It is well known that treating the imputed values as if they were true values could lead to serious un...

A Model for Open Populations Subject to Periodical Re-Classifications

Abstract: Populations with periodical re-classifications occur in many fields, such as Insurance Companies, Bank Institutions, Pension Funds, Epidemiology and Health/Disease studies, where population elements are periodically reclassified. p]We show how to carry out inference for such populations, under general assumptions. Our treatment differs from previous ones since we consider open populations and focus on the sizes of sub-populations, availing ourselves of the Stochastic Vortices theory, see Guerreiro and Mexia (2004) and Guerreiro and Mexia (2008).

Fast Bayesian Estimation for VARFIMA Processes with Stable Errors

Abstract: We present an approach via a multivariate preconditioned conjugate gradient (MPCG) algorithm for Bayesian inference for vector ARFIMA models with sub-Gaussian stable errors. This approach involves solution of a block-Toeplitz system, and treating the unobserved process history and the underlying positive stable process as unknown parameters in the joint posterior. We use Gibbs sampling with the Metropolis-Hastings algorithm. We illustrate our approach on time series of daily average temperatures measured over several years at different U.S. cities.

Nesting Segregated Mixed Models

Abstract: A mixed model has segregation when its random effects part is segregated as a sub-model. It will be shown that under orthogonality condition, nesting a random effects model inside a segregated mixed model or a segregated mixed model inside a fixed effects model the result will be a segregated mixed model. Unbiased estimators will be obtained for the variance components in both classes of models which are UMVUE, once normality is assumed.

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

Abstract: 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.

Nonparametric Predictive Category Selection for Multinomial Data

Abstract: A new method is presented for selecting a single category or the smallest subset of categories, based on observations from a multinomial data set, where the selection criterion is a minimally required lower probability that (at least) a specific number of future observations will belong to that category or subset of categories. The inferences about the future observations are made using an extension of Coolen and Augustin’s nonparametric predictive inference (NPI) model to a situation with multiple future observations.

Computer Aided Construction of Efficient Multi-level Supersaturated Designs

Abstract: Motivated by the computer search algorithms for constructing two-level supersaturated designs by Heavlin and Finnegan (1993), Li and Wu (1997), Nguyen (1996), Lejeune (2003) and Gupta, Parsad, Kole and Bhar (2008), this paper develops an algorithm to generate multi-level supersaturated designs. Popular E(f NOD) and E(χ2) criterion have been used as a measure of non-orthogonality for the designs generated. The algorithm also ensures that no two columns in the designs generated are fully aliased. A catalogue of 120 optimal supersaturated designs for different number of factors m, design runs n, with 5 ≤ n ≤ 16 runs, and different number of factor levels q, with 3 ≤ q ≤ 6, has been prepared. All the designs generated are f NOD-optimal; some designs are χ2-optimal too.

Batch Arrival Vacation Queue with Second Optional Service and Random System Breakdowns

Abstract: A single server queue with a second optional service, Bernoulli schedule server vacations, and random system breakdowns was analyzed. It is assumed that customers arrive to the system in batches of variable size, but served one by one. As soon as the first service of a customer is completed, then with probability k he may opt for the second service. After a customer is served, the server may decide to leave the system with probability p, or to continue serving customers with probability 1 - p. If the system breaks down, it enters a repair process immediately. The repair time and the vacation time both are assumed to have general distributions. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue. Some special cases of interest are presented and a numerical illustration is provided.

A Single Server Retrial Queue with Bernoulli Feedback and Collisions

Abstract: A non-Markovian feedback single-server retrial queue with collisions and general retrial times is investigated. A necessary and sufficient condition for the system to be stable is studied. Using the supplementary variable technique, the joint distribution of the server state and the orbit length under steady-state is obtained. Some interesting and important performance measures of the system are provided. Finally, numerical illustrations are presented.

Kernel Regression Estimators for Nonparametric Model Calibration in Survey Sampling

Abstract: This paper introduces new kernel regression estimators with strictly non-negative smoothing weights that are iteratively adjusted. One estimator shares the “optimal” asymptotic bias and variance of the local linear regressor. Other estimators have zero sum of residuals, a desirable property in many applications. In a survey sampling context these estimators can easily be adjusted so that they are internally bias calibrated, which is a property with intuitive appeal. We demonstrate in simulations that one of the estimators with zero sum residuals has bias and variance properties that are very close to “optimal”. In addition, we propose a potentially useful refinement to the usual orders of asymptotic approximations for bias and variance of kernel regression smoothers. The smoothers are illustrated using two examples from fisheries applications, one of which involves data from a stratified random bottom-trawl survey.

Statistical Inference for Non-linear Models involving Ordinary Differential Equations

Abstract: In the context of nonlinear fixed effect modeling, it is common to describe the relationship between a response variable and a set of explanatory variables by a system of nonlinear ordinary differential equations (ODEs). More often such a system of ODEs does not have any analytical closed form solution, making parameter estimation for these models quite challenging and computationally very demanding. Two new methods based on Euler’s approximation are proposed to obtain an approximate likelihood that is analytically tractable and thus making parameter estimation computationally less demanding than other competing methods. These methods are illustrated using a data on growth colonies of paramecium aurelium and simulation studies are presented to compare the performances of these new methods to other established methods in the literature.

Nonlinear Time Series Modeling and Forecasting for Periodic and ARCH Effects

Abstract: 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 serie...

Stochastic Volatility Models with Application in Option Pricing

Abstract: In this paper, we develop a new method for pricing derivatives under stochastic volatility models by viewing the call price as an expected value of a truncated log normal distribution under the risk neutral measure. We also obtain the formula for the estimate of the variance of the call price with the stochastic volatility. Using return data, we estimate the mean and variance of the stochastic volatility of the Black-Scholes model. An extensive empirical analysis of the European call option valuation for S & P 100 Index shows: (i) our method outperforms other compelling stochastic volatility pricing models, (ii) the pricing errors when using our method are quite small even though our estimation procedure is based only on historical return data. Formulas for option pricing and variances derivation for Heston's (1993) continuous time stochastic volatility model and for Taylor's (1973) discrete time stochastic volatility model are also discussed in some detail.

On Distribution-Free Runs Test for Symmetry Using Extreme Ranked Set Sampling with an Application Involving Base Deficit Score

Abstract: Most statistical inferences, which are essential for decision making and research in the area of biomedical sciences, are valid only under certain assumptions. One of the important assumptions in the literature is the symmetry of the underlying distribution of a study population. Several tests of symmetry are found in the literature. Most of these tests suffer from low statistical power which fails to detect a small but meaningful asymmetry in the population. Many investigators have attempted to improve the power of some of these tests. This paper examines several ranked set sample designs for the runs test of symmetry. Our investigation reveals that an optimal ranked set sample design for runs test of symmetry is the extreme ranked set sample (extreme ordered statistics sampling) (ERSS). This design of sampling increases the power and improves the performance of the runs test of symmetry and hence reduces the sample size needed in the study and the cost of the study. Intensive simulation is cond...

Statistical Inference for Spatial Auto-Linear Processes

Abstract: A new class of simultaneous auto-models that clothe naturally the weak dependence of spatial processes, is presented. the parameeters of the so-called auto-linear process are best linear prediction coefficients. With a finite transformation on the original process, the new process has auto-correlations equal to the parameters of interest. New method of moments estimators are proposed and they are consistent and asymptotically normal. Their variance matrix may be written down explicitly in terms of the auto-linear parameters and the result is distribution free. A simulation study and a data example are presented to support the use of the auto-linear model for spatial processes provding convenience for the statistical inference.

The Probability of an Out of Control Signal from Nelson's Supplementary Zig-Zag Test

Abstract: Nelson's ‘supplementary runs’ tests are widely used to augment the standard ‘out of control’ test for an [xbar] control chart, or a chart with individual values, to determine if any special causes exist. The fourth of Nelson's tests gives an out-of-control signal when fourteen points in a row follow a zig-zag pattern (alternating up and down); it is thus a signal that the process has negative autocorrelation. Using a recursive formula, die exact probability of a zig-zag sequence of length 14 is calculated for an in control process. This value does not appear in the SQC literature, but can be simply determined from results of Andre (1879, 1881, 1883). rediscovered by Entringer (1966), which long precede the development of SQC. Two curious properties, relating die probabilities of zig-zag sequences of successive lengths, are also demonstrated.

Product-Convolution of R-Symmetric Unimodal Distributions: An Analogue of Wintner's Theorem

Abstract: Gnedenko and Kolmogorov (1949) in their acclaimed monograph. Limit Distributions for Sums of Independent Random Variables, claimed that the convolutions of unimodal distributions are unimodal. Kai Lai Chung, in an appendix of his English translation of the monograph, by a counterexample, refuted the claim and further noted Wintner's (1938) result that the convolutions of symmetric unimodal distributions are symmetric unimodal. In this note, it is shown that the product-convolutions of unimodal distributions are not unimodal either. Furthermore, an analogue of Wintner's result based on the relatively recent notion of R-symmetry (Mudholkar and Wang, 2007) is offered by showing that the product-convolutions of R-symmetric unimodal distributions are R-symmetric unimodal.