Balanced Confidence Regions Based on Tukey's Depth and the Bootstrap

We propose and study the bootstrap confidence regions for multivariate parameters based on Tukey's depth. The bootstrap is based on the normalized or Studentized statistic formed from an independent and identically distributed random sample obtained from some unknown distribution in R ^q . The bootstrap points are deleted on the basis of Tukey's depth until the desired confidence level is reached. The proposed confidence regions are shown to be second order balanced in the context discussed by Beran. We also study the asymptotic consistency of Tukey's depth-based bootstrap confidence regions. The applicability of the method proposed is demonstrated in a simulation study.     

ESTIMATION OF A DISTRIBUTION FUNCTION DOMINATING STOCHASTICALLY A KNOWN DISTRIBUTION FUNCTION

This paper considers the problem of estimating a cumulative distribution function (cdf), when it is known a priori to dominate a known cdf. The estimator considered is obtained by adjusting the empirical cdf using the prior information. This adjusted estimator is shown to be consistent, its limiting distribution is found, and its mean squared error (MSE) is shown to be smaller than the MSE of the empirical cdf. Its asymptotic efficiency (compared to the empirical cdf) is also found.     

Atmospheric Dispersion of Inflammable Substances for Estimating Vulnerable Zones in Hydrocarbon Industry

The present study utilizes an operational model as well as simple empirical relationships for estimating hazard zones due to fire, explosion, and toxic vapor cloud dispersion. The empirical relationships are based on giving appropriate weightage to each of the parameters on which the hazard in question (viz, fire, explosion, toxic vapour dispersion) is dependent. Results from these two approaches [i.e., an operational model FLAMCALC of U.K. Health and Safety Executive (HSE) and an empirical model named FIREX] have been compared with the data obtained from the Mexico City disaster in 1984. In general, results from the empirical approach and FLAMCALC are comparable to the observed effects.     

Estimation of Vulnerable Zones Due to Accidental Release of Toxic Materials Resulting in Dense Gas Clouds

Heavy gas dispersion models have been developed at IIT (hereinafter referred as IIT heavy gas models I and II) with a view to estimate vulnerable zones due to accidental (both instantaneous and continuous, respectively) release of dense toxic material in the atmosphere. The results obtained from IIT heavy gas models have been compared with those obtained from the DEGADIS model [Dense Gas Dispersion Model, developed by Havens and Spicer (1985) for the U.S. Coast Guard] as well as with the observed data collected during the Burro Series, Maplin Sands, and Thorney Island field trials. Both of these models include relevant features of dense gas dispersion, viz., gravity slumping, air entrainment, cloud heating, and transition to the passive phase, etc. The DEGADIS model has been considered for comparing the performance of IIT heavy gas models in this study because it incorporates most of the physical processes of dense gas dispersion in an elaborate manner, and has also been satisfactorily tested against field observations. The predictions from IIT heavy gas models indicate a fairly similar trend to the observed values from Thorney Island, Burro Series, and Maplin experiments with a tendency toward overprediction. There is a good agreement between the prediction of IIT Heavy Gas models I and II with those from DEGADIS, except for the simulations of IIT heavy gas model-I pertaining to very large release quantities under highly stable atmospheric conditions. In summary, the performance of IIT heavy gas models have been found to be reasonably good both with respect to the limited field data available and various simulations (selected on the basis of relevant storages in the industries and prevalent meteorological conditions performed with DEGADIS). However, there is a scope of improvement in the IIT heavy gas models (viz., better formulation for entrainment, modification of coefficients, transition criteria, etc.). Further, isotons (nomograms) have been prepared by using IIT heavy gas models for chlorine, which provide safe distance for various storage amounts for 24 meteorological scenarios prevalent in the entire year. These nomograms are prepared such that a nonspecialist can use them easily for control and management in case of an emergency requiring the evacuation of people in the affected region. These results can also be useful for siting and limiting the storage quantities.     

A non-normal class of distribution function for dose-binary response curve*

A non-normal class of distribution (Edgeworth Series distribution) function in three and four parameters has been considered for dose-binary response relationship. This class accounts for the non-normality (expressed in terms of skewness and kurtosis) present in the relationship in addition to the usual location and scale parameters (generally considered by two parameter models). We present the maximum likelihood method of estimation of the parameters and test of probit (normal distribution) hypothesis. Edgeworth Series distribution when fitted to the data of Milicer & Szczotka (1966) showed an excellent closeness to the observed values, significant improvement over probit and logit fit (Aranda-Ordaz, 1981), and better fit compared to Prentice (1976) model.     

Testing order restricted hypotheses with proportional hazards

Inferences for survival curves based on right censored continuous or grouped data are studied. Testing homogeneity with an ordered restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods are obtained by applying order restricted procedures to the estimates of the regression coefficients. Ordered analogues to the log rank test which are based on the score statistics are considered also. Chi-bar-squared distributions, which have been studied extensively, are shown to provide reasonable approximations to the null distributions of these tests statistics. Using Monte Carlo techniques, the powers of these two types of tests are compared with those that are available in the literature.     

REGRESSION ANALYSIS USING SCRAMBLED RESPONSES

This paper investigates the general linear regression model Y = Xβ+e assuming the dependent variable is observed as a scrambled response using Eichhorn & Hayre's (1983) approach to collecting sensitive personal information. The estimates of the parameters in the model remain unbiased, but the variances of the estimates increase due to scrambling. The Wald test of the null hypothesis H₀: β=β₀, against the alternative hypothesis H_a: β#β₀, is also investigated. Parameter estimates obtained from scrambled responses are compared to those from conventional or direct-question surveys, using simulation. The coverage by nominal 95% confidence intervals is also reported.     

A NOTE ON OPTIMUM STRATIFICATION IN SAMPLING WITH VARYING PROBABILITIES1

Singh and Sukhatme [4] have considered the problem of optimum stratification on an auxiliary variable x when the units from the different strata are selected with probability proportional to the value of the auxiliary variable and the sample sizes for the different strata are determined by using Neyman allocation method. The present paper considers the same problem for the proportional and equal allocation methods. The rules for finding approximately optimum strata boundaries for these two allocation methods have been given. An investigation into the relative efficiency of these allocation methods with respect to the Neyman allocation has also been made. The performance of equal allocation is found to be better than that of proportional allocation and practically equivalent to the Neyman allocation.     

A new class of distribution having decreasing,increasing, and bathtub-shaped failure rate

In this article, we propose a new class of distribution which is based on the concept of exponentiated generalization with some modification so as to provide a better result in terms of flexibility. Our proposed distribution accommodates various shapes of hazard rate including the bathtub. Exponential distribution has been taken as the baseline distribution. Various statistical properties of the proposed distribution have been studied. We have used the method of maximum likelihood for estimation of the parameters of the proposed model. Last, we have analyzed four real datasets to illustrate the flexibility of the model in comparison to eight existing well-known distributions.     

Estimation of population mean using imputation techniques in sample surveys

To deal with the problems of non-response, one-parameter classes of imputation techniques have been suggested and their corresponding point estimators have been proposed. The proposed classes of estimators include several other estimators as a particular case for different values of the parameter. A design based approach is used to compare the proposed strategy with the existing strategies. Theoretical results have been verified through simulation studies handling real data examples.