Research Projects
(a
) Improved Estimation
Traditionally the
unknown population mean is estimated by the sample mean. Improved estimators,
in the sense of admissibility, accuracy and efficiency are recent phenomenon in
statistical inference. Improved estimators such as the preliminary test,
shrinkage and positive-rule shrinkage estimators, perform better than the
traditional estimators based on normal models. When a number of alternative
estimators are available to estimate an unknown parameter (scalar or vector) a
natural question is, which one should be used and why? The choice obviously
depends on the objective of the study and some appropriate criteria to judge
the relative performance of the estimators. Generally, in the classical theory
of statistics several criteria are employed to judge the characteristics of
good estimators. Most common/popular of these criteria include unbiasedness, mean squared error (mse), and quadratic risk. Although the level of
emphasis on these criteria varies from application to application, it is
desirable that a good estimator will meet the most important/appropriate criterion determined by the
researcher, and over perform the rest.
(b ) Improving Tests
The idea of using non-sample prior information in the form of pre-testing for improving properties of estimators is applied in the testing regime to achieve better power of the ultimate test in this paper. For example, to test the intercept of a simple
regression model, prior information from previous investigations or expert knowledge on the suspected value of the slope is potentially beneficial. Any uncertainty on the value of the slope is removed by performing a pre-test before testing the significance of the intercept. The impact of the pre-test on the performance (power and size) of the ultimate test is studied. Defining unrestricted test (UT), restricted test (RT) and pre-test test (PTT) corresponding to the unrestricted (UE), restricted (RE), and preliminary test estimators (PTE) in the estimation case, the critical region and power functions are derived. Analytical and graphical comparisons of the three tests are obtained by studying the power functions with respect to size and power of the tests. It is shown that PTT achieves a reasonable dominance over the others asymptotically.
The problem can be addressed for both parametric and
non-parametric set ups. Robust procedure based on M-estimator can also be used
to formulate a test and deriving its power function. In comparison to the other
non-pre-test based test, the PTT based on pre-test performs better and its
power function behaves similar to the quadratic risk function of the
preliminary test estimator (PTE). Guidelines in choosing appropriate value of
nominal sizes of pre-test for appropriate value of size of the PTT subject to
the values of the slope can also be investigated.
(c) Estimation of Parameters of Regression
Model Under Uncertain Prior Information
The estimation of
the slope and the intercept parameters of linear regression model under the
constraint of suspected equality of the slopes of two simple regression models is handled. The suspected equality slopes is presented by a
null hypothesis and is
tested by using an appropriate test statistic. Different estimators of the
slope and intercept are defined. Some important statistical properties of the
unrestricted, restricted and preliminary test estimators are investigated.
Conclusions about the performances of different estimators under various
conditions are discussed to explore the optimal choice. Another constraint in
terms of the coefficient of distrust on the null hypothesis is introduced in
the estimation process, especially for the definition of shrinkage restricted and shrinkage preliminary
test estimators. Estimators are compared under various conditions for different
values of the coefficient of distrust. Applications in the medical, biological and
agricultural sciences are considered.
(d ) Estimation
Under Alternative Test Statistics
The preliminary
test estimator depends of the choice of the statistic to remove the uncertainty
in the prior information as well as the preselected
level significance. Three different tests, namely the World (W) Test,
Likelihood Ratio (LR) Test and Lagrange Multiplier (LM) Tests, are used in the
literature to test many linear hypotheses. The exact distribution of the test
statistics are complicated and often researchers use asymptotic equivalent of
the test statistics for inferential purposes. The use of the same approximate
chi-square test does not have the correct significance level. This causes
conflict in the statistical inference. Use of the approximate chi-square test
statistic in the definition of the preliminary test estimator causes conflict
in the biases and mean squared errors of the estimators. The use of modified
test statistic reduces the conflicts to some extent but does not remove it.
Edgeworth size corrected test does a lot better job than the modified test to
reduce the conflict. This fact has been studied for various models under the
framework of improved estimation.
(e ) The
Multivariate Student-t Distribution
The customary use of the normal
model is under serious question when the population distribution is symmetric
but have heavier tails that the normal distribution. Also, the normal model
fails to incorporate dependent but uncorrelated responses. In such cases the
multivariate Student-t distribution provides an appropriate model for the
population. Such a model can be viewed as a mixture of normal and inverted
gamma distributions. Using this result we obtained the maximum likelihood
estimators of the mean and scale parameters of multivariate Student-t
distribution. The model has been used to find appropriate test statistic to
test the mean vector. The non-null distribution of the test statistic has been
derived. The distribution of the sum of squares and product
matrix for the multivariate Student-t model as well as the predictive distribution
of future model have been proposed. Similar results for the matrix T
model are also obtained.
(f ) Predictive Inference
Prediction distribution is the basis for many predictive inferences. Unlike the common practice of estimating parameters of a model or performing tests of hypotheses regarding the parameters involved, often the aim of a researcher/practitioner is to predict the value of a (or a set of) future response(s) from a given model. The technique of prediction is used in many real world situations as it has a common sense appeal and simple interpretation. The prediction distribution is the probability distribution of one or more future (unobserved) responses, conditional on a set of observed responses from the same model. The method is useful in both univariate and multivariate problems. Predictive inference is possible for models with independent as well as dependent and correlated responses. Bayesian and other approaches are adopted for the purpose of predictive inference. Available methods can handle the conventional normal model and non-normal robust models. Application of predictive inference includes problems in areas such as tolerance regions, model selection, process control, optimisation, perturbation and many others.
(g) Testing after Pre-test
Testing of a parameter of a model is a common task in statistical inference. Like, the preliminary test estimation, improved tests can be defined for the test of a parameter in the presence of uncertain non-sample information on the value of another parameter. Combining both the sample data and the non-sample prior information, we define preliminary test based improved test. The power function as well as the size and power of the improved test are derived and analysed. Performance of the proposed test is compared with the commonly used test based on sample data only. Both parametric and nonparametric methods are used for the project.
(h) Meta-analysis for
RCT
This is a statistical method to combine data from several independent studies conducted using randomised control trails for making inferences. Analyses are done for relative risks and odd ratios for binary data, and weighted mean difference for ratio/scale data. Both classical and Bayesian approached can be used.
(i ) Occupational Mobility
Mobility
models are very useful to explain movements of people over socio-economic and
job categories. Occupational mobility deals with the movements of individuals
over job categories during their employment periods. Since the time interval
between successive job changes is a random variable, different occupational
mobility models have been developed by scientists using modified Markov and
semi-Markov processes. This phenomenon can be modelled by considering the
underlying factors such as job satisfaction, salary, distance of the work place
from the home, family requirements and others. Unlike most of the previous
works in this area, our study suggests a new measure of occupational mobility
based on the distribution of wages. Here a general occupational mobility model
has been developed to study the pattern of mobility during the service life of
employees. First the probability distribution of the number of job changes in
the entire employment life of individuals has been obtained considering the
inter-job offer times (within an interval) and the associated wages as random
variables. Then a measure of occupational mobility based on this distribution
has been developed. The results are obtained under both frequentist and
Bayesian frameworks. As an application
of the proposed model the results in this paper have been illustrated by using
data from a recent survey among the staff members of the