Statistical inference

Often scientific information on various data generating processes are presented in the from of numerical and categorical data. Except for some very rare occasions, generally such data represent a small part of the population, or selected outcomes of any data generating process. Although, valuable and useful information is lurking in the array of scientific data, generally, they are unavailable to the users. Appropriate statistical methods are essential to reveal the hidden jewels in the mess of the row data. Exploratory data analysis methods are used to uncover such valuable characteristics of the observed data. Statistical inference provides techniques to make valid conclusions about the unknown characteristics or parameters of the population from which scientifically drawn sample data are selected. Usually, statistical inference includes estimation of population parameters as well as performing test of hypotheses on the parameters. However, prediction of future responses and determining the prediction distributions are also part of statistical inference. Both Classical or Frequentists and Bayesian approaches are used in statistical inference. The commonly used Classical approach is based on the sample data alone. In contrast, increasingly popular Bayesian approach uses prior distribution on the parameters along with the sample data to make inferences. The non-parametric and robust methods are also being used in situations where commonly used model assumptions are unsupported. In this chapter, we cover the philosophical and methodological aspects of both the Classical and Bayesian approaches. Moreover, some aspects of predictive inference are also included. In the absence of any evidence to support assumptions regarding the distribution of the underlying population, or if the variable is measured only in ordinal scale, non-parametric methods are used. Robust methods are employed to avoid any significant changes in the results due to deviations from the model assumptions. The aim is to provide an overview of the scientific domain of statistical inference without going in to minute details of any particular statistical method. This is done by considering several commonly used multivariate models, following both normal and non-normal, including elliptically contoured, distributions for the responses