Non-linear and Linear Estimators in Ultrastructural Model with Replicated Observations

Abstract

The least square estimation provides best linear unbiased estimator for the regression coefficients in a linear regression model only when the observation are recorded without any error. In case, the observations on variables are contaminated by measurement errors, the same least squares estimator not only becomes biased but inconsistent also. The inconsistency of this estimator can be removed if some additional information on parameters of the model in some suitable form, e.g., measurement error variances associated with dependent and independent variables or their ratio etc., can be incorporated in the estimation procedure. Under this proposition, the question arises that from where we can get this additional information? The simple answer is - from some past experience, long association with the experiment etc. In practice, it is very difficult to know such values. Even if they are known, their reliability is always questionable which in turn will change the value of the estimated parameter. A solution to this problem is the option to use replicated observations on the true values of variables. Such replicated observations can be utilized to estimate the unknown parameters and a feasible version of estimators can be obtained which will be consistent under the influence of measurement errors. But this approach will increase the variability in the estimators as they are the feasible version of the true estimators. Another alternative way to use such replicated observations is that if they can be used in such a way that the resulting estimator is independent of unknown parameters, information or even does not involve the estimation of unknown parameters from the sample values. An attempt has been made in this direction in this talk. An ultrastructural model with replicated observations is considered. The replicated observations are utilized in two types of regressions - regression of original observations and regression of group means. A non-linear combination of these two estimators is considered and a new estimator is proposed. This estimator is independent of any unknown value or any value needed to be estimated from the sample as well as consistent. Further, most of the literature dealing with measurement errors assumes that the probability distribution of all the errors is normal. In practice, this assumption may be violated. So in our set up we have considered the non- normal distribution of measurement errors as well as the random error component. The properties of this estimator are derived by employing large sample asymptotic approximation theory and studied. The effect of not necessarily distributed errors is also analyzed. So the results are very general and can be used for any probability distribution whose first four moments are finite. The estimator is exposed to some real life data and its performance is studied.