In this work, we propose two information generating functions: general weighted information and relative information generating functions, and study their properties. It is shown that the general weighted information generating function (GWIGF) is shift-dependent and can be expressed in terms of the weighted Shannon entropy. The GWIGF of a transformed random variable has been obtained in terms of the GWIGF of a known distribution. Several bounds of the GWIGF have been proposed. We have obtained sufficient conditions under which the GWIGFs of two distributions are comparable. Further, we have established a connection between the weighted varentropy and varentropy with proposed GWIGF. An upper bound for GWIGF of the sum of two independent random variables is derived. The effect of general weighted relative information generating function (GWRIGF) for two transformed random variables under strictly monotone functions has been studied. Further, these information generating functions are studied for escort, generalized escort and mixture distributions. Specially, we propose weighted β-cross informational energy and establish a close connection with GWIGF for escort distribution. The residual versions of the newly proposed generating functions are considered and several similar properties have been explored. A non-parametric estimator of the residual general weighted information generating function is proposed. A simulated data set and two real data sets are considered for the purpose of illustration. Finally, we have compared the non-parametric approach with a parametric approach in terms of the absolute bias and mean squared error values.

There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: (i) i.i.d. sampling, (ii) record values, (iii) Type-II censoring and (iv) progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.