On ‘Useful’ R-norm Relative Information Measures and Applications Oriental Journal of Physical Sciences

In this communication a new ‘useful’ R-norm relative information measure is introduced and characterized axiomatically. Its in equalities with particular cases are described. This new information measure has also been applied to study the status of the companies with regard to their loss and profit and that has been illustrated by considering empirical data and drawing figures. Ad joint of the relative information measure is also defined with the illustration of its application in share market with examples.


Introduction
Information theory as a separate subject is about 70 years old. Since information is energy, therefore it is measured, managed, regulated and controlled for the sake of welfare of humanity. The role of information function is to remove uncertainty and the amount of uncertainty removed is a measure of information.
The concept of information proved to be very important and universally useful. These days language used in telephones, business management, and cybernetics falls under the name "Information Processing". In addition to this, information theory particularly measures of information have applications in physics, statistical inference, data processing and analysis, accountancy, psychology, etc.
Shannon 24 was the first who developed a measure of uncertainty. He was interested in communicating information across the channel in which some information is lost in the process of communication and that was called a noisy channel. His objective was to measure the amount of information lost. He defined a measure of uncertainty of a probability distribution as given below: H(P)= -kΣ P i log P i , ... (1.1) where k is an arbitrary positive constant. The measure(1.1) was called entropy. Thereafter, Shannon's entropy wascharacterized by various researchers like Khinchin, 16 Fadeev. 8 Teverberg, 25 Chandy and Mcleod, 5 Kendal, 15 Lee, 20 Berges, 2 Cziszar, 7 Cheng, 6 etc. on using different sets of postulates.
The quantity (1.1) measures the amount of information of probability distribution P when effectiveness or importance of the events is not taken into account. In addition to this; some probabilistic problems also play important role. Considering effectiveness of the outcomes, Belis and Guiasu 1 introduced U= (u 1 ,u 2 ,……,u n ) as a utility distribution, where u i >0, is the usefulness of an event having probability of occurrence pi and consequently, "self useful information' is defined as given below: The measure (1.2) is based two postulated as given below: P1. In case all the events of an random experiment have the same utility u>0, then the self used information generated by the product of two statistical independent events E 1 and E 2 can be expressed as the sum of the self-useful information provided by E 1  In this communication , the 'useful' relative information measure is defined and characterized axiomatically in section 2. The new measure thus introduced is generalized in section 3 with its and its particular cases are studied in section 4. The applications of new R-norm information measure are described in section 5. In section 6 its ad joint by taking empirical data is studied with its illustration graphically. In the end the conclusion is given along with an exhaustive list of references.

Useful' Relative Information Measure
Let X be a random variable in an experiment and be its probability distribution having U=( u 1 , u 2 …….., u n ) as a utility distribution, where ui>0 for each i, is the utility of an event having probability p i .
A 'useful' directed divergence measure was defined by Bhaker and Hooda [3] and characterized as given below:  Kumaret al. [18] also defined the following 'useful' R-norm relative measure: ... (2.8) There are many other generalizations of (2.8) also and one of them is where A 'Useful' R-Normrelative Information Measure. Theorem 3.1 Let P and Q be two probability distributions attached with a utility distribution U, then the following holds: or R-β >0, (2.6) is positive.
It implies that and are convex functions of P in view of .
Similarly for, is also aconvex function of P.
For R-β<0, (3.6) is negative, so and are concave functions of P, since .
On same lines we can prove that D β R ( U;P: Q) is a convex function of Q for R-β<0 and R-β>0 provided .

A Generalized 'Useful' R-norm Relative information Measure of Degree β
We consider the following function: ... In particular when β=1 and R→1, then reduces to which is (2.1)

Particular Cases
When then (4.11) reduces to ... (4.13) and in case P = C; D(P;C;U) =0,. Further, it can be verified that D(P:C;U) is a convex function of P. In case u i = 1 for each i in (4.10), it reduces ... (4.24) which is well known Renyi's [23] entropy of order R.

Illustration with an Example
In this section we consider production data of different companies due to Nager and Singh 22 represented in Table 5.1. We calculate D β R (P:Q;U) in Table 5  Next we compute these values of the generalized 'useful' r-norm relative measure when R = 2 and β =0.5 in the following table: Now,the graphically representation of the new 'useful' R-norm relative information of degree β when R=2 and β=0.5 is given in fig. 5.1.  The amount of divergence values can be arranged for forecasting the profit maximization in a table as Thus (6.1) is called the ad joint of (2.9). Similarly, we compute these values of the ad joint of generalized measure at R = 2, β =0.5 and representedt able (6.1) as given below: Considering the above table 6.1, the graph is drawn as given in the following figure 6.1:  The data for forecasting the profit maximization is arranged as given below:

Interpretation
The adjoint of' useful' R-norm relative information of degree in decreasing order in the table (6.1 to suggest the investor to make investment in the company of maximum divergence

Conclusion
In this paper we have defined and characterized the generalized 'useful' R-norm relative information of degree β and discussed its particular cases also. The application of this information measure has been studied. The adjoint of this measure is defined and its application in share market and decision making problems are described graphically.
The 'Useful' R-norm relativei nformation measures of degree and its ad joint are defined and studied in this communication can further be generalized parametrically and applied in planning, forecasting, agriculture, etc.