Atasets which have a different structure with respect towards the GYY4137 medchemexpress deviation from the DS model, Ando et al. [10] showed that the all values in the index DS applied to these datasets are the identical, whereas all of the values of your two-dimensional index are distinct. As a result, this two-dimensional index offers extra detailed results than the index DS .On the other hand, existing indexes S , PS and DS are constructed applying energy divergence, while the two-dimensional index is constructed utilizing only Kullback-Leibler info, which can be a special case of power divergence. Additionally, the power divergence includes quite a few divergences, as an example, the energy divergence with = -0.five is equivalent towards the Freeman-Tukey variety divergence, the power divergence with = 1 is equivalent to the Pearson chi-squared sort divergence. For details on power divergence, see Cressie and Read [11], Study and Cressie [12]. Earlier research (e.g., [7,8]) pointed out that it truly is vital to make use of several indexes of divergence to accurately measure the degree of deviation from a model. This study proposes a two-dimensional index that may be constructed by combining existing indexes S and PS depending on energy divergence. The rest of this paper is organized as follows. In Section 2, we propose a generalized two-dimensional index for measuring the degree of deviation from DS. In Section three, we create an approximate confidence region for the proposed two-dimensional index. We then use numerical examples to show the utility with the proposed two-dimensional index in Section 4. We also present results obtained by applying the proposed two-dimensional index to actual data. We close with concluding remarks in Section 5. two. Two-Dimensional Index to Measure Deviation from DS We propose a generalized two-dimensional index for measuring deviation from DS in square contingency tables. The proposed two-dimensional index can concurrently measure the degree of deviation from S and PS. The proposed two-dimensional index is according to energy divergence. Assume that ij ji 0 for all i = j, and ij i j 0 for all (i, j) E, where E= (i, j) (r is odd), i, j = 1, . . . , r (r is even).As a way to measure the degree of deviation from DS, we take into account the Alvelestat tosylate following two-dimensional index: = S PS( -1),Symmetry 2021, 13,three ofwhere indexes S and PS are these thought of by Tomizawa et al. [7] and Tomizawa et al. [8], respectively (see the Appendixes A and B for the particulars of those indexes). Note that the is really a real worth and is chosen by the user. We advise deciding upon the (e.g., -0.5, 0, 1) corresponding for the famous divergence. When = 0, the proposed two-dimensional index is equivalent towards the index by Ando et al. [10]. Therefore, is a generalization of the index by Ando et al. [10]. The two-dimensional index has the following qualities: (i) = (0, 0) if and only if the DS model holds; (ii) = (1, 1) if and only if the degree of deviation from DS is maximum, within the sense that ij = j i = 0 (then ji 0 and i j 0) or ji = i j = 0 (then ij 0 and j i 0) for all i = j, and either ii = 0 or i i = 0 for i = 1, . . . , r/2 (when r is even) or i = 1, . . . , (r – 1)/2 (when r is odd); (iii) = (1, ) if and only if the degree of deviation from S is maximum along with the degree of deviation from PS will not be maximum, inside the sense that ij = 0 (then ji 0) for all i = j; and (iv) = (, 1) if and only in the event the degree of deviation from PS is maximum and the degree of.