Atasets which have a distinct structure with respect to the deviation in the DS model,

Atasets which have a distinct structure with respect to the deviation in the DS model, Ando et al. [10] showed that the all values of the index DS applied to these datasets will be the very same, whereas all the values from the two-dimensional index are unique. Thus, this two-dimensional index provides far more detailed benefits than the index DS .On the other hand, existing indexes S , PS and DS are constructed utilizing energy divergence, although the two-dimensional index is constructed applying only Kullback-Leibler information and facts, that is a special case of power divergence. Moreover, the power BI-0115 web divergence involves a number of divergences, as an example, the power divergence with = -0.five is equivalent towards the Freeman-Tukey type divergence, the power divergence with = 1 is equivalent for the Pearson chi-squared variety divergence. For information on energy divergence, see Cressie and Read [11], Study and Cressie [12]. Preceding research (e.g., [7,8]) pointed out that it is actually crucial to make use of various 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 determined by energy divergence. The rest of this paper is organized as follows. In Betamethasone disodium Data Sheet Section 2, we propose a generalized two-dimensional index for measuring the degree of deviation from DS. In Section three, we create an approximate self-assurance area for the proposed two-dimensional index. We then use numerical examples to show the utility with the proposed two-dimensional index in Section four. We also present results obtained by applying the proposed two-dimensional index to genuine information. We close with concluding remarks in Section 5. 2. 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, exactly where E= i, j = 1, . . . , r; (i, j) = ((r 1)/2, (r 1)/2) (r is odd), i, j = 1, . . . , r (r is even).In order to measure the degree of deviation from DS, we contemplate the following two-dimensional index: = S PS( -1),Symmetry 2021, 13,3 ofwhere indexes S and PS are these regarded by Tomizawa et al. [7] and Tomizawa et al. [8], respectively (see the Appendixes A and B for the information of these indexes). Note that the can be a actual worth and is selected by the user. We advise deciding upon the (e.g., -0.5, 0, 1) corresponding to the popular divergence. When = 0, the proposed two-dimensional index is equivalent for the index by Ando et al. [10]. Thus, is often 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 when the degree of deviation from DS is maximum, in 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 when the degree of deviation from S is maximum along with the degree of deviation from PS is just not maximum, inside the sense that ij = 0 (then ji 0) for all i = j; and (iv) = (, 1) if and only when the degree of deviation from PS is maximum and the degree of.