Deviation from S just isn’t maximum, in the sense that ij = 0 (then i j 0) for all (i, j) E. 3. Approximate Confidence Area for the Proposed Two-Dimensional Index Let n = (n11 , n12 , . . . , n1r , n21 , n22 , . . . , n2r , . . . , nr1 , nr2 , . . . , nrr ) , = (11 , 12 , . . . , 1r , 21 , 22 , . . . , 2r , . . . , r1 , r2 , . . . , rr ) . Assume that n has a multinomial distribution with sample size N and probability vector . The N ( p – ) has an asymptotically Gaussian distribution with imply zero and covariance matrix D – , where p = n/N and D is actually a diagonal matrix using the elements of around the main diagonal (see, e.g., Agresti [13]). We estimate by ^ ^ ^ ^ ^ = (S , PS ) , exactly where S and PS are offered by S and PS with ij replaced ^ by pij , respectively. Utilizing the delta technique (see Agresti [13]), N ( – ) has an asymptotically bivariate Gaussian distribution with imply zero and covariance matrix = = 11 D – 12,with 12 = 21 . Let = ij ,i=j=(i,j) Eij .The elements 11 , 12 , and 22 are expressed as follows:= =S 1D – ijSiji=j- S,= =SD – ij – S PSiji=jWij- PS,Symmetry 2021, 13,4 of=PSD -PS=where for -1 ij Wij two (i,j)E- two PS , ij=1 log 2ac ij log 2 1 c c (2aij ) – 1 ac (2aij ) – (2ac ) ji ji 2 -( = 0),( = 0), ( = 0),Wij=1 log 2cc ij log two 2 1 c c (2cij ) – 1 cic j (2cij ) – (2cic j ) -( = 0),withc aij =ij , ij jic cij =ij . ij i j Note that the asymptotic variances 11 and 22 of S and PS , respectively, have already been offered by Tomizawa et al. [7] and Tomizawa et al. [8], however, the asymptotic covariance 12 of S and PS is initial derived in this study. An approximate bivariate 100(1 – ) self-assurance region for the index is offered by ^ N ( – ) -1 ^ ( – ) 21-;2) , (exactly where 21-;two) is the upper 1 – percentile in the central chi-square distribution with two ( degrees of freedom and is provided by with ij replaced by pij . four. Examples four.1. Utility of your Proposed Two-Dimensional Index In this section, we demonstrate the usefulness employing many divergences to PHA-543613 MedChemExpress examine the degrees of deviation from DS in AZD4625 manufacturer various datasets. We take into consideration the two artificial datasets in Table 1. We evaluate the degrees of deviation from DS for Table 1a,b utilizing the self-assurance area for . Table 2 offers the estimated values of and for Table 1a,b.Table 1. Two artificial datasets. (a) 137 291 1 22 71 605 450 645 948 400 268 639 986 997 361 124 (b) 801 964 85 809 247 973 952 697 132 56 333 625 104 406 393Symmetry 2021, 13,five of^ ^ Table 2. Estimated indexes S and PS and estimated covariance matrix of applied for the information in Table 1a,b. (a) For Table 1a Index 0 1 (b) For Table 1b Index 0 1 ^ S 0.287 0.348 ^ PS 0.259 0. ^Covariate Matrix ^ PS 0.341 0.370 ^^ S 0.346 0.^^0.471 0.0.278 0.0.417 0.Covariate Matrix ^^0.853 1.0.488 0.0.538 0.From Figure 1, we see that the self-assurance regions for usually do not overlap for the information in Table 1a,b. We are able to conclude that Table 1a,b has a distinct structure within the degree of deviation from DS. That is, Table 1a,b has a various structure with regard towards the degree of deviation from S or PS. From Figure 1, when = 0, we are able to conclude that the degree of deviation from DS for Table 1a is higher than that for Table 1b, but when = 1, we can’t conclude this. We should, as a result, examine the value of your two-dimensional index employing many to compare the degrees of deviation from DS for many datasets.0.0.40 1a0.1a0.35 1bPS0.PS0.1b 0.0.0.20 0.20 0.25 0.30 S 0.35 0.0.20 0.20 0.