Christian H Weiß1. 1. Department of Mathematics and Statistics, Helmut Schmidt University, 22043 Hamburg, Germany.
Abstract
The family of cumulative paired ϕ-entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen's κ, with the weights being related to the actual choice of ϕ. Again, the asymptotic distribution of the corresponding sample κϕ is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel ϕ-entropy-related measures are applied to an ordinal time series on the daily level of air quality.
The family of cumulative paired ϕ-entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen's κ, with the weights being related to the actual choice of ϕ. Again, the asymptotic distribution of the corresponding sample κϕ is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel ϕ-entropy-related measures are applied to an ordinal time series on the daily level of air quality.
Entities:
Keywords:
Cohen’s κ; dispersion; entropy; ordinal time series; serial dependence
During the last years, ordinal data in general [1] and ordinal time series in particular [2] received a great amount of interest in research and applications. Here, a random variable X is said to be ordinal if X has a bounded qualitative range exhibiting a natural order among the categories. We denote the range as with some , and we assume that . The realized data are denoted as with . They are assumed to stem either from independent and identically distributed (i. i. d.) replications of X (then, we refer to as an ordinal random sample), or from a stationary ordinal stochastic process (then, are said to be an ordinal time series).In what follows, we take up several recent works on measures of dispersion and serial dependence in ordinal (time series) data. Regarding ordinal dispersion, the well-known measures such as variance or mean absolute deviation cannot be used as the data are not quantitative. Therefore, several tailor-made measures for ordinal dispersion have been developed and investigated in the literature, see, among others, [3,4,5,6,7,8,9,10,11,12,13,14,15]. The unique feature of all these measures is that they rely on the cumulative distribution function (CDF) of X, i.e., on with for ( is omitted in as it necessarily equals one). They classify any one-point distribution on as a scenario of minimal dispersion, i.e., if all probability mass concentrates on one category from (maximal consensus):By contrast, maximal dispersion is achieved exactly in the case of the extreme two-point distribution (polarized distribution), , where we have 50 % probability mass in each of the outermost categories (maximal dissent). Further details on ordinal dispersion measures are presented in Section 2 below.Building upon earlier works by Klein [16], Yager [17], it was recently shown by Klein Doll [18], Klein et al. [19] that the aforementioned ordinal dispersion measures can be subsumed under the family of so-called “cumulative paired -entropies” (see Section 2), abbreviated as , which constitutes the starting point of the present article. Our first main task is to derive the asymptotic distribution of the corresponding sample version, , for both i. i. d. and time series data, and to check the finite sample performance of the resulting approximate distribution, see Section 3 and Section 5.In the recent paper by Weiß [20] on the asymptotics of some well-known dispersion measures for nominal data (i.e., qualitative data without a natural ordering), it turned out that the corresponding dispersion measures—if these are applied to time series data—are related to specific measures of serial dependence. Therefore, our second main task is to explore for a similar relation in the ordinal case, if considering the -family for measuring dispersion. Ordinal dependence measures can be defined in analogy to the popular autocorrelation function (ACF) for quantitative time series, namely by using the lagged bivariate CDF for time lags as their base [14]. For the novel family of measures, which cover the existing ordinal Cohen’s [14,15] as a special case, we derive the asymptotics under the null hypothesis of i. i. d. time series data, see Section 4. This result is used in Section 5 to test for significant serial dependence, in analogy to the application of the sample ACF to quantitative time series. In Section 6, we discuss an illustrative real-world data example from an environmental application, namely regarding the daily level of air quality. The article concludes in Section 7.
2. The Family of Cumulative Paired -Entropies
Klein Doll [18], Klein et al. [19] proposed and investigated a family of cumulative paired -entropies. Although their main focus was on continuously distributed random variables, they also referred to the ordinal case and pointed out that many well-known ordinal dispersion measures are included in this family. Here, we exclusively concentrate on the ordinal case as introduced in Section 1, and we define the (normalized) cumulative paired -entropy as (see Section 2.3 in Klein et al. [19])
The entropy generating function (EGF) is defined on , it satisfies , and it is assumed to be concave on . Later in Section 3, when deriving the asymptotic distribution of the sample counterpart , we shall also require that is (twice) differentiable. As pointed out in Section 2.3 and 3.1 of Klein et al. [19], several well-known measures of ordinal dispersion can be expressed by (1) with an appropriate choice of .Leik’s ordinal variation [11] corresponds to the choice (which is not differentiable in ) because of the equality :An analogous argument applies to the whole family of ordinal variation measures, with [3,9,10,13]. Choosing with , we have the relationNote that leads to the LOV, and the case is known as the coefficient of ordinal variation, [4,8].Related to the previous case with , becomes the widely-used index of ordinal variation [3,7,8] if choosing :The cumulative paired (Shannon) entropy [12] corresponds to the choice (with the convention ):can be embedded into the family of a-entropies [21,22],
as the boundary case . Plugging-in (6) into (1), one obtainsNote that both and in (7) lead to the IOV (4).The EGFs involved in (2)–(4) are symmetric around , i.e., they satisfy . This is also illustrated by Figure 1, where the cases (left) and (right; both in bold black) agree with each other except a scaling factor. The plotted EGFs are maximal in with . The EGF is maximal in with for , whereas in the boundary case , takes its maximal value at . However, since in (1) is normalized, it is not necessary to care for a possible rescaling of if computing .
Figure 1
Plot of EGFs against z. (Left): ; (right): .
The dotted curve in the right part of
Using that
The Lambert W function is defined to solve the equation
Thus, sinceLet us conclude this section with some examples of measures (see Figure 2). For all examples, we set , i.e., we have five ordinal categories like, for example, in the case of a simple Likert scale. In the left part of Figure 2, f was computed according to the binomial distribution , which has maximal dispersion for . This is also recognized by any of the plotted measures , with their maximal dispersion values varying around 0.6. This medium level of dispersion is plausible because is far away from the extreme two-point distribution. The right part of Figure 2, by contrast, shows the for the two-point distribution with (). So corresponds to a one-point distribution in (minimal dispersion), and to the extreme two-point distribution (maximal dispersion). Accordingly, all measures reach their extreme values 0 and 1, respectively. It is interesting to note that outside these extreme cases, the dispersion measures judge the actual dispersion level quite differently; see the related discussion in Kvålseth [10], Weiß [13].
Figure 2
Plot of against p for specific cases of and . (Left): binomial distribution ; (Right): two-point distribution with .
3. Asymptotic Distribution of Sample
From now on, we focus on the sample version of from (1), i.e., on , where denotes the vector of cumulative relative frequencies computed from . To derive the asymptotic distribution of , which is to be used as an approximation to the true distribution of , we recall the following results from Weiß [14]. Provided that the data-generating process (DGP) satisfies appropriate mixing conditions, e.g., -mixing with exponentially decreasing weights (which includes the i. i. d.-case), it holds that
For an analogous result in the presence of missing data, see Theorem 1 in Weiß [15]. In (8), finite (co)variances are ensured if we require that holds for all (“short memory”). In particular, all sums vanish in the i. i. d.-case. Otherwise, they account for the serial dependence of the DGP. This can be seen by considering the trace of , which equals
Here, the term agrees with the IOV in (4) except the normalizing factor , i.e., it corresponds to with . The term , in turn, is the ordinal Cohen’s [14] defined by
It is a measure of signed serial dependence, which evaluates the extent of (dis)agreement between and by positive (negative) values.Based on Taylor expansions of in , we shall now study its asymptotic behavior. To establish asymptotic normality and to derive the asymptotic variance of , we need to be differentiable. For an asymptotic bias correction, which relies on a second-order Taylor expansion, even has to be twice differentiable (then, the concavity of implies that ).From the examples discussed inAssuming to be (twice) differentiable, the partial derivatives of according to (1) are
We denote the gradient (Jacobian) of by , and the Hessian equals . Note that if is symmetric around , i.e., if , then and .Let us compute the derivatives required in (For
and for
Note that bothFor
For
Note that these derivatives are continuous inPartial derivatives (10) for EGFs discussed in Example 1.Using (10), the second-order Taylor expansion equals
According to (8), the linear term in (11) is asymptotically normally distributed (“Delta method”), provided that does not vanish (see Remark 3 below). Then, we conclude from (8) that
The approximate bias correction relies on the quadratic term in (11), because . Using (8), we conclude that
Let us summarize the results implied by (12) and (13) in the following theorem.Under the mixing assumptions stated before (
In addition, the bias-corrected mean ofNote that the second-order derivatives are negative due to the concavity of , so exhibits a negative bias. expresses the effect of serial dependence on . For i. i. d. ordinal data, , so Theorem 1 simplifies considerably in this case, namely to . The bias of is affected by serial dependence via , which is a -type measure reflecting the extent of (dis)agreement between lagged observations, recall (9). More precisely, can be interpreted as weighted type of , where the weights depend on the particular choice of . It thus provides a novel family of measures of signed serial dependence, the asymptotics of which are analyzed in Section 4 below.In the special case
Hence, the bias is determined by both the serial dependence and the dispersion of the process.As another simple example, consider the choice
Thus, we have a unique i. i. d.-bias, independent of the actual marginal CDF
So, besides the pairA few examples are plotted inThe newly obtained measure from (14) constitutes a counterpart to the nominal measures in Weiß [20]. It is worth pointing out that the latter measures were derived from the nominal entropy and extropy, respectively, while the in (5) can be interpreted as a combination of cumulative entropy and extropy. It has to be noted that also shares a disadvantage with : if only one of the equals 0 or 1, we suffer from a division by 0 in (14). For according to (9), by contrast, a division by 0 only happens in the (deterministic) case of a one-point distribution. As a workaround when computing , it is recommended to replace the affected summands in (14) by 0.If For example, plugging-in [14].
4. Asymptotic Distribution of Sample
The bias equation in Theorem 1 gives rise to a novel family of serial dependence measures for ordinal time series, namely
for a given EGF . Some examples are plotted in the left part of Figure 4, where the DGP assumes that the rank counts follow the BAR model with marginal distribution and dependence parameter ; recall the discussion of Figure 3. So, the rank counts have the first-order ACF , whereas the plotted have absolute value .
Figure 4
Plots for specific cases of and : against BAR’s dependence parameter with marginal (left); against p of marginal (right).
Figure 3
Plots for specific cases of and : (left) and (right) against p of marginal , where i. i. d. DGP in (a,b), and BAR DGP with dependence parameter in (c,d).
In practice, the sample version of this measure, , is particularly important, where the cumulative (bivariate) probabilities are replaced by the corresponding relative frequencies. For uncovering significant deviations from the null hypothesis of serial independence (then, ), the asymptotic distribution of under the null of i. i. d. time series data is required. For its derivation, we proceed in an analogous way as in Section 3. As the starting point, we have to extend the asymptotics of the marginal sample CDF in (8) by also considering the bivariate sample CDF . Let , and denote its sample version by . Then, under the same mixing conditions as in Section 3, Weiß [14] established the asymptotic normality
and he derived general expressions for the asymptotic (co)variances . Analogous results for the case of missing data can be found in Supplement S.3 of Weiß [15]. For the present task, the asymptotics of the i. i. d.-case are sufficient. Then, for all , and the covariances in (16) are given by
see Weiß [14] (as well as p. 8 in Supplement S.3 of Weiß [15] if being concerned with missing data).Next, we derive the asymptotics of under the i. i. d.-null, and this requires to derive the second-order Taylor expansion for ; details are postponed to Appendix A.1. As higher-order derivatives of , which are initially used while deriving a bias correction of , cancel out, the final result still relies on derivatives of up to order 2 only.Under the null hypothesis of i. i. d. data, i.e., ifIn addition, the bias-corrected mean ofNote that we have a unique bias correction for any of the measures , independent of the choice of the EGF . Thus, for application in practice, it remains to compute the asymptotic variance in Theorem 2. This only requires knowledge about to evaluate the , but not about higher-order derivatives of the EGF (see Example 3 for illustration). Further examples are plotted in the right part of Figure 4, where was computed for the marginal distribution . The oscillating behavior of for with is quite striking. It is also interesting to note that among the plotted -measures, the novel (case ) has the lowest variance.While we have a unique bias correction forforwhile forImportant special cases are:For any other choice ofForThus, we getsee Theorem 7.2.1 in Weiß [ForThus, we get
5. Simulation Results
In what follows, we discuss results from a simulation study, being tabulated in Appendix B, where replications per scenario were used throughout. In view of our previous findings, achieved when discussing the asymptotics plotted in Figure 3 and Figure 4, we do not further consider the choice for the EGF , but we use instead. The latter choice, in turn, was not presented before as the resulting asymptotic curves could hardly be distinguished from the case . So, altogether, as well as were taken into account for simulations. The ordinal data were generated via binomial rank counts, with , which either exhibit serial dependence caused by a BAR DGP with dependence parameter , or which are i. i. d. (corresponding to ).Let us start with the ordinal dispersion measures . Table A1 presents the simulated means (upper part) and SEs (lower part) for the case of i. i. d. ordinal data, and these are compared to the asymptotic values obtained from Theorem 1. Generally, we have an excellent agreement between simulated and asymptotic values, i.e., the derived asymptotic approximation to the true distribution of works well in practice. This is even more remarkable as also the low sample size is included. There is a somewhat larger deviation only for the mean in the case , i.e., for the CPE (5), if and . In this specific case, the simulated sample distribution might be quite close to a one-point distribution, which might cause computational issues for (5); recall that the convention has to be used. However, as the approximation quality is good throughout, a pivotal argument for the choice of in practice might be that the least SEs are observed if using with .
Table A1
Simulated vs. asymptotic mean and SE of for specific cases of and , where i. i. d. rank counts and sample size n.
Simulated Mean
Asymptotic Mean
p
n
a=1
a=1.5
a=2
a=2.5
q=4
a=1
a=1.5
a=2
a=2.5
q=4
0.1
50
0.305
0.282
0.273
0.272
0.336
0.301
0.282
0.273
0.272
0.337
100
0.310
0.285
0.276
0.274
0.341
0.308
0.285
0.276
0.274
0.341
250
0.313
0.287
0.278
0.276
0.343
0.312
0.287
0.278
0.276
0.343
500
0.314
0.288
0.278
0.277
0.344
0.314
0.288
0.278
0.277
0.344
1000
0.314
0.288
0.279
0.277
0.344
0.315
0.288
0.279
0.277
0.344
0.3
50
0.538
0.500
0.484
0.481
0.610
0.538
0.500
0.484
0.481
0.610
100
0.546
0.506
0.490
0.486
0.618
0.545
0.505
0.489
0.486
0.617
250
0.550
0.508
0.492
0.488
0.622
0.550
0.509
0.492
0.489
0.622
500
0.551
0.509
0.493
0.489
0.624
0.551
0.510
0.493
0.490
0.624
1000
0.552
0.510
0.494
0.490
0.625
0.552
0.510
0.494
0.490
0.625
0.5
50
0.601
0.554
0.536
0.532
0.679
0.602
0.555
0.536
0.532
0.679
100
0.609
0.560
0.541
0.537
0.687
0.609
0.561
0.541
0.537
0.688
250
0.613
0.564
0.544
0.540
0.693
0.614
0.564
0.545
0.540
0.693
500
0.616
0.566
0.546
0.542
0.696
0.615
0.565
0.546
0.542
0.695
1000
0.616
0.566
0.546
0.542
0.696
0.616
0.566
0.546
0.542
0.696
0.1
50
0.048
0.044
0.043
0.043
0.053
0.049
0.044
0.043
0.043
0.054
100
0.034
0.031
0.030
0.030
0.037
0.034
0.031
0.030
0.030
0.038
250
0.022
0.020
0.019
0.019
0.024
0.022
0.020
0.019
0.019
0.024
500
0.015
0.014
0.014
0.014
0.017
0.015
0.014
0.014
0.014
0.017
1000
0.011
0.010
0.010
0.009
0.012
0.011
0.010
0.010
0.010
0.012
0.3
50
0.053
0.051
0.051
0.050
0.059
0.053
0.051
0.051
0.051
0.058
100
0.037
0.036
0.036
0.036
0.041
0.037
0.036
0.036
0.036
0.041
250
0.024
0.023
0.023
0.023
0.026
0.024
0.023
0.023
0.023
0.026
500
0.017
0.016
0.016
0.016
0.018
0.017
0.016
0.016
0.016
0.018
1000
0.012
0.011
0.011
0.011
0.013
0.012
0.011
0.011
0.011
0.013
0.5
50
0.056
0.055
0.054
0.054
0.065
0.055
0.055
0.054
0.054
0.065
100
0.039
0.038
0.038
0.038
0.046
0.039
0.039
0.038
0.038
0.046
250
0.025
0.025
0.025
0.025
0.029
0.024
0.025
0.024
0.024
0.029
500
0.017
0.017
0.017
0.017
0.021
0.017
0.017
0.017
0.017
0.021
1000
0.012
0.012
0.012
0.012
0.015
0.012
0.012
0.012
0.012
0.015
Table A2 considers exactly the same marginals as before, but now in the presence of additional serial dependence (). Comparing Table A1 and Table A2, it becomes clear that the additional dependence causes increased bias and SE. However, and this is the crucial point for practice, the asymptotic approximations from Theorem 1 work as well as they do in the i. i. d.-case. If there are visible deviations at all, then these happen again mainly for and low sample sizes. Overall, we have an excellent approximation quality throughout, but with least SEs again for .
Table A2
Simulated vs. asymptotic mean and SE of for specific cases of and , where BAR rank counts with and sample size n.
Simulated Mean
Asymptotic Mean
p
n
a=1
a=1.5
a=2
a=2.5
q=4
a=1
a=1.5
a=2
a=2.5
q=4
0.1
50
0.298
0.276
0.268
0.266
0.330
0.292
0.275
0.267
0.266
0.330
100
0.306
0.282
0.273
0.271
0.337
0.304
0.282
0.273
0.271
0.337
250
0.312
0.286
0.277
0.275
0.342
0.311
0.286
0.277
0.275
0.342
500
0.313
0.287
0.278
0.276
0.343
0.313
0.287
0.278
0.276
0.343
1000
0.314
0.288
0.279
0.277
0.344
0.314
0.288
0.278
0.277
0.344
0.3
50
0.529
0.491
0.476
0.473
0.598
0.528
0.491
0.476
0.472
0.597
100
0.540
0.501
0.485
0.481
0.611
0.540
0.501
0.485
0.481
0.611
250
0.547
0.506
0.490
0.487
0.620
0.548
0.507
0.490
0.487
0.620
500
0.551
0.509
0.493
0.489
0.623
0.550
0.509
0.492
0.489
0.623
1000
0.552
0.510
0.493
0.490
0.624
0.551
0.510
0.493
0.490
0.624
0.5
50
0.590
0.545
0.527
0.523
0.667
0.592
0.545
0.527
0.523
0.666
100
0.604
0.555
0.537
0.532
0.682
0.604
0.556
0.537
0.533
0.682
250
0.612
0.562
0.543
0.539
0.691
0.612
0.562
0.543
0.539
0.691
500
0.614
0.564
0.545
0.540
0.694
0.614
0.564
0.545
0.541
0.694
1000
0.615
0.565
0.546
0.542
0.695
0.615
0.566
0.546
0.542
0.695
0.1
50
0.066
0.062
0.061
0.061
0.070
0.070
0.065
0.064
0.064
0.073
100
0.047
0.045
0.044
0.044
0.050
0.049
0.046
0.045
0.045
0.052
250
0.031
0.029
0.028
0.028
0.032
0.031
0.029
0.029
0.029
0.033
500
0.022
0.020
0.020
0.020
0.023
0.022
0.021
0.020
0.020
0.023
1000
0.015
0.014
0.014
0.014
0.016
0.016
0.015
0.014
0.014
0.016
0.3
50
0.064
0.061
0.061
0.060
0.071
0.065
0.063
0.062
0.062
0.073
100
0.046
0.044
0.043
0.043
0.050
0.046
0.044
0.044
0.044
0.051
250
0.029
0.028
0.027
0.027
0.032
0.029
0.028
0.028
0.028
0.032
500
0.021
0.020
0.019
0.019
0.023
0.021
0.020
0.020
0.020
0.023
1000
0.014
0.014
0.014
0.014
0.016
0.015
0.014
0.014
0.014
0.016
0.5
50
0.065
0.063
0.062
0.062
0.074
0.064
0.064
0.064
0.064
0.076
100
0.046
0.046
0.045
0.045
0.054
0.045
0.046
0.045
0.045
0.054
250
0.029
0.029
0.029
0.029
0.034
0.029
0.029
0.029
0.028
0.034
500
0.020
0.020
0.020
0.020
0.024
0.020
0.020
0.020
0.020
0.024
1000
0.014
0.014
0.014
0.014
0.017
0.014
0.014
0.014
0.014
0.017
While the -type dispersion measures and their asymptotics perform well, essentially for any choice of , the gap becomes wider when looking at the serial dependence measure . The asymptotics in Theorem 2 refer to the i. i. d.-case, which is used as the null hypothesis () if testing for significant serial dependence. Thus, let us start by investigating again the mean and SE of for i. i. d. data (same DGPs as in Table A1); see the results in Table A3. For the asymptotic mean, we have the unique approximation , and this works well except for and low sample sizes. In particular, for with , i.e., for , we get notable deviations. The reason is given by the computation of , where division by zero might happen (in the simulations, this was circumvented by replacing a zero by ). In a weakened form, we observe a similar issue for the case ; generally, we are faced with the zero problem if because of the second-order derivatives of . Analogous deviations are observed for the SEs. Here, generally, the simulated SEs tend to be larger than the asymptotic ones. As a consequence, if using the asymptotic SEs for calculating the critical values when testing , we expect a tendency to oversizing.
Table A3
Simulated vs. asymptotic mean and SE of for specific cases of and , where i. i. d. rank counts and sample size n.
Simulated Mean
Asymptotic Mean
p
n
a=1
a=1.5
a=2
a=2.5
q=4
a=1
a=1.5
a=2
a=2.5
q=4
0.1
50
−0.020
−0.019
−0.019
−0.019
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
100
0.005
−0.008
−0.008
−0.008
−0.009
−0.010
−0.010
−0.010
−0.010
−0.010
250
0.003
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
500
0.000
−0.001
−0.002
−0.002
−0.001
−0.002
−0.002
−0.002
−0.002
−0.002
1000
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
0.3
50
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
100
−0.010
−0.010
−0.010
−0.010
−0.011
−0.010
−0.010
−0.010
−0.010
−0.010
250
−0.003
−0.004
−0.004
−0.004
−0.003
−0.004
−0.004
−0.004
−0.004
−0.004
500
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
1000
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
0.5
50
−0.020
−0.021
−0.021
−0.021
−0.020
−0.020
−0.020
−0.020
−0.020
−0.020
100
−0.010
−0.010
−0.010
−0.010
−0.010
−0.010
−0.010
−0.010
−0.010
−0.010
250
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
−0.004
500
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
−0.002
1000
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
−0.001
0.1
50
0.124
0.124
0.130
0.132
0.134
0.074
0.105
0.120
0.122
0.103
100
0.134
0.082
0.088
0.089
0.085
0.053
0.074
0.085
0.086
0.073
250
0.086
0.050
0.055
0.056
0.050
0.033
0.047
0.054
0.055
0.046
500
0.045
0.034
0.038
0.039
0.034
0.023
0.033
0.038
0.039
0.033
1000
0.022
0.024
0.027
0.027
0.023
0.017
0.023
0.027
0.027
0.023
0.3
50
0.098
0.097
0.101
0.101
0.101
0.079
0.089
0.096
0.097
0.088
100
0.066
0.067
0.070
0.071
0.067
0.056
0.063
0.068
0.068
0.062
250
0.040
0.040
0.043
0.043
0.040
0.035
0.040
0.043
0.043
0.040
500
0.026
0.028
0.030
0.031
0.028
0.025
0.028
0.030
0.031
0.028
1000
0.018
0.020
0.021
0.021
0.020
0.018
0.020
0.021
0.022
0.020
0.5
50
0.084
0.092
0.098
0.099
0.089
0.080
0.086
0.092
0.094
0.080
100
0.058
0.063
0.067
0.068
0.059
0.057
0.061
0.065
0.066
0.057
250
0.036
0.039
0.042
0.043
0.037
0.036
0.039
0.041
0.042
0.036
500
0.025
0.027
0.029
0.030
0.025
0.025
0.027
0.029
0.030
0.025
1000
0.018
0.019
0.021
0.021
0.018
0.018
0.019
0.021
0.021
0.018
If looking at the simulated rejection rates in Table A4, first at the size values () being printed in italic font, we indeed see sizes being slightly larger than the nominal 5%-level, as long as . For larger sample sizes, by contrast, the -test satisfies the given size requirement quite precisely. Here, we computed the critical values by plugging-in the respective sample CDF into the formula for . In Table A4, power values for are also shown. Note that for a BAR process, can take any positive value in , but the attainable range of negative values is bounded from below by [2]. Thus, only are considered in Table A4. Generally, the -tests are powerful regarding both positive and negative dependencies, but the actual power performance differs a lot for different . The worst power is observed for with , followed by with . Regarding the remaining -cases, does slightly worse than , and we often have a slight advantage for , especially for negative dependencies.
Table A4
Simulated rejection rate (: size; : power) of -test at 5 %-level (: , i.e., i. i. d.-case) for specific cases of and , where BAR rank counts and sample size n.
Simulated Rejection Rate
Simulated Rejection Rate
p
n
a=1
a=1.5
a=2
a=2.5
q=4
a=1
a=1.5
a=2
a=2.5
q=4
ρ=0.2
50
0.227
0.265
0.263
0.262
0.222
100
0.367
0.447
0.451
0.450
0.370
250
0.721
0.830
0.830
0.829
0.723
500
0.950
0.982
0.982
0.981
0.947
1000
1.000
1.000
1.000
1.000
0.999
ρ=−0.4
ρ=0.4
50
0.484
0.565
0.585
0.588
0.312
50
0.663
0.749
0.756
0.757
0.625
100
0.807
0.878
0.891
0.893
0.672
100
0.917
0.959
0.962
0.962
0.897
250
0.989
0.999
1.000
1.000
0.987
250
1.000
1.000
1.000
1.000
0.999
500
0.997
1.000
1.000
1.000
1.000
500
1.000
1.000
1.000
1.000
1.000
1000
1.000
1.000
1.000
1.000
1.000
1000
1.000
1.000
1.000
1.000
1.000
ρ=−0.2
ρ=0.6
50
0.151
0.188
0.195
0.196
0.094
50
0.956
0.979
0.981
0.981
0.923
100
0.298
0.365
0.374
0.375
0.231
100
1.000
1.000
1.000
1.000
0.998
250
0.643
0.744
0.751
0.751
0.584
250
1.000
1.000
1.000
1.000
1.000
500
0.925
0.965
0.966
0.965
0.900
500
1.000
1.000
1.000
1.000
1.000
1000
0.994
1.000
1.000
1.000
0.998
1000
1.000
1.000
1.000
1.000
1.000
ρ=0
ρ=0.8
50
0.061
0.056
0.054
0.054
0.061
50
0.997
1.000
1.000
1.000
0.981
100
0.056
0.060
0.058
0.057
0.057
100
1.000
1.000
1.000
1.000
0.999
250
0.047
0.048
0.046
0.046
0.049
250
1.000
1.000
1.000
1.000
1.000
500
0.049
0.050
0.050
0.050
0.052
500
1.000
1.000
1.000
1.000
1.000
1000
0.053
0.047
0.048
0.049
0.050
1000
1.000
1.000
1.000
1.000
1.000
To sum up, while the whole -family is theoretically appealing, and while there are hardly any noteworthy problems with the sample dispersion measures , the performance of clearly depends on the choice of . It is recommended to use the family of a-entropies (6), and there, is preferable. The measure from (14), for example, although theoretically appealing as a combination of entropy and extropy, has a relatively bad finite-sample performance. The probably most well-known pair, , has a good performance, although there appears to be a slight advantage if choosing a somewhat larger than 2, such as (recall that leads back to the case ).
6. Data Application
Ordinal time series are observed in quite diverse application areas. Economic examples include time series on credit ratings [14] or on fear states at the stock market [20], and a climatological example is the level of cloudiness of the sky [23]. Health-related examples are time series of electroencephalographic (EEG) sleep states [24], the pain severity of migraine attacks, and the level of perceived stress [15]. In this section, we are concerned with an environmental application, namely the level of air quality. Different definitions of air quality have been reported in the literature. In Chen Chiu [25], the air quality index (AQI) is used for expressing the daily air quality, with levels ranging from to . Another case study is reported by Liu et al. [26], who use the classification of the Chinese government, which again distinguishes levels, but now ranging from to . The latter article investigates daily time series from thirty Chinese cities for the period December 2013–July 2019, i.e., the sample size equals . In what follows, we use one of the time series studied by Liu et al. [26], namely the daily air quality levels in Shanghai, for illustrating our novel results about cumulative paired -entropies.The considered time series is plotted in the top panel of Figure 5. The bottom left graph shows the sample version of the probability mass function (PMF) , i.e., the relative frequencies of the categories. It exhibits a unimodal shape with mode (=median) in . The serial dependence structure is analyzed in the bottom right graph, where with a-entropy having is used, as this is the recommended choice according to Section 5. All of the plotted -values are significantly different from 0 at the 5 %-level, where the critical values (plotted as dashed lines in Figure 5) are computed as according to Theorem 2 (and by plugging-in the sample CDF). We recognize a medium level of dependence (), which quickly decreases with increasing time lag h, similar to an AR-type process.
Figure 5
Daily air quality level in Shanghai: plot of time series in top panel; plot of sample PMF (left) and (right) in bottom panel; uses a-entropy with .
Let us now have a closer look at the dispersion properties of the Shanghai series. The different choices of the -measure considered so far provide slightly different results regarding the extent of dispersion. In accordance with Figure 2, the largest point estimates are computed for (0.514) and (0.465), followed by with 0.394, whereas (0.349), (0.332), and (0.328) lead to similar but clearly lower values. Comparing the sample PMF in Figure 5 to the extreme scenarios of a one-point and an extreme two-point distribution, the PMF appears to be more close to a one-point than to a two-point distribution, i.e., the lower ones among the above dispersion values seem to be more realistic here.The novel asymptotics of Theorem 1 allow to judge the estimation uncertainty for the above point estimates. To keep the discussion simple, let us focus again on the case . In the first step, we compute the i. i. d.-approximations of bias and SE, and , respectively. By plugging-in the sample CDF, these are obtained as and , respectively. However, these i. i. d.-results are misleading in the present example as the data exhibit significant serial dependence (recall Figure 5). As we know from Theorem 1, the bias has to be increased by the factor , and the SE by . These factors shall now be computed based on the so-called “ZOBPAR model” proposed by Liu et al. [26], which constitutes a rank-count approach, . In view of the AR-like dependence structure and the high frequency for , namely 0.560, the conditional distribution of is assumed to be a truncated Poisson distribution, truncated to the range , with time-varying Poisson parameter and additional one-inflation parameter 0.3463 ([26] Table III). For this model fit, we compute
Thus, an approximate 95 %-confidence interval (CI) for is given by . CIs for the remaining -measures are computed analogously, leading to for , to for , to for , to for , and to for .
7. Conclusions
In this article, we considered the family of cumulative paired -entropies. For each appropriate choice of the EGF , an ordinal dispersion measure is implied. For example, particular choices from the families of a-entropies or q-entropies, respectively, lead to well-known dispersion measures from the literature. The first main contribution of this work was the derivation of the asymptotic distribution of the sample version for ordinal time series data. These asymptotics can be used to approximate the true distribution of , e.g., to compute approximate confidence intervals. Simulations showed that these asymptotics lead to an excellent finite-sample performance. Based on the obtained expression for the asymptotic bias of , we recognized that each EGF also implies a -type serial dependence measures, i.e., altogether, we have a matched pair for each EGF . Again, we analyzed the asymptotics of the sample version , and these can be utilized for testing for significant serial dependence in the given ordinal time series. This time, however, the finite-sample performance clearly depends on the choice of . Choosing based on an a-entropy with , such as , ensures good finite-sample properties. The practical application of the measures and their asymptotics was demonstrated with an ordinal time series on the daily level of air quality in Shanghai.
Table 1
Partial derivatives (10) for EGFs discussed in Example 1.
Figure 1
Figure 2
Figure 4
Figure 3
Figure 5