Abstract
Additional properties and generalizations are explored for a recently introduced concentration index CK. The CK is based on both the distribution of a set of proportions (probabilities) as well as their ranks. The CK is closely related to and proposed as a preferred alternative to the widely used Q that equals the sum of quadratic terms (proportions). Besides the use of CK and Q as measures of market or industry concentration, with the proportions being market shares, CK or its potential transformations can be used as alternative measures in a variety of real measurement situations for which Q has been applied. The extended analysis of CK includes the proof that CK is a convex function, which makes it capable of decomposition analysis. The sensitivity and transfer effect of CK due to changes in the distribution of the proportions is studied. Derivation is given for the so-called numbers equivalent of CK and for its probability interpretation. Generalizations of CK are considered for changing the relative emphasis of the component proportions. Randomly generated distributions exemplify the limited effect on CK from excluding the smallest proportions that are often unavailable in real situations. Numerical comparisons between CK and other concentration indices are presented for a wide variety of firms or industries. A statistical inference procedure is presented for appropriate situations.
Citation: Kvålseth TO (2026) The new rank-based concentration index: Further analysis and properties. PLoS One 21(2):
e0343034.
https://doi.org/10.1371/journal.pone.0343034
Editor: Muntazir Hussain, Air University, PAKISTAN
Received: December 5, 2024; Accepted: January 30, 2026; Published: February 24, 2026
Copyright: © 2026 Tarald O. Kvålseth. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data is within the manuscript and its Supporting Information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The importance of measuring concentration, especially market or industry concentration, is evidenced by the number and variety of measures or indices that have been proposed over the years (e.g., [1]). Those measures have all been defined in terms of proportions or probabilities 
, or market shares in the case of market (industry) concentration, with 
for 
and 
(or 100%). The most popular one is simply the following sum of quadratic terms:
(1)As a measure of market concentration, for example, it is known as the Herfindahl-Hirschman index after Herfindahl [2] and Hirschman [3].
However, since those various proposed indices, including (1), lack an important property, the value-validity property, an alternative index based on the ranked components of the distribution 
has recently been introduced by Kvålseth [1] as follows:
(2)That is, with the 
’s arranged in descending order (tied or equal 
’s may be arranged in any order), 
is simply defined as the largest 
divided by 1, plus the second largest 
divided by 2, etc. Thus, 
is seen to be simply the weighted mean reciprocal rank, i.e., the reciprocal of the ranks 1,...,n weighted with the respective ranked 
’s.
This index in (2) was first briefly introduced by Kvålseth [4] as a general measure of homogeneity for categorical data. A particular form of the expression in (2) with 
and referred to as the mean reciprocal rank is also being used for information retrieval and ranking systems (e.g., [5,6]). Besides being a weighted mean of the reciprocal ranks, the 
in (2) can also be interpreted in terms of a statistical expectation as the expected reciprocal rank of a randomly chosen observation. Thus, in terms of a random variable X that can take on values 
with the respective ordered probabilities 
, 
becomes the expected value of X.
While 
in (2) was primarily introduced as a measure of market concentration, the versatility of 
in (1) also extends to 
because of a close approximate functional relationship between the two indices [1]. Such variety of applications include 
as a measure of biological species concentration [7], coincidence in cryptology [8], political consensus (e.g., [9]), and accounting harmonization and standardization (e.g., [10]). The complement 1 − Q(Pn) has been used as a measure of biological diversity (e.g., [11, Ch. 4]), qualitative variation ([12, pp. 70–71]), linguistic diversity ([13,14]), ethnic fractionalization ([15]), political fractionalization ([16, Ch. 2]), quadratic entropy ([17,18, pp. 174–176]; [19,20]). Other functions of Q(Pn) include 1/Q(Pn) as a measure of diversity (e.g., 11, Ch. 4]) and −logQ(Pn) as both the collision entropy [21] and as a measure of biological diversity ([22, p. 311]). Thus, being an approximate function of 
, but having the advantage of the value-validity property, the 
in (2) may have other potential applications besides as a measure of market concentration.
With the recent introduction of 
, its various properties were defined and discussed [1]. Those properties included such generally well-known characteristics as continuity, symmetry, zero-indifference (adding one or more zero-probability components does not affect 
, Schur-convexity, and value validity. With emphasis on market (industry) concentration, the 
was compared with other frequently used concentration indices using both randomly generated data and real market-share data. The 
was also considered in terms of economic theory and market competition, leading to merger implications equivalent to those based on the Herfindahl-Hirschman index [2,3]. Since these properties, discussions, and results were given in a readily available open-access publication [1], there is no need for a repetition.
Rather, the purpose of the present paper is to identify and prove additional properties with relevant and important implications. One such property is the convexity of 
, which permits subsystem decompositions. A simple probability interpretation of 
is defined from its functional relationship with 
. Expressions will be derived for computing the lower bound on 
and also the so-called numbers equivalent. The transfer and elasticity properties of 
will be considered as will be its potential parameterized generalizations. Randomly generated distributions 
will be used to demonstrate the effect of ignoring the smallest 
’s, which are often excluded from real data. Real market-share data for a variety of firms (industries) are used to compare the types of numerical values taken on by 
and other concentration indices. A statistical inference procedure will also be derived and exemplified.
2 Properties of 
Before discussing the properties of 
, a point should be made about the notation used throughout this paper. Thus, with any concentration index C being a function of the distribution 
, it would be mathematically most correct to use 
to denote the value of the index (function) C. However, as a matter of simplicity and convenience and when there is no chance of ambiguity, C will be used to denote both the index and its value for 
.
In the introductory paper on 
in (2), various properties discussed can be concisely outlined as follows ([1]):
(P1) 
is simple, comprehensible, and meaningfully interpretable;
(P2) 
takes on its extreme values for the distributions
(3)with
(4)(P3) CK is (permutation) symmetric with respect to p1,..., pn;
(P4) CK is zero indifferent (expansible), i.e., 
;
(P5) CK is strictly Schur-convex;
(P6) CK has the value-validity property.
2.1 Convexity of 
The fact that 
is strictly Schur-convex (Property (P5)) does not imply that 
is convex, which is a stronger requirement. In order to prove the convexity of 
, a more general formulation will be used and of which 
is a particular member.
Thus, consider a general class of concentration measures defined as
(5)where 
and 
are both arranged in descending order for 
. Then, for any ordered distributions 
and 
and for any constant 
, it follows immediately from the definition in (5) that
(6)where the last equality follows from the (permutation)symmetry of C, i.e., the value of C is invariant with respect to any permutations of the unordered 
’s and 
is always ordered (ranked) as 
. Next, as an immediate consequence of the definition of majorization ([23, p. 8]),
(7)where the symbol 
means that the left side of (7) is majorized by the right side. Then, from (7) and the fact that the type of expression as in (5) is (strictly) Schur-convex [23, pp 160, 639],
(8)Finally, from (6) and (8),
(9)which completes the proof that C in (5) is a convex function of 
. With 
in (2) being a particular member of C in (5), with 
for all i, this result proves that 
is convex.
Note that C in (5) is convex, but not strictly convex. Had C been strictly convex, then the inequality in (9) would have been strict and C in (5) and hence 
in (2) could not have complied with the value-validity requirement [1]. With 
and 
in (6)-(9) replaced with 
and 
in (3), respectively, the value-validity property requires that
(10)which is clearly satisfied by C in (5).
2.2 Comment on 
The 
component of 
in (4) is recognized as the n-th harmonic number and is of considerable mathematical interest. It is also well known that the logarithmic expression 
, with 0.5772 being the Euler’s constant (to 4 decimal places), converges quite rapidly to 
with increasing n. Therefore, an approximate expression for 
can be defined as
(11)This approximation is adequate for all practical purposes. In fact, it is found to be correct to at least 3 decimal places when 
.
It could, of course, be argued that although 
is mathematically interesting and easily computable from (11), a more convenient and intuitively reasonable lower bound on a concentration index would be 1/n (as in the case of Q in (1)). In order to be defined over the interval [1/n, 1], 
could be transformed as follows:
(12)Note, however, that the transformed index in (12) lacks the zero-indifference property (P4). Also, the true n may not necessarily be known in all real situations.
2.3 Numbers equivalent of 
Some prefer that a concentration measure should be of a so-called number equivalent or effective number form (e.g., [24]). For any distribution 
, the numbers equivalent 
of 
in (2) can most concisely be defined by the approximate expression
(13)where 
is the nearest integer that makes this approximation as accurate as possible. For any given 
, the value of 
in (13) could be determined by means of a search procedure or by trial and error based on the 
expression in (4) or (11).
An alternative approach to obtain 
would be to explore some potential approximate functional relationship between 
in (4) and n. Based on exploratory graphical analysis and statistical regression analysis, with parameter estimates rounded off to convenient fractions, the following fitted model has been derived:
(14)For the fitted model in (14), with the 25 data points for n = 2,4,6,...,50, the coefficient of determination 
, when properly computed [25], is found to be
. When the predicted 
is rounded off to the nearest integer, it is found that 
for n = 1,2,...,50.
The expressions in (13)-(14) can then be used to determine the numbers equivalent 
as
(15)for any given distribution 
. It needs to be emphasized, however, that 
being a non-linear function of 
does not meet the value-validity condition in (10) with 
for 
. Nevertheless, 
does provide an alternative interesting interpretation of 
. For example, consider a market with 30 firms and 
for which 
from (15), which means that this 0.45 concentration is equivalent to that of a market with 5 firms of equal size (market share).
2.5 Sensitivity and transfer of 
A potentially interesting characteristic of 
is its sensitivity to the individual components and the form of the distribution 
or of its rank ordered form 
This can simply be done by taking partial derivatives of the relative terms 
for i = 1,...,n and 
and by treating each 
as a continuous variable for mathematical purpose. Thus, the sensitivity of 
to a small change in 
, with all other 
kept fixed, can be defined as
(18)With the interest being the change in 
, irrespective of being an increase or decrease, the absolute value 
from (18) provides some clear indications of the sensitivity of 
to small changes in 
. A most striking overall observation would seem to be that C is most sensitive to changes in the distribution 
towards its upper and lower ends. That is, this sensitivity increases with decreasing i for 
and with increasing i for 
. For specific components, 
is particularly sensitive to changes in the extreme components 
and 
.
A related characteristic of 
is the so-called transfer, i.e., the effect on 
when transferring a small amount 
from a smaller 
to a larger 
. Such a transfer will cause the value of 
to increase as a consequence of the Schur-convexity of 
(Property (P5)) [23, Ch. 1]. With the restriction that 
and 
, the transfer effect on 
may be defined as
(19)which is a similar form of definition to that used by Cowell [26, pp. 57, 154–156] for measures of inequality, except for the relative (versus absolute) difference used in (19). This expression shows that the relative effect on 
from a small transfer from 
to 
depends on their rank difference, but not explicitly on their values. The extreme effect occurs with a transfer from 
to 
.
It may be of interest to compare these sensitivity and transfer effects of 
with those corresponding to Q in (1). Therefore, when the equivalent expressions to (18) and (19) are applied to Q, the following formulations are obtained:
(20)The general results from (18)-(20) show that the form of the sensitivity to changes in 
and the transfer effects are comparable for the two measures, with the difference that those of 
are determined by the ranks while those of Q depend on the ranked 
‘s.
3 m-Category 
It is clear from the definition of 
in (2) that when a number of 
’s are very small, ignoring those from the computation of 
only marginally affects the value of 
. In practice, this characteristic of 
is in effect an advantage since reported data often ignore very small 
’s or group them into an “all others” category. It is therefore worth determining more closely the effect of such exclusion on the value of 
.
Therefore, expressing 
as
(21)the concern is basically with the size of the “error” term 
. From majorization theory [23, Ch. 1], the following majorization applies:
(22)so that from (22) and the Schur-convexity of 
in (21), the following inequality is obtained:
(23)It is clear from (23), especially from the upper bound, that little information is lost by disregarding the smallest 
’s if m is not small.
The extent of 
in (23) can also be examined empirically by using various distributions 
. Thus, a random sample of such distributions was generated using the computer algorithm described in [1] in which n and each 
were generated as random numbers within specified intervals. Specifically, for each randomly generated integer n, each 
was generated as a random number (to the desired decimals) in decreasing order within the following intervals:
A total of 30 such distributions were randomly generated for 
and, for each distribution, computations were made for the values of 
and 
in (21) for the chosen m = 5 and 10 as given in Table 1.
As could reasonably be expected, it is apparent from Table 1 that a substantial amount of information may be lost when using a value of m as small as m = 5. For example, for the error term 
in (21), it is seen that 
for Data Sets 3, 8, 14, 16, and 21, with a mean error value of 0.030 for the 30 data sets. By comparison, for m = 10, Table 1 shows that the errors are substantially lower and values of 
and 
are generally quite comparable (except for, say, Data Sets 3 and 16). The linear regression of 
on 
is found to be 
with 
, indicating the considerable agreement between the two indices.
A conservative conclusion from these results would be as follows: utilize all 
when computing the value of 
, but if some of the smaller 
’s are ignored, the effect on 
is likely to be rather negligible.
4 Generalizations of 
The 
in (2) could be potentially generalized in a number of different ways by introducing some additional parameter 
. One such generalization would be the following 
-order weighted mean of the reciprocal ranks:
(24)of which 
is the particular member 
. Another rather obvious generalization would be
(25)with 
being the member 
. Note that 
is also a member of the concentration class in (5) with 
for 
.
From a property of generalized means (e.g., [27, Ch. III]), the index (family) in (24) is strictly increasing in 
, whereas the parameterized index in (25) is seen to be decreasing in 
for any given 
. An example of these two generalized indices as functions of the parameter 
for the distribution 
is given in Fig 1. As 
, 
and 
. The two curves cross at 
when 
in (2).
The effect on these two families of indices from changing 
is basically one of changing the weights or emphasis given to the different 
‘s. While increasing 
places increasing weights on the larger 
’s for 
in (24), the effect on 
in (25) is the reverse. The value-validity condition in (10) can be seen to be satisfied by 
for all 
, but only for 
in the case of 
when 
in (2).
In spite of the flexibility offered by these two families of potential concentration indices, there seems to be no compelling reason to prefer any particular alternative over 
in (2) as a single measure. Using such curves as in Fig 1 to provide comparison between the concentration for two distributions 
and 
is restricted by by the fact that the two curves may potentially cross such that 
for some 
and similarly for 
. Nevertheless, such generalized formulations may provide some useful information in real applications. In the case of Fig 1, for example, with 
being a compromise single concentration measure, both curves show precisely how changing emphasis on the larger or smaller component 
’s affect the concentration measurement.
5 Statistical inferences about 
In situations when 
are multinominal random sample probabilities with 
for sample size 
, it may be of interest to make statistical inferences about CK in (2), especially confidence-interval construction. That is, if Pn = (p1,.., pn) is the sample probability distribution and Πn = (π1,..., πn) is the corresponding population distribution, one may want to make inferences about the population index CK(Πn). Besides resampling methods such as bootstrap and jackknife, such statistical inferences can be done by means of the delta method. The delta method is a useful and powerful results of statistical limit theory that is widely discussed in textbooks on categorical data (e.g., [28, Ch. 16], [29, Ch. 14]).
Concisely stated, it follows from the delta method applied to 
that the following convergence-in-distribution holds:
(26)so that for a large multinomial sample of size N, the estimator CK(Pn) is approximately normal with mean 
and variance 
. The accuracy of this asymptotic result depends, of course, on the sample size N. By taking the partial derivatives of 
with respect to 
and then substituting those with the corresponding sample estimates 
for 
, the estimated variance in (26) becomes
(27)From the definition of 
in (2), the expression in (27) becomes
(28)Instead of performing the statistical inferences directly on CK(Πn), it is preferable to use the following logarithmic transformation and its inverse:
(29)since this transformation provides a more rapid convergence to normality and ensures that a confidence internal will always fall inside the [0, 1]-interval (e.g., [28, pp. 70, 618]; [30, p. 106]). The estimated variance of L(Pn) in (29) becomes
(30)An approximate confidence interval 
for 
then becomes
(31)where 
is the standard normal quartile (e.g., 
for 
and for 95% confidence). The corresponding CI for CK(Πn) is then obtained by applying the inverse transformation in (29) to each side of the interval in (31).
As a numerical example, let 
be a sample distribution based on a sample size 
. With 
, it follows from (28) that 
so that, from (30), 
. Then, with 
from (29), a 95% confidence interval for L(
from (31) becomes 
, or [0.6501, 1.2063]. Then, by applying the inverse transformation in (29), a 95% confidence interval for CK(Πn) becomes [0.66, 0.77].
6 Discussion
6.1 Illustrative example
As a real example of the computation of the new index 
defined in (2) and its interpretations, consider the results of the national elections in Norway in 2025. By including only the parties that received at least one percent of the votes (i.e., excluding 13 parties), the following percentage votes were obtained: 28.0, 23.8, 14.6, 5.6, 5.6, 5.3, 4.7, 4.2, 3.7 (for a total of 95.5%). Since these results are given in descending order with 
, etc., the value of 
becomes
or, in terms of proportions instead of percentages, 
. Note that for the two tied results of 5.6%, they are divided by consecutive ranks (rather than their mean). As a limiting case, had all 9 parties’ votes been tied, then 
or 31%. By comparison, a slightly larger concentration value than that of Norway’s election with 
is obtained for the 2025 general election in the Czech Republic with 
(56%) for 7 parties.
As a measure of political dominance (vote concentration), how can the 
(50%) be interpreted in terms of the extent of such dominance? Since 
belongs to the interval (0, 1], 
would seem to imply a political dominance that is neither high nor low. However, had the votes of the top three parties been combined, the resulting value of 
(74.2%) could arguably be interpreted as a high degree of political dominance.
In terms of a meaningful interpretation, what does the election result 
actually mean? The answer lies in (17). That is, with 
in (17), 
. This means that the probability is about 0.18 that two randomly chosen individuals voted for the same party. By comparison, the corresponding value of 
in (1) applied directly to the above voting results, gives 
.
As another meaningful interpretation of the election result 
, consider the numbers equivalent defined in (15). Then, with 
in (15), it is found that 
or about 4. This result means that the political dominance of 
would be the same as if Norway had had about 4 different political parties with equal party support.
6.2 Empirical comparison with other indices
Among the various alternative concentration indices with similar properties that have been proposed over the years [1], their numerical values may differ greatly for the same data sets. Consequently, the results and conclusions from any data analysis can depend strongly on the index being used. What sets the new index 
apart from other concentration indices is the so-called value-validity property (Property (P6) stated above). This property imposes a condition specifically on the numerical values taken on by a concentration index to ensure that those values can be justified as providing realistic, true, or valid representations of the concentration characteristic or attribute [1].
Therefore, it is of interest to compare values of 
with those of other indices for the same data sets. Consider, for example, the following indices:
(32)where 
is the well-known 4-firm concentration ratio, HHI is the popular Herfindahl-Hirschman index [2, 3], RHT is the index by Rosenbluth [31] and Hall and Tideman [32], and 
is a member of a parameterized family of indices by Davies [33]. For this comparative analysis, real market-share data were used as in [1] for a variety of different types of firms or industries. The results are summarized in Table 2.
These results show clearly how the values of various indices can differ substantially for the same sets of data. These indices range in potential value from 0 to 1, except for 
which can range from 0 to n, between the two extreme distributions in (3). Values of 
are seen to be consistently larger than those of HHI and RHT and frequently larger than those of 
(except for examples 12, 14, and 20) even though the potential range of values is greatest for 
. As expected from its definition in (32), 
values are seen to be consistently greater than those of 
, HHI, and RHT.
For each index, the results in Table 2 show that the market (industry) with the highest market concentration was that of the leading search engines in Norway (in 2020) (Example 12) while the lowest concentration occurred for the best-selling cars in Britain (in 2018) (Example 8). Although each of these indices provide the same rank order for these two extreme cases, other order (“larger than”) comparisons vary considerably between the indices. In the case of 
and RHT, for example, 
shows greater concentration for airline travel (number of flights per weekday between London and New York by 30 different carriers in 2000) (Example 2) than for global pharmaceutical products (Example 5) while this order is reversed when using RHT. Note that these two indices are the only rank-based ones. Similarly, when comparing 
and HHI, for instance, reverse order occurs between Examples 5 and 6 and between Examples 10 and 15.
Even though 
is highly correlated with the other indices, with Pearson’s correlation coefficient 
between 
and 
, HHI, RHT, and 
, respectively, for the data in Table 2, it is clear from these exemplary data that different indices can provide substantially different and contradictory assessment of concentration. Although these data are based on the market shares 
of a wide variety of markets or industries, similar results can be expected of any other real situations or applications involving some distribution 
. In spite of the fact that many proposed concentration indices share several of the above properties (P1)-(P6), as well as the convexity property, they lack one important property: the value-validity property (P6), which only 
has.
7 Conclusion
When considering the results derived in this paper together with those previously reported [1], one conclusion would seem to be clear: 
has the various types of properties required of an appropriate concentration measure. One of the interesting features of 
is its close functional relationship to the quadratic index Q in (1), implying that the various applications of Q or its functions can also be considered for 
.
There is, however, an important difference between 
and Q: 
has the value-validity property (Property(P6)), but Q does not since it cannot satisfy the equivalent of the equality part of (10) (with C = Q and 
, 
) because its convexity is strict. The value-validity property is considered to be necessary in order to make true and reliable difference comparisons for the concentration characteristic. The validity of such comparisons is essential for determining trend information such as changes in concentration over time periods. Without the value-validity property, different indices can produce widely differing results and conclusions as demonstrated by using real market-share data for a variety of markets (industries). In that analysis, the popular Herfindahl-Hirschman index HHI in (32) equals Q in (1).
The parameterized generalizations in (24)-(25) do provide for some potentially interesting assessment of the effect on the concentration values caused by varying the relative weight or emphasis assigned to the ordered distribution components 
. However, as a choice for a single concentration index, there would seem to be no particular reason for a preference other than 
.
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