In various research areas related to decision making, problems and their solutions frequently rely on certain functions being monotonic. In the case of non-monotonic functions, one would then wish to quantify their lack of monotonicity. In this paper we develop a method designed specifically for this task, including quantification of the lack of positivity, negativity, or sign-constancy in signed measures. We note relevant applications in Insurance, Finance, and Economics, and discuss some of them in detail.

In various research areas such as Economics, Finance, Insurance, and Statistics, problems and their solutions frequently rely on certain functions being monotonic, as well as on determining the degree of their monotonicity, or lack of it. For example, the notion of profit seekers in Behavioural Economics and Finance is based on increasing utility functions, which can have varying shapes and thus characterize subclasses of profit seekers (e.g., [

Due to these and a myriad of other reasons, researchers quite often restrict themselves to function classes with pre-specified monotonicity properties. But one may not be comfortable with this element of subjectivity and would therefore prefer to rely on data-implied shapes when making decisions. To illustrate the point, we recall, for example, the work of Bebbington et al. [

Monotonicity may indeed be necessary for certain results or properties to hold, but there are also many instances when monotonicity is just a sufficient condition. In such cases, a natural question arises: can we depart from monotonicity and still have valid results? Furthermore, in some cases, monotonicity may not even be expected to hold, though perhaps be desirable, and so developing techniques for quantifying the lack of monotonicity becomes of interest. Several results in this direction have recently been proposed in the literature (e.g., [

We have organized the rest of the paper as follows. In Section

Insurance losses are non-negative random variables

As an example, consider the following ‘dual’ version of the weighted pcp ([

When dealing with insurance losses, researchers usually choose non-decreasing weight functions in order to have non-negative loading of

Note that the pcp

In general, the function

Hence, we are interested when

As already noted, broader contexts than that of classical insurance suggest various shapes of probability distortions and thus lead to functions

Those familiar with asset pricing will immediately see how inequality (

The above interpretations have shaped our considerations in the present paper, and have led toward the construction of monotonicity indices that we introduce and discuss next.

We are interested in assessing monotonicity of a function

Let

Theorem

The index

The index

An illustrative example follows.

Consider the functions

Example

The use of the integral

The index values in Example

If, however, the total variations of

Recall that we are dealing with the functions

Next are properties of the normalized indices.

The three indices

The three indices

The three indices

Moreover,

We next use the above indices to introduce three new orderings:

The function

The function

The function

We see that on the interval

One of the fundamental notions of ordering random variables is that of first-order stochastic dominance (e.g., [

We now take a path in the direction of general Measure Theory. Namely, let

Since

Similarly to the LOP index, the index of lack of negativity (LON) of

In this paper, we have introduced indices that, in a natural way, quantify the lack of increase, decrease, and monotonicity of functions, as well as the lack of positivity, negativity, and sign-constancy in signed measures. In addition to being of theoretical interest, this research topic also has practical implications, and for the latter reason, we have also introduced a simple and convenient numerical procedure for calculating the indices without resorting to frequently unwieldy closed-form expressions. The indices satisfy a number of natural properties, and they also facilitate the ranking of functions according to their lack of monotonicity. Relevant applications in Insurance, Finance, and Economics have been pointed out, and some of them discussed in greater detail.

We are indebted to Professor Yu. Mishura and two anonymous reviewers for insightful comments and constructive criticism, which guided our work on the revision. We gratefully acknowledge the grants “From Data to Integrated Risk Management and Smart Living: Mathematical Modelling, Statistical Inference, and Decision Making” awarded by the Natural Sciences and Engineering Research Council of Canada to the second author (RZ), and “A New Method for Educational Assessment: Measuring Association via LOC index” awarded by the national research organization Mathematics of Information Technology and Complex Systems, Canada, in partnership with Hefei Gemei Culture and Education Technology Co. Ltd, Hefei, China, to Jiang Wu and RZ.