The Weibull distribution is one of the most important generalizations of the exponential distribution. Despite of the loss of the important memorylessness feature, the Weibull distribution has several advantages which determine its wide use in many real life areas including engineering sciences [29], meteorology and hydrology [28], communications and telecommunications [26], energetics [22], chemical and metallurgical industry [4], epidemiology [16], insurance and banking [6], etc. For more theoretical and practical details related to this distribution we refer to the original work [27] as well as the recently published books [18, 20, 17], and [15].

A significant modification of the Weibull distribution known as a Kies distribution was firstly proposed in [9] and recently considered in many studies. It reduces the positive real half-line support of the Weibull to the interval 0 , 1 changing the variables as t = x x + 1 . Later, many authors discussed several modifications. Refs. [11, 21] and [30] examine four parameter distributions whose domain is translated to an arbitrary positive interval. Refs. [12, 13] take a power of the Kies cumulative distribution function (CDF, hereafter) to define a new family.

A composition approach for the Kies distribution is presented in [2]. This way the new distribution is defined after the change of variables t = H x where H x is the CDF of an auxiliary distribution. Ref. [25] introduces the Fréchet distribution for this purpose and later in [24] the resulting Kies-Fréchet distribution is applied to model the COVID 19 mortality. Alternatively, [1] and [3] use the exponential and Lomax distributions, respectively. See also [5] for another composition based on the generalized uniform distribution.

In the present article we define a new Kies family by its CDFs which are constructed as a composition of two other Kies CDFs on the interval 0 , 1 , G H t . Note that this definition is possible due to this domain. We name the distributions G · and H · the original one and the correction. Thus we have a four-parameter family – two parameters for each initial distribution. We investigate the probabilistic properties of the resulting distribution in the light of its compositional essence. We derive many relations between the corresponding terms – CDF, probability density function, quantile function, mean residual life function, different expectations and moments – of the resulting and the initial distributions. Also we investigate the tail behavior by the use of three risk measures arising in the modern capital markets, namely VaR (abbreviated from Value-at-Risk), AVaR (Average-Value-at-Risk, also known as CVAR and TVAR), and expectile based VaR.

Other important results we derive are related to the so-called Hausdorff saturation. It presents the distance between the CDF and a Γ-shaped curve connecting its endpoints. In fact, the saturation measures how the distribution mass is located in the domain – the distribution is more left-placed when the saturation is lower, and vice versa. Also, when studying specific classes of cumulative distribution functions it is important to know their intrinsic characteristics – the saturation in the Hausdorff sense is namely such one. This characteristic is important for researchers in choosing an appropriate model for approximating specific data from different branches of scientific knowledge such as Biostatistics, Population dynamics, Growth theory, Debugging and Test theory, Computer viruses propagation, Insurance mathematics. In addition, the use of composite Kiess families can also be useful in the study of reaction-kinetic models – a similar study of the dynamics of the classical Kiess model is discussed in [14]. We obtain in this paper an interval evaluation of the saturation and investigate its power w.r.t. the four distribution parameters. Moreover, we prove a formula of a semiclosed form for the Hausdorf saturation. Many numerical experiments are provided.

The next question we discuss is related to the inverse problem – the calibration w.r.t. some empirical data. It is accepted in the present literature that the maximum likelihood estimation is a good approach for the Kies style distributions due to the available closed form estimator. However, our numerical simulations do not support this opinion. For this we construct an algorithm based on the least square errors. It turns out that this method produces very believable outcomes.

To illustrate our results we explore an empirical data from the real financial markets, namely, for the S&P500 index. It is well known that there are high- and low-volatility periods. In fact, this is the ubiquitous phenomenon of volatility clustering. We extract the periods between two market shocks and examine the distribution of their lengths. We compare the results which the corrected Kies distribution returns with the outcome of its ancestors, namely, the exponential, the Weibull, and the original Kies distributions.

The paper is organized as follows. Section 2 defines the new class of distributions and discusses their probabilistic properties. The tail behavior is examined in Section 3 trough the measures VaR, AVaR, and expectile based VaR. The Hausdorff distance and the related saturation are considered in Section 4. We discuss the calibration problem in Section 5. Finally, we present a numerical example based on the S&P500 index in Section 6.

We shall use for convenience the following notations in the whole paper. The cumulative distribution function (CDF, as we mentioned above) of a distribution will be denoted by an uppercase letter, the overlined letter will be used for the complementary cumulative distribution function (CCDF), the corresponding lowercase letter is preserved for the probability density function (PDF), and finaly the letter Q indexed by the CDF’s letter will mean the quantile function (QF). For example, if F t is the CDF, then F ‾ t , f t , and Q F t are the corresponding CCDF, PDF, and QF, respectively. Also, we shall use the Greek letter ξ for random variables, and we shall mark it by the corresponding CDF letter in the subscript, i.e. ξ F means a random variable the CDF of which is F t .

The standard Kies distribution is defined on the domain 0 , 1 by its CDF (1) H t : = 1 − e − k t 1 − t b for some positive parameters b and k. Inverting CDF (1) we can derive the quantile function for t ∈ 0 , 1 : (2) Q H t = − ln 1 − t 1 b k 1 b + − ln 1 − t 1 b . Differentiating equation (1) we obtain the probability density function (3) h t = b k e − k t 1 − t b t b − 1 1 − t b + 1 . The following proposition describes the shape of PDF (3).

We introduce and investigate a new class of distributions for which the correction is presented by another Kies distribution (1).

As a corollary of Definition 2.2 we can establish the quantile function.

Differentiating equation (8) we obtain for the PDF (12) f t = a b λ k e − λ e k t 1 − t b − 1 a e k t 1 − t b − 1 a − 1 e k t 1 − t b t b − 1 1 − t b + 1 . More informative is another form of PDF (12) presented in the following proposition.

First we have to derive the PDF value of the H-corrected Kies distribution at the left domain endpoint t = 0 (obviously, the value at the right one, t = 1, is zero). Analogously to the original Kies distributions, one can expect that 0 < f 0 < ∞ when a = b = 1. This is true, but the values a = b = 1 are far from exhausting the cases in which the left endpoint of the PDF is finite and nonzero. The proposition below characterizes the PDF’s behavior near the zero.

The shape of the PDF of the H-corrected Kies distribution is a consequence of Propositions 2.1 and 2.6. Obviously, it has to be more various than the PDF of the original Kies distribution. Various examples are presented in Figure 1. In each of all six subfigures we vary the coefficients λ and a for the original distribution G as λ ∈ 0.5 , 1 , 1.5 and a ∈ 0.5 , 2 . Namely, the original Kies distribution G is colored by blue. The rest of the plotted PDFs are produced by the following parameters for the correcting distribution H: k ∈ 1 , 2 and b ∈ 0.5 , 1 , 2 .

The following proposition for the expectations of the corrected Kies random variables holds.

The following corollaries hold.

Let us consider now the mean residual life function (MRLF, hereafter) of an H-corrected Kies distribution. Usually it is defined as the conditional expectation (21) m F t : = E ξ F − t | ξ F > t ] . t].\]]]> We shall use an alternative presentation stated in [7]: (22) m F t : = 1 F ‾ t ∫ t 1 F ‾ s d s . The following proposition for the MRLF stands.

Let us consider three measures arising from the risk management – VaR, AVaR, and expectile based VaR; we shall use the notation EX for the last one. By its original definition, the VaR at level α of a random variable is just the opposite of the quantile function VaR ( α ) = − Q ( α ). Since the domain of the Kies family is the interval 0 , 1 we shall think VaR ( α ) : = Q ( α ). As its name shows, AVaR is an average VaR in some sense – it is defined as (25) AVaR ( α ) : = 1 α ∫ 0 α VaR ( u ) d u . Also, we consider the right tail behavior by defining the following term (26) AVaR ‾ ( α ) : = 1 1 − α ∫ α 1 VaR ( u ) d u . The expectile function is related to the quantiles in the following way. The α-quantile of the random variable ξ can be viewed as the lower solution of the optimal problem (27) Q α = arg min x ∈ R E α ξ − x + + 1 − α ξ − x − , where z + and z − are notations for max z , 0 and max − z , 0 , respectively. For more details, see, for example, [10]. Analogously, the expectile is defined in [19] as the solution of the following quadratic problem (28) EX ( α ) : = arg min x ∈ R E α ξ − x + 2 + 1 − α ξ − x − 2 . Note that the expectiles are well defined when the random variable has a finite second moment. For the corrected Kies distributions this is true due to Corollary 2.9. It can be easily proven that expectile (28) is the solution of the following equation w.r.t. the variable x (29) α E ξ − x + = 1 − α E ξ − x − . We derive AVaRs and the expectile based VaR in the following two propositions.

Next we discuss the expectile based VaR. It can be obtained through both of equations presented in the following proposition. Proposition 3.2.

The α-expectile based VaR, EX ( α ), is the solution of the following equivalent equations (w.r.t. the variable t) (37) 1 − 2 α m F t F ‾ t + 1 − α t − μ 1 = 0 t G ‾ H t − α − E Q H ξ G 1 − α − 1 − 2 α I ξ G > H t = 0 , H\left(t\right)}}\right)\right]=0,\end{aligned}\]]]> where μ 1 is the first moment given in Corollary 2.9 and m F · is the MRLF.

Let us consider the max-norm in R 2 , i.e. if A and B are the points A = t A , x A and B = t B , x B , then ‖ A − B ‖ : = max t A − t B , x A − x B . We define the Hausdorff distance, also known as a H-distance, in a sense of [23].

We can define now the saturation of a distribution.

Having in mind that the domain of the Kies distribution is the interval 0 , 1 , we can prove the following corollary for its saturation.

We shall prove now the following formula of a semiclosed form for the saturation of CDF (8).

The behavior of the corrected Kies CDFs can be seen in Figures 2a–2d together with the saturation d ‾; it is presented by the red points. The red lines form squares which in fact confirms Corollary 4.3. The used quadruplet for the parameters are ( λ , k , a , b ) ∈ 2 , 2 , 5 , 1 , 2 , 2 , 1 , 0.5 , 2 , 1 , 2 , 1 , 0.5 , 1 , 0.5 , 0.5 .

Next we discuss a useful in practice interval approximation of the Hausdorff saturation d ‾. Let us consider first the case λ = a = b = 1. The saturation d ‾ has to satisfy equation (44) which now can be written as (52) μ d = e k d 1 − d − 1 + ln d = 0 . The function μ d can be approximated very well for small ds by the function (53) μ 1 d = e k d − 1 + ln d . Taking the exponent in the Taylor series we see that the function (54) μ 1 d = k d + ln d + ∑ n = 2 ∞ k d n n ! . can be approximated as O d 2 for small enough values of d by the function (55) μ 2 d = k d + ln d . Let d 1 and d 2 be defined as d 1 : = 1 k and d 2 : = ln k k . We shall check when μ 2 d 1 < 0 < μ 2 d 2 . Obviously the first inequality holds when k > ee$]]>. Assuming that this restriction holds, we see that μ 2 d 2 = ln ln k > 00$]]> and hence the second inequality holds, too. Thus we conclude that the function μ 2 d has a unique root in the interval 0 , 1 and it belongs to the subinterval d 1 , d 2 when k > ee$]]>, since this function is strictly increasing.

In the next proposition we discuss the general case assuming that λ k a > 11$]]>.

Let us return to the saturation of the corrected Kies distribution. We have shown above that it is the solution of equation (44) which is equivalent to (61) e k d 1 − d b − 1 − − ln d λ 1 a = 0 . Analogously to the case a = b = λ = 1 , formula (54), we can see that after the Taylor expansion of the exponent, the left hand-side of equation (61) can be approximated by the function μ 2 d near zero as O d 2 b . Hence, its root can be used as an approximation of the corrected Kies distribution’s saturation when it is small enough. On the other hand, the function μ 2 d is a lower approximation of μ d and therefore μ 2 d < μ d . Thus the saturation is below the root of the function μ 2 d and hence d ‾ < d 2 . The question stands, when d 2 < 1. Let us define b 1 as (62) b 1 : = ln λ k a a λ k a = b ‾ λ k a . Note that b 1 < b ‾. We shall show that if b < b ‾, then d 2 < 1 only when b > b 1 {b_{1}}$]]>. Using again the notations K = λ k a and B = a b and having in mind formula (58) we see that d 2 < 1 when 1 > ln K B K \frac{\ln K}{BK}$]]> which is equivalent to b > b 1 {b_{1}}$]]>. Note that K > 11$]]>.

On the contrary, d 1 is not always below the saturation d ‾. It turns out that there exists a value, say b 2 , dependent on the other parameters, such that d 1 < d ‾ for b < b 2 , and vice versa. To see this, we consider function (44). Obviously, it is increasing in the distribution domain 0 , 1 . Also, d 1 increases w.r.t. the parameter b because λ k a > 11$]]>. Therefore μ ‾ b : = μ d 1 b is an increasing function, too; note that d 1 b < 1. Having in mind μ ‾ 0 = − ∞, μ ‾ + ∞ = + ∞, and μ d ‾ b = 0, we conclude that indeed d 1 b < d ‾ b for b < b 2 , where b 2 is the unique solution in the interval 0 , ∞ of the equation (63) μ ‾ b = λ e λ − 1 a 1 − λ − 1 a b k − 1 b − b − 1 a − ln λ a b − ln k b .

We shall show that b 2 < b ‾, too. Let us mark the dependence on b in the terms d 1 and d 2 . We can easily check that d 1 b ‾ = d 2 b ‾ and hence d ‾ < d 1 b ‾ = d 2 b ‾ < 1 (because d ‾ < d 2 b ‾ and d 1 b ‾ < 1). Therefore μ ‾ b ‾ = μ d 1 b ‾ > μ d ‾ = 0\mu \left(\overline{d}\right)=0$]]>, since μ b is an increasing function. Thus we see that b 2 < b ‾.

We can formulate these results in the following proposition.