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 $\left(0,1\right)$ changing the variables as $t=\frac{x}{x+1}$. 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 $t=H\left(x\right)$ where $H\left(x\right)$ 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 $\left(0,1\right)$, $G\left(H\left(t\right)\right)$. Note that this definition is possible due to this domain. We name the distributions $G\left(\cdot \right)$ and $H\left(\cdot \right)$ 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 $F\left(t\right)$ is the CDF, then $\overline{F}\left(t\right)$, $f\left(t\right)$, and ${Q_{F}}\left(t\right)$ 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. ${\xi _{F}}$ means a random variable the CDF of which is $F\left(t\right)$.

The standard Kies distribution is defined on the domain $\left(0,1\right)$ by its CDF for some positive parameters b and k. Inverting CDF (1) we can derive the quantile function for $t\in \left(0,1\right)$: Differentiating equation (1) we obtain the probability density function 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 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\left(0\right) < \infty $ 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 $\lambda \in \left\{0.5,\hspace{2.5pt}1,\hspace{2.5pt}1.5\right\}$ and $a\in \left\{0.5,\hspace{2.5pt}2\right\}$. 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: $k\in \left\{1,\hspace{2.5pt}2\right\}$ and $b\in \left\{0.5,\hspace{2.5pt}1,\hspace{2.5pt}2\right\}$.

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 We shall use an alternative presentation stated in [7]: The following proposition for the MRLF stands.

Let us consider three measures arising from the risk management – $\mathit{VaR}$, $\mathit{AVaR}$, and expectile based VaR; we shall use the notation $\mathit{EX}$ for the last one. By its original definition, the $\mathit{VaR}$ at level α of a random variable is just the opposite of the quantile function $\mathit{VaR}(\alpha )=-Q(\alpha )$. Since the domain of the Kies family is the interval $\left(0,1\right)$ we shall think $\mathit{VaR}(\alpha ):=Q(\alpha )$. As its name shows, $\mathit{AVaR}$ is an average $\mathit{VaR}$ in some sense – it is defined as Also, we consider the right tail behavior by defining the following term 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 where ${z^{+}}$ and ${z^{-}}$ are notations for $\max \left(z,0\right)$ and $\max \left(-z,0\right)$, respectively. For more details, see, for example, [10]. Analogously, the expectile is defined in [19] as the solution of the following quadratic problem 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 We derive $\mathit{AVaR}$s and the expectile based $\mathit{VaR}$ 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.

Let us consider the max-norm in ${\mathbb{R}^{2}}$, i.e. if A and B are the points $A=\left({t_{A}},{x_{A}}\right)$ and $B=\left({t_{B}},{x_{B}}\right)$, then $\| A-B\| :=\max \left\{\left|{t_{A}}-{t_{B}}\right|,\left|{x_{A}}-{x_{B}}\right|\right\}$. 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 $\left(0,1\right)$, 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 $\overline{d}$; 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 $(\lambda ,k,a,b)\in \left\{\left(2,2,5,1\right),\hspace{2.5pt}\left(2,2,1,0.5\right),\hspace{2.5pt}\left(2,1,2,1\right),\hspace{2.5pt}\left(0.5,1,0.5,0.5\right)\right\}$.

Next we discuss a useful in practice interval approximation of the Hausdorff saturation $\overline{d}$. Let us consider first the case $\lambda =a=b=1$. The saturation $\overline{d}$ has to satisfy equation (44) which now can be written as The function $\mu \left(d\right)$ can be approximated very well for small ds by the function Taking the exponent in the Taylor series we see that the function can be approximated as $\mathcal{O}\left({d^{2}}\right)$ for small enough values of d by the function Let ${d_{1}}$ and ${d_{2}}$ be defined as ${d_{1}}:=\frac{1}{k}$ and ${d_{2}}:=\frac{\ln k}{k}$. We shall check when ${\mu _{2}}\left({d_{1}}\right)<0<{\mu _{2}}\left({d_{2}}\right)$. Obviously the first inequality holds when $k>e$. Assuming that this restriction holds, we see that ${\mu _{2}}\left({d_{2}}\right)=\ln \ln k>0$ and hence the second inequality holds, too. Thus we conclude that the function ${\mu _{2}}\left(d\right)$ has a unique root in the interval $\left(0,1\right)$ and it belongs to the subinterval $\left({d_{1}},{d_{2}}\right)$ when $k>e$, since this function is strictly increasing.

In the next proposition we discuss the general case assuming that $\lambda {k^{a}}>1$.

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 Analogously to the case $a=b=\lambda =1$ , 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 ${\mu _{2}}\left(d\right)$ near zero as $\mathcal{O}\left({d^{2b}}\right)$. 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 ${\mu _{2}}\left(d\right)$ is a lower approximation of $\mu \left(d\right)$ and therefore ${\mu _{2}}\left(d\right)<\mu \left(d\right)$. Thus the saturation is below the root of the function ${\mu _{2}}\left(d\right)$ and hence $\overline{d}<{d_{2}}$. The question stands, when ${d_{2}}<1$. Let us define ${b_{1}}$ as Note that ${b_{1}}<\overline{b}$. We shall show that if $b<\overline{b}$, then ${d_{2}}<1$ only when $b>{b_{1}}$. Using again the notations $K=\lambda {k^{a}}$ and $B=ab$ and having in mind formula (58) we see that ${d_{2}}<1$ when $1>\frac{\ln K}{BK}$ which is equivalent to $b>{b_{1}}$. Note that $K>1$.

On the contrary, ${d_{1}}$ is not always below the saturation $\overline{d}$. It turns out that there exists a value, say ${b_{2}}$, dependent on the other parameters, such that ${d_{1}}<\overline{d}$ for $b<{b_{2}}$, and vice versa. To see this, we consider function (44). Obviously, it is increasing in the distribution domain $\left(0,1\right)$. Also, ${d_{1}}$ increases w.r.t. the parameter b because $\lambda {k^{a}}>1$. Therefore $\overline{\mu }\left(b\right):=\mu \left({d_{1}}\left(b\right)\right)$ is an increasing function, too; note that ${d_{1}}\left(b\right)<1$. Having in mind $\overline{\mu }\left(0\right)=-\infty $, $\overline{\mu }\left(+\infty \right)=+\infty $, and $\mu \left(\overline{d}\left(b\right)\right)=0$, we conclude that indeed ${d_{1}}\left(b\right)<\overline{d}\left(b\right)$ for $b<{b_{2}}$, where ${b_{2}}$ is the unique solution in the interval $\left(0,\infty \right)$ of the equation

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

We can formulate these results in the following proposition.

As we can see from definition (58), ${d_{1}}$ can be viewed as an increasing function w.r.t. the parameter b. Let us consider the second value ${d_{2}}$. It can be written as ${d_{2}}=\alpha \left(\beta \right)={\left(\beta c\right)^{\beta }}$, where $\beta =\frac{1}{ab}$ and $c=\frac{\ln k}{k}$. The function $\alpha \left(\beta \right)$ decreases in the interval $\beta \in \left(0,\frac{1}{ec}\right)$ and increases for $\beta >\frac{1}{ec}$ because its derivative can be written as ${\alpha ^{\prime }}\left(\beta \right)=\alpha \left(\beta \right)\left(\ln \beta c+1\right)$. Thus, we conclude that ${d_{2}}$, considered as a function of the parameter b, ${d_{2}}\left(b\right)$, decreases for $b<{b^{\ast }}$ and increases otherwise, where Some calculus shows that ${b^{\ast }}<\overline{b}$ when $\lambda >\frac{e}{{k^{a}}}$, and ${b^{\ast }}>\overline{b}$ otherwise. Note that ${b_{1}}<{b^{\ast }}$.

The interval approximations of the saturation are presented in Figures 2e and 2f. The values of $\overline{d}$, ${d_{1}}$, and ${d_{2}}$ considered as functions of the parameter b are colored in blue, red, and orange, respectively. The parameters for the first figure are $\lambda =2$, $a=1$, and $k=20$. The related important values for the parameter b are ${b_{1}}=0.0922$, ${b_{2}}=1.9126$, ${b^{\ast }}=0.2507$, and $\overline{b}=3.6889$. We mark ${b_{1}}$, ${b_{2}}$, and ${b_{3}}$ by black, green, and blue points, respectively. In this case ${b_{1}}<{b^{\ast }}<{b_{2}}$ and thus the first case of Proposition 4.6 holds. Also ${b^{\ast }}<\overline{b}$, since $\lambda >\frac{e}{{k^{a}}}$. We can see in Figure 2e that the interval $\left({d_{1}},{d_{2}}\right)$ is a good evaluation for the saturation $\overline{d}$ when $b\in \left({b^{\ast }},{b_{2}}\right)$. Otherwise, if $b>{b_{2}}$, then the limitation is $\overline{d}<{d_{1}}$.

We choose parameters $\lambda =2$, $k=1$, and $a=5$ for Figure 2f. Now the important values are ${b_{1}}=0.0693$, ${b_{2}}=0.0134$, ${b^{\ast }}=0.1884$, and $\overline{b}=0.1386$. Note that ${b^{\ast }}>\overline{b}$ because $\lambda <\frac{e}{{k^{a}}}$. Also, ${b_{1}}>{b_{2}}$ and thus the second case of Proposition 4.6 is actual. We have that $\overline{d}>{d_{1}}$ for $b<{b_{2}}$, but both values are very close. Otherwise $\overline{d}<{d_{1}}$ when $b\in \left({b_{2}},\overline{b}\right)$.

Let us mention that the relation $\overline{d}<{d_{2}}$ still holds if we remove the restriction $b<\overline{b}$. In this case we have $\overline{d}<{d_{2}}<{d_{1}}<1$.

The defined corrected Kies distributions depend on four parameters λ, k, a, and b. The maximum likelihood estimator can be obtained in a closed form – for original Kies distributions, see [13]. Unfortunately, it turns out that this method does not work efficiently either in terms of speed or precision. For this we construct a least square errors (LSqE) type algorithm. It falls in the large class of generalized methods of moments (GMM), since it is based on curve fitting to a histogram. Hence we can use the existing results for the GMM; we refer to [8]. These methods produce consistent and asymptotically normal estimators for the Kies distributions since the whole Kies family exhibits finite moments.

Let us have n observations ${t_{1}}$, ${t_{2}},\dots ,{t_{n}}$. First we calculate the empirical PDF at $m\left(=50\right)$ bins as Then we derive the PDF values of the corrected Kies distribution with parameters $\left(\lambda ,k,a,b\right)$ in the centers of the bins, say ${l_{i}^{Kies}}\left(\lambda ,k,a,b\right)$, via formula (12). The usual LSqE criterion for minimization is We introduce a little logarithmic modification to minimize the impact of the extremely large values of the PDF. We make this because the PDF is infinitely large at the zero for some values of the parameters. Thus we define the cost function as We have to minimize the corresponding criterion – (66) or (67) – over all possible parameters $\left\{\lambda ,k,a,b\right\}$. The additional constant ϵ is introduced, because some empirical values may be equal to zero. We set this constant to be $\epsilon ={10^{-5}}$. Also, we can use criterion (66) if the empirical PDF seems to be finite at its left endpoint. We provide some experiments to validate this algorithm. We generate n corrected Kies distributed random numbers as ${t_{i}}={Q_{F}}\left({r_{i}}\right)$, where ${Q_{F}}\left(\cdot \right)$ is the quantile function given in equation (11), and ${r_{i}}$ are $\left(0,1\right)$-uniformly distributed random numbers. Our choice of n is among $n=1\hspace{2.5pt}000$, $n=10\hspace{2.5pt}000$, $n=100\hspace{2.5pt}000$, and $n=1\hspace{2.5pt}000\hspace{2.5pt}000$. We fix the coefficients λ and k to one, $\lambda =k=1$, and vary a and b among 0.5, 1, and 2. We report in Table 1 the results which are returned by our calibration algorithm. The fits can be seen in Figure 3. It turns out that this simple LSqE algorithm is quite fast and accurate.

We investigate now the behavior of the S&P500 index. It is one of the most used indicators in the financial markets and provides an important information for the world economy. We use daily observations for the period between January 2, 1980 and July 01, 2022 – totally 10717 ones. We derive the so-called log-returns, denoted by ${r_{i}}$, via the equation where ${S_{i}}$ are the observed S&P500 values. The log-returns are presented in Figure 4a. It can be seen that there are periods of calm trading as well as high-volatility periods. This is a well observed phenomenon at all financial markets – the so-called volatility clustering. The highest downward peak happens at October 19, 1987 (1971th observation) – the Black Monday. The S&P500 index loses more than twenty percents – this is the highest one-day loss ever. We are interested in the length of the periods between the shocks. We derive them obtaining the dates at which the index falls by more than two percents – there are 357 such dates – and then we calculate the lengths of the periods between these days. The longest such period contains 950 days – between May 19, 2003 and February 26, 2007. We mark these days with red points in Figure 4. We may view the derived lengths as survival times and we examine their distribution. We divide all observations by 1000 to fit the Kies domain, because the maximal value is 950. We calibrate the parameters of four distributions – corrected Kies and its ancestors, namely, exponential, Weibull, and original Kies. We use the following parametrization:

The derived parameters are reported in the first part of Table 2. Immediately after them we provide the results which are returned by the LSqE algorithm described in Section 5. The constant ϵ in cost function (67) is chosen to be $\epsilon =0.01$. It turns out that the corrected Kies distribution is significantly closer to the real observations – its error is 23.1820. The value of this error for the original Kies distribution is 25.3491, whereas for the exponential and Weibull distributions it is 26.6652 and 29.4037, respectively.

Having in mind Propositions 2.1 and 2.6 (third statements) we conclude that the initial value of PDF for both Kies style distributions is the infinity, because $b=0.7120<1$ for the original distribution and $ab=0.4509<1$ for the corrected one. Hence, a lot of mass is in the left part of the domain. This fact confirms the mentioned above financial phenomenon of volatility clustering. This is true for the Weibull distribution, too, since its parameter k is less than one; $k=0.8327<1$. Also, the initial value of the exponential PDF is relatively large; it is 34.1236.

Additionally, the shape of the calibrated distributions means that the right tails are important. In fact, they present the probabilities of large calm periods in the markets. We compare the results which the four distributions generate for the tail measures $\mathit{VaR}$, $\mathit{AVaR}$, and $\mathit{EX}$ with the empirical ones – see again Table 2. The levels we have chosen are 0.9, 0.925, 0.95, and 0.975; the values of $\mathit{VaR}$, $\mathit{AVaR}$, and $\mathit{EX}$ are derived via equations (11), (30), and (37). Note that here the meaning of these measures is quite different from their traditional use in finance. We can see again that the corrected Kies distribution produces more realistic values in a comparison with the other distributions.

Here is the place to mention another purpose for which we can use these distributions. The available historical data generates relatively small number of observations for the dates with shocks. Note that the lower empirical values for all tail measures reported in Table 2 can be explained namely by the lack of enough observations. Therefore we can use the theoretical distributions as a tool to fulfill the missing information. In this light the closest tail behavior of the corrected Kies distribution has an additional importance.