The purpose of this paper is to explore two probability distributions originating from the Kies distribution defined on an arbitrary domain. The first one describes the minimum of several Kies random variables whereas the second one is for their maximum – they are named min- and max-Kies, respectively. The properties of the min-Kies distribution are studied in details, and later some duality arguments are used to examine the max variant. Also the saturations in the Hausdorff sense are investigated. Some numerical experiments are provided.

Several extensions of the exponential distribution are available in the stochastic literature. They are very useful in different practical areas including engineering sciences, meteorology and hydrology, communications and telecommunications, energetics, chemical and metallurgical industry, epidemiology, insurance and banking, etc. One of the most important is the Weibull distribution firstly introduced in [

Later many authors turn to different modifications of these distributions. Refs. [

Alternatively, we propose a polynomial transformation to be used before the fractional linear one leading to the Kies distribution. The importance of the arising distribution is due to the following property. It turns out that if

Analogously to the min-Kies distribution we define the max-Kies one – it describes the random variable

Finally, we present some numerical results using a real statistical data for the S&P500 financial index on the one side, and for the unemployment insurance issues on the other. Note that we can describe a left-skewed sample as well as a right-skewed one by the min- or max-Kies distributions.

The paper is structured as follows. We present the base we use later in Section

The following notations shall be used hereafter: an uppercae letter for the cumulative distribution function of a distribution, the overlined letter for the complementary cumulative distribution function, the corresponding lowercase letter for the probability density function, and the letter

Let

The PDF values at the endpoints can be derived directly from formula (

Suppose that

Suppose now that

Suppose that

We establish the min-Kies family through the following definitions.

Let the set

We shall denote by

The following lemma holds.

Note first that the inverse function really exists, because the functions

Let

Alternatively, we may define the new Kies style distribution as its CCDF.

The following theorem discloses the theoretical importance of the min-Kies distribution and motivates its name.

We need to slightly modify the proof of the well known fact that the product of the CDFs of two independent random variables is the CDF of their maximum. Using the lower index to distinguish the distribution terms in Corollary

The corollary below gives the PDF of the min-Kies distribution.

The behavior of the min-Kies PDF given in formulas (

The results when

Suppose now that

If

Otherwise, if

It remains to investigate the case

We present in Figure

Min- and max- Kies PDFs

Parameters

parameter | (A1) | (A2) | (A3) | (B1) | (B2) | (B3) |

2 | 2 | 2 | 5 | 10 | 5 | |

1 | 1 | 1 | 48.8656 | 3.8367 | 7.7266 | |

2 | 2 | 2 | 33.4292 | 9.5941 | 0.8782 | |

– | – | – | 5.6038 | 2.2675 | 7.4407 | |

– | – | – | 2.9209 | 2.7862 | 0.5300 | |

– | – | – | 2.4441 | 6.3067 | 4.4290 | |

– | – | – | – | 2.6047 | – | |

– | – | – | – | 8.7562 | – | |

– | – | – | – | 9.5910 | – | |

– | – | – | – | 2.0228 | – | |

– | – | – | — | 0.9302 | – | |

2 | 2 | 2 | 48.5290 | 41.8723 | 39.8993 | |

0.5 | 1 | 1.5 | 38.6619 | 37.1788 | 35.9901 | |

– | – | – | 34.0970 | 34.0846 | 32.7791 | |

– | – | – | 5.8855 | 31.9772 | 27.8533 | |

– | – | – | 1.0243 | 29.3628 | 1.6168 | |

– | – | – | – | 26.4715 | – | |

– | – | – | – | 23.3149 | – | |

– | – | – | – | 16.1187 | – | |

– | – | – | – | 1.8369 | – | |

– | – | – | – | 0.8744 | – | |

saturation, prime | 0.3678 | 0.4806 | 0.5469 | 0.4753 | 0.5164 | 0.4891 |

saturation, dual | 0.9611 | 0.9635 | 0.9655 | 0.9999 | 0.9999 | 0.9999 |

compl. sat., prime | 0.9611 | 0.9635 | 0.9655 | 0.9999 | 0.9999 | 0.9999 |

compl. sat., dual | 0.3678 | 0.4806 | 0.5469 | 0.4753 | 0.5164 | 0.4891 |

Next we discuss the quantile function.

We have to prove

An important property of the min-Kies distributions is their finite moments.

Let

Although the expectations cannot be derived in closed form, Lemma

Changing the variables as

The following relation between the min-Kies distribution and the gamma function stands.

Applying Proposing

We investigate in this section the tail properties of the min-Kies distribution. We shall present some results based on several risk measures arising from the financial markets, namely, Value-at-Risk (

Let

The left

The left and right

The

The

In fact the original definition of the

The original definition of the expectile is the minimizer of the following quadratic problem

Definition

Let the constants

We need the following lemma before to continue with the expectile tail measure and the mean residual life function.

We have to rewrite truncated expectations (

The statements below for the

The first statement holds due to Definition

We define the Hausdorff distance in a sense of [

Let us consider the max-norm in

We can view the Hausdorff distance as the highest optimal path between the curves. Next we define the

Let

The following corollary holds for the saturation.

Having in mind that the distribution’s left endpoint is

Obviously

Intuitively, the saturation has to depend only on the domain’s length

Using equation (

Next two theorems provide a semi-closed form formula for the saturation.

Suppose that presentation (

Equation (

The CDFs with parameters given in Table

We first introduce a specific distributional duality.

Let

The following corollary explain the essence of the dual distributions. In fact it moves the left behavior of a distribution to the right side and vice versa.

The max-Kies distribution is defined as dual of the min-Kies one.

The name max-Kies is motivated by the following proposition.

The proof is based on the fact that the CDF of the max-Kies distribution is the product of the CDFs of the underlying original Kies ones. See the proof of Theorem

The probability density and the quantile functions of the max-Kies distribution can be presented as

The PDF’s shape can be deduced through Proposition

Let the functions

We discuss now briefly the tail behavior of the max-Kies distribution. The results are similar to their min-Kies analogous and we omit the proofs. The

Having in mind that formulas (

Finally, let us turn to the saturation of the max-Kies distribution. We formulate the analogues of the results derived in Section

Equation (

Let us consider the left hand-side of equation (

We need to define a saturation of another style to continue the investigation on the max-Kies distribitions.

The complementary saturation when

The following analogue of Corollary

The proof is very similar to the proof of Corollary

The following theorem allows us to derive the complementary saturation for the min- and max-Kies distributions.

Equation (

The PDFs of the dual distributions are presented in Figures

We shall provide now two numerical experiments to check how the defined in this paper distributions describe real statistical samples. The first one is based on the historical data for the S&P 500 financial index whereas the second one is for the unemployment insurance issues. We shall use the min-Kies distribution because both are left-skewed.

We shall calibrate now the min-Kies distribution to the statistical data used in [

We calibrate the min-Kies distribution with one, two, three, and four components minimizing cost function (

Applications

Numerical estimations

par. | (A0) | (A1) | (A2) | (A3) | (A4) | (B1) | (B2) | (B3) | (B4) |

1 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |

15.7857 | 15.1377 | 3053.03 | 19444234.9 | 11623604.4 | 101.708 | 13.4539 | 74.0164 | 193.5577 | |

– | – | 6.6251 | 32.1833 | 15.6489 | – | 7.2452 | 0.0123 | 0.0078 | |

– | – | – | 3.9017 | 3.2997 | – | – | 8.5378 | 0.0111 | |

– | – | – | – | 0.3394 | – | – | – | 9.1257 | |

0.7120 | 0.6800 | 4.8953 | 11.8106 | 9.6112 | 2.3473 | 13.7467 | 13.3751 | 14.4112 | |

– | – | 0.3609 | 1.8377 | 1.3580 | – | 1.2393 | 6.1671 | 6.9765 | |

– | – | – | 0.2259 | 0.2136 | – | – | 1.2787 | 5.9488 | |

– | – | – | – | 0.1186 | – | – | – | 1.2987 | |

0 | 0.0048 | 0.0064 | 0.0077 | 0.0080 | 0.0257 | 1.1141 | 1.1132 | 1.1125 | |

1 | 0.9900 | 1.0076 | 1.0165 | 0.9912 | 18.8619 | 8.3748 | 8.9268 | 9.1328 | |

error | 25.3491 | 25.2108 | 22.1423 | 21.3855 | 21.0382 | 4.7166 | 3.4270 | 3.3980 | 3.3908 |

We shall use now a monthly historical data for the unemployment insurance issues in the period between 1971 and 2018 – totally 574 observations. The data can be found at

We can see a major difference between these examples – the initial point for the first PDF is the infinity whereas it is zero for the second one. This is due to the coefficient

The authors would like to express sincere gratitude to the editor Prof. Yuliya Mishura and to the anonymous reviewers for the helpful and constructive comments which substantially improve the quality of this paper.