A Two-Parameter Inverted Exponentiated Skewed Student-T Distribution: Theory and Application

In this study, a new two-parameter generalization of the skew-t distribution called the Inverted Exponentiated Skew t distribution, has been proposed. Some properties of the model like order statistics, entropy, asymptotic behavior, moments, characteristic function, and quantile function were derived. Parameter estimates using maximum likelihood estimation were consistent and the behavior of the estimates with increase in sample size was studied using Montecarlo simulation. The flexibility of the Inverted Exponentiated Skew Student t model was demonstrated by application to four financial datasets and the results revealed that the Inverted Exponentiated Skew Student t model yielded a better fit

where ( ) Gy is the CDF of any baseline distributions.To obtain the PDF of the above CDF is by taking differentiated with respect to y to give; ( ) ( ) ( ) where 0 u  , the shape parameter and ( ) ( ) dG y gy dy =

Inverting a probability density function
The procedure of inverting the density function used in this study was adapted from that described by Chesneau et al.,, (2020).The CDF of the exponentiated family distribution is given by (1).Let ( ) which is the inverted CDF of the Inverted Exponentiated Distribution (IED).

Inverted Exponentiated Skewed Student-t Distribution (IESStD)
The Inverted Exponentiated generator as derived and given by the expression in (3) and the CDF of the Skewed Student-t Distribution is given as; ; (Jones, and Faddy, 2007) (4) Substituting (4) into equation (3), the CDF of IESStD is obtained as; ( )  ( ) ( ) ~, y IESStD u  , its PDF is given by; Let y be a random variable from an arbitrary baseline distribution.Hence, y is said to follow the proposed Inverted Exponentiated Skewed student-t distribution (IESStD) if it has its CDF and PDF as presented in equation ( 5) and (6).In Figure 1, the PDF plot of the IESStD reveals crucial insights into the distribution's shape and likelihood of different values.Asymmetry is evident with a prolonged tail on one side and a shorter, steeper tail on the other.The peak of the curve signifies the mode or the most probable value, while the heavier tails, characteristic of the IESStD, suggest a greater probability of extreme values.In Figure 2, the CDF plot of the IESStD.The gradual increase from 0 to 1 (on the y axis) indicates the increasing likelihood of higher values, with any changes in slope corresponding to critical points.The point where the CDF reaches 0.5 marks the median of the distribution, highlighting its central tendency.

Asymptotic behaviour of IESStD Proposition 1
The limit of Inverted Exponentiated Skewed student-t density function as y → is 0 and the limit as y → − is 0.

Proof:
These can be shown by getting the limit of the Inverted Exponentiated Skewed student-t density function For y →: and let y → − ( ) ( ) This is an indication that the proposed IESStD is unimodal.

Properties of IESStD Binomial Expansion of the Proposed Probability Distributions
In the following, the PDF and CDF of the proposed IESStD was we expand using series expansion in order to easily compute the some of the properties of the distribution.Thus, if "a" is a positive real non-integer and y 1  , we can consider the power series expansion given as;  ; and also the sum of the binomial series given by; ( ) ( )

Binomial expansion of PDF of IESStD
The PDF of IESStD, is given as; ( ) ( ) ( )

Quantile function of IESStD
The corresponding quantile function ( ) where ( )

Moment function of IESStD
By definition, the r th moment of a continuous random variable y is given by; ( ) Let ỹ IESStD , then the r th moment of the random variable is derived as follows.On substituting ( 18) into(24) , we have that Substitute ( 27) into ( 26) Hence the moment of the IESStD is given as;

Characteristic function of IESStD
The characteristic function for a random variable y with PDF ( )  Substitute ( 34) into (33) Recall from the moment expression in (31), substituting into (35) yields the characteristic function given as;

Order Statistic
Order statistics are the sorted values of a set of random variables.These statistics are useful in statistical inference, providing information about the distribution and characteristics of the underlying population (Dey, Raheem, and Mukherjee 2017).Let Y1 < Y2 < ... < Yc be an ordered random sample from a valid continuous density function, then, the c th order statistic is given as; Therefore, equation (37) becomes The above expression in (39) will be used to derive the smallest and largest order statistic for the proposed probability model, and to do that, we will utilize the sum of the binomial series of for the PDF and CDF of the distributions in ( 40) and (41) for IESStD.

Order statistic for IESStD
To obtain the c th order statistic for the IESStD, we make the substitution (42) for g(y) and ( 43)for ( ) Largest order statistic

Reliability Functions
Reliability analysis in distribution theory is basically concerned with the analysis of time to failure for a system under study and in this section, the reliability measures/functions for the proposed distributions will be derived.

Survival (S(y)) and cumulative hazard (H(y)) function
One of the functions usually estimated in reliability analysis is the survival function.The survival function is defined as the probability that the time to the event is greater than a specified time t.This is obtained by subtracting 1 from the CDF of a distribution.Mathematically, it can be expressed as; The hazard function, represents the instantaneous rate of occurrence of an event at time t, given survival up to that time.The cumulative hazard function is given by; In the subsequent subsections, the H(y) and S(y) for the proposed probability distribution in the will be derived.

S(y) and H(y) for IESStD
Substituting ( 5) into (47), the survival function of IESStD is obtained and expressed as follows; ( )  Information measures are quantities used to quantify various aspects of information, uncertainty, and entropy in information theory.These measures have had different applications in various fields including communication theory, cryptography, machine learning, and statistics, where understanding and quantifying information and uncertainty are crucial (Pierce 2012).In this study the Renyi entropy and its Arimoto measure will be derived in this subsection.

Renyi entropy
In information theory, the Rényi entropy family of entropy measures bear the name of the mathematician Alfréd Rényi.It generalizes the concept of Shannon entropy, providing a parameterized family of entropy measures.This can be defined as follows; ( ) ( ) ( ) via the binomial expansion, the following is obtained; ) Where ( ) ( )

Arimoto measure of entropy
The Arimoto measure of entropy, also known as the "information radius entropy" or "information divergence entropy," provides a way to compare the uncertainty or disorder in probability distributions.This is defined by the following equation;  ( ) By substituting the transformation in ( 27) into (61) we have that; Therefore, on transforming (62) into the form in (29), the Arimoto measure of entropy of the IESStD is given as; Parameter Estimation and Stability Analysis for the IESStD This section presents the MLE estimates and a stability analysis of the estimates of the proposed model.The method of maximum likelihood was used to estimate the values of the model's parameters.This will be illustrated in the following subsections.

Maximum Likelihood Estimators (MLE)
The Maximum likelihood estimation method (MLE) have been known to yield estimators with properties viz; minimum bias, efficiency, consistency etc. (Banerjee and Bhunia 2022).Let random variable y1,y2,...,yn of random sample of size n be obtained from IESStD with probability density function in equation ( 6), then, the likelihood function is given by: ( ) ) The Mean Squared Error (MSE) and Root Mean Square Error (RMSE) serve as critical metrics in statistics and data analysis.These metrics offer a comprehensive evaluation of the accuracy and precision of predictions generated by statistical models or estimators (Chai andDraxler 2014, Hodson 2022).For the proposed probability distributions, the stability of the Maximum Likelihood Estimators (MLE) was studied (with the aid of r software) by observing the behaviors of the MSE and RMSE at different sample sizes via Monte Carlo simulations.Random numbers were generated using the quantile functions and the uniform distribution with fixed parameter values for the simulations.

Stability Analysis Via Montecarlo Simulations for IESStD
Table 4, 6, and 8 presents the results of fitting the Ethereum, Dow Jones Industrial Average, SandP500 return series to IESStD, ESStD, GD, Logistic D, and SStD probability models.The results include the AIC, MLE, and loglikelihood values.The results revealed that the IESStD has the smallest AIC and a corresponding high log-likelihood values.A larger likelihood value indicates a better model fit, while a lower AIC which also suggests better fit indicating that the IESStD model provides a better fit when compared with other probability models.

Conclusion
IESSt distribution was derived in this study by inverting the CDF generator of the exponentiated family of distribution.Probability density plot and the corresponding cumulative distribution function were plotted for some selected parameter values.Some statistical properties of the model such as the moments, the moment generating function, the quantile function, the c th order statistic were derived for the proposed distribution.The model parameters were estimated using the maximum likelihood estimation (MLE) method.Through the use of AdequacyMlodel and Maxlik packages in r software, a Montecarlo simulation of the the MLE was performed and the results revealed the MLE to be consistent and as expected, with increased sample size, the RMSE almost converges to zero.The model potential of the in financial data analysis was illustrated using four financial data (return of daily price of Ethereum, ASI, SandP500 and Dow Jones Industrial Average) datasets.The results revealed that the IESSt distribution fitted well among other distributions in terms of having the lowest AIC and the largest loglikelihood values.For modeling financial data, the application of IESSt distribution should be highly considered.Also, in future research, this distribution may contribute in the area of financial asset modeling such as the Generalized Autoregressive Conditional Heteroscedasticity (GARCH), in reliability and survival analysis especially in observations that are right skewed.
theory of calculus, the PDF of the IESStD is obtained from the CDF by differentiating with respect to (wrt) the random variable y, given as;

Figure 1 :
Figure 1: Probability Density Function plot of IESStD at different parameter values where l  =

Figure 2 :
Figure 2: Cumulative Density Function plot of IESStD at Different Parameter values where l  = CDF of IESStDUsing (13) the CDF of IESStD can be represented as a sum of series as follows.


The c th order statistic for the IESStD is thus expressed as; cumulative hazard function of IESStD, we make the substitution of equation (49) into equation (48) and in figure3, the hazard function plot is presented.©2024 Department of Mathematics, Modibbo Adama University

Figure 4 :
Figure 4: Hazard Function plot of IESStD at Different Parameter values where l  = Information MeasuresInformation measures are quantities used to quantify various aspects of information, uncertainty, and entropy in information theory.These measures have had different applications in various fields including communication theory, cryptography, machine learning, and statistics, where understanding and quantifying information and uncertainty are crucial(Pierce 2012).In this study the Renyi entropy and its Arimoto measure will be derived in this subsection.


, then the degree of uncertainty can be estimated by substituting (6) for g(y) in equation (53) as follows;

Danjuma et al. (2024) International Journal of Development Mathematics Vol 1 Issue 2, 131 -152
© 2024 Department of Mathematics, Modibbo Adama University Gy is the CDF of the density function.Then,

Table 6 :
Goodness of fit for Dow Jones Industrial Average