The unit log–log distribution: a new unit distribution with alternative quantile regression modeling and educational measurements applications

In this paper, we propose a new distribution, named unit log–log distribution, defined on the bounded (0,1) interval. Basic distributional properties such as model shapes, stochastic ordering, quantile function, moments, and order statistics of the newly defined unit distribution are studied. The maximum likelihood estimation method has been pointed out to estimate its model parameters. The new quantile regression model based on the proposed distribution is introduced and it has been derived estimations of its model parameters also. The Monte Carlo simulation studies have been given to see the performance of the estimation method based on the new unit distribution and its regression modeling. Applications of the newly defined distribution and its quantile regression model to real data sets show that the proposed models have better modeling abilities than competitive models. The proposed unit quantile regression model has targeted to explain linear relation between educational measurements of both OECD (Organization for Economic Co-operation and Development) countries and some non-members of OECD countries, and their Better Life Index. The existence of the significant covariates has been seen on the real data applications for the unit median response.


Introduction
In educational measurement, it should be considered that the measurement process is done for an educational purpose. Educational measurement is the tool in order to obtain information about the characteristics of students and the educational politics of the countries as well as it provides information about the development levels of the countries. Among these educational measurements, the results obtained from international exams, educational attainment percentage of countries, education and school living conditions, data about education and training can be given as examples. These measurements of countries are easily accessible by the OECD. OECD has examined the metrics of countries related to education on two themes. The first of these is the education and learning theme, and the other is the Social Protection and Well-being theme. Using the data sets from OECD countries, there are many papers in the literature. It can be seen [1,2,4,7,13,14,18,21,28,30,52] for some of them. These data sets, which make up the educational measurements, vary according to the development status of the countries. Therefore, some countries' measurements may appear as outliers in the sets. For such situations, using statistically robust methods would be more appropriate for inferences. For example, quantile regression, introduced by [34], models can explain the linear relationship between response variable and independent variable with more robust results than ordinary regression models. One may see [2,13,14,23,37,54] for applications of the educational measurements based on the regression modeling.
On the other hand, modeling the real data sets by the proposed probability models is a very important issue in terms of statistical inferences. This modeling process draws attention to the importance of the probability distributions and it has received great attention by many statisticians. Especially, the modeling approaches on the unit interval have increased recently since they related to specific issues such as the recovery rate, mortality rate, proportion of the educational measurements etc. So, it is needed a unit distribution defined on the (0, 1) interval to model random variable (rv) belonging to the above important issues. When the unit modeling is wanted to apply, no doubt that the beta distribution is firstly remembered by the researchers. Although the beta distribution has different density shapes to fit data on the unit support, in order to obtain better results based on inferences from data sets, the new alternative models have been defined on the unit interval in the statistical distribution literature. These alternative models generally have been obtained via transformations of the well-known rv. For example, the Johnson S B distribution, introduced by [32], has been obtained via the inverse function of the logit transformation of the standard normal rv. Further, the Johnson S B [33], unit logistic [57], logit slash, [9] and unit Johnson S U [25] distributions have been proposed with the same transformation method via standard Laplace, logistic, slash, and unbounded Johnson S U rvs respectively. The unit gamma [15], log-Lindley [29], log-xgamma [5], unit Weibull [43], unit Birnbaum-Saunders [40], unit inverse Gaussian [26], unit Gompertz [41], unit generalized half-normal [10] and log-weighted exponential [2] distributions have been defined by the transformation of the negative exponential function of the baseline rvs. These proposed unit distributions have more flexible density shapes on the (0,1) interval than those in their baseline distribution. This result emphasizes that the suitable transformations, which transform the baseline rvs to the unit interval, propose the distinguishing features in the new density on the (0,1) interval. For example, the ordinary slash distribution has the heavy-tailed and symmetric bell-shaped density on the real support. So, this distribution models only the data sets with the unimodal heavy-tailed density shape on the unit interval. If the data set has different properties, it does not model this data. However, the logit slash distribution [9], which has been defined on the unit interval by the inverse function of the logit transformation of the ordinary slash distribution, has very flexible density shapes such as the w-shaped, N-shaped, inverse N-shaped, bimodal, unimodal, increasing and decreasing. This transformation has brought into the open the different density properties that the ordinary slash distribution does not have on the unit interval. This situation has been seen for both the transformation distribution of the negative exponential function of the baseline rv and the above all proposed distributions. The Topp-Leone [58], Kumaraswamy [35], generalized beta [44], simplex [6], standard two-sided power [59], unit Lindley [39], unit improved Lindley [3] distributions have been suggested as the other alternative models to beta distribution. It is noted that the Kumaraswamy distribution can be obtained with the transformation of the negative exponential function with respect to the exponentiated exponential distribution [27]. Further, for the above some distributions, the alternative regression models such as the beta regression [24], Kumaraswamy quantile regression [46], log-Lindley regression [29], unit-Lindley regression [41], log-weighted exponential regression [2], improved Lindley regression [3] and unit-Weibull quantile regression [42] models have been proposed.
On the other hand, the log-log (LL) distribution has been proposed by Pham [48]. If T rv is the LL rv then its cumulative distribution function (cdf), probability density function (pdf) and hazard rate function (hrf) are respectively given by and where t > 0, α > 0 and β > 1. The pdf can be decreasing and unimodal. The author has also shown that the hrf is the decreasing if t ≤ t 0 and it is the Hence, the hrf of the model has the bathtub shape. Some statistical properties of the LL distribution such as hrf shapes, mean residual lifetime and estimation of the parameters have been studied by the author. Pham [48] has used this model for evaluating the reliability of several helicopter parts as well as it has been modeled for the mortality rate with the data year 2005 of the United States by Pham [49,50]. The [56] has derived the Bayes estimations of the model parameters via the Markov chain Monte Carlo technique. This distribution is also known as Pham distribution in the literature. Under the above information, the goal of this study is to propose a new unit alternative distribution and its quantile regression modeling to model the percentages and proportions. A new alternative bounded statistical model, which has a very flexible density in terms of the data modeling, to well-known bounded unit models such as beta and Kumaraswamy has been aimed by the current paper. Further, it is to show its modeling ability with their applications based on the proportion of the educational measurements' OECD countries. To obtain a new unit distribution, we use the negative exponential transformation of the LL distribution. In this way, we will transport its applicability and work-ability to the unit interval. Hence, we will bring into the different density and hrf properties that the LL distribution has not on the unit interval.
The paper has been planned as follows. The proposed distribution has been defined in Section 2. Its basic distributional properties and procedures of the maximum likelihood estimation (MLE) for model parameters have been introduced by Supplemental File.
The new quantile regression model based on the newly defined distribution, its MLE processes for the model coefficients, and its residual analysis are introduced in Section 3. The simulation studies are given to see the performance of the MLE estimators of the model parameters in Section 4 based on the new unit distribution and new quantile regression model. Three real data illustrations, one is related to the univariate data modeling and others are the quantile regression modeling, have been illustrated in Section 5. Finally, the paper is ended with concluding in Section 6.

Definition of the new unit distribution
The new unit distribution is defined as follows: Let T ∼ LL(α, β) and define the X = exp(−T) rv. Then, the corresponding cdf and pdf are respectively given by and where x ∈ (0, 1), β > 1 and α > 0 are the model parameters. These equations can be obtained with the transformation method. We denote call it the unit log-log distribution and denote it with the ULL(α, β). The shapes of the pdf and hrf are given by Supplemental File in detail. In addition, for the newly defined unit distribution, Supplemental File provides the basic distributional properties such as stochastic ordering, quantile function, moments, and order statistics as well as the MLE processes for the model parameters. Related references are [55] and [45] for the stochastic ordering and moments subsections respectively.

The ULL quantile regression model
If the support of the response variable is defined on the unit interval, using a unit regression model based on the unit distribution can be used for modeling the conditional mean or quantiles of the response variable via independent variables (covariates). Ferrari and Cribari-Neto [24] proposed the beta regression model for continuous variates that assume values in the standard unit interval, e.g. rates, or proportions. When the response variable has the outliers in the measurements, robust estimation results based on the regression model are needed for model inference. The quantile regression model is a good robust alternative model to the ordinary LSE and beta regression models. Whereas the method of least squares and beta regression models estimate the conditional mean of the response variable, the quantile regression estimates the conditional median or other quantiles of the response variable. With this approach, [11][12][13][14]38,42,46] have introduced the Kumaraswamy, unit Weibull, exponentiated arcsecant normal, unit Birnbaum-Saunders, unit Burr-XII, transmuted unit Rayleigh, and arcsecant Weibull quantile regression models respectively. Now, we introduce an alternative quantile regression based on the ULL distribution. The quantile function of the ULL distribution is given by where u ∈ (0, 1). Now, the pdf of the ULL distribution can be given with re-parameterized based on its quantile function. Let μ = x u (α, β) and β = (1 − log u) (− log μ) −α . Then, the pdf and cdf of the re-parametrized distribution is given by and respectively, where α > 0 is the shape parameter, the parameter μ ∈ (0, 1) represents the quantile parameter, and u is known. We call it as quantile unit log-log distribution and denote it with Y ∼ QULL(α, μ, u). Some possible shapes of the QULL distribution are shown in Figure 1. We see that the QULL distribution has the skewed shapes as well as bathtub shaped, inverse N-shaped, decreasing, increasing and unimodal. Now, it can be focused on the quantile regression model based on the QULL distribution with pdf (3). Let y 1 , y 2 , . . . , y n random observations from QULL distribution with unknown parameters, i , and α. Note that the parameter u is known. The QULL quantile regression model is defined as are the unknown regression parameter vector and known ith vector of the covariates. The function g(x) is the link function that is used to link the covariates to the conditional quantile of the response variable. For instance, when the parameter u = 0.5, the covariates are linked to the conditional median of the response variable. Since the QULL distribution is defined on the unit interval, we use the logit-link function which is given by

Parameter estimation via MLE method
We point out the unknown parameters of the QULL quantile regression model to obtain them via the MLE method. Consider below link function From (5), the following equation is obtained Inserting the equations (6) in (3), the log-likelihood function of the QULL quantile regression model is given by where = (α, δ) is the unknown parameter vector. Since (7) includes nonlinear function according to model parameters, this log-likelihood function can be maximized directly by the software such as R, S-Plus, and Mathematica. It can be noticed that when u = 0.5, it is equivalent to model the conditional median. Under mild regularity conditions, the asymptotic distribution of (ˆ − ) is multivariate normal N p+2 (0, I −1 (ˆ )), where I −1 (ˆ ) is the expected information matrix. One may use the (p + 2 × p + 2) observed information matrix for the I −1 (ˆ ). The elements of observed information matrix are evaluated numerically by the software. We use the maxLik function [31] of R software to maximize (7). This function also gives asymptotic standard errors numerically, which are obtained by the observed information matrix.

Residual analysis
The residual analysis can be needed to check whether the regression model is suitable. In order to see this, we will point out the randomized quantile residuals [22] and the Cox-Snell residuals [16].

Randomized quantile residuals
The randomized quantile residuals are defined bŷ where the G(y i ,α,δδδ) is the cdf of the QULL distribution, which is given by (4). If the fitted model successfully deals with the data set, the distribution of the randomized quantile residuals will distribute standard normal distribution.

Cox-Snell residuals
Cox and Snell residuals are given bŷ If the model fits data accordingly, the distribution of theê i 's will distribute exponential distribution with scale parameter 1.

Simulation studies
In this Section, we separately point out the simulations studies about the MLE method for ULL distribution and proposed quantile regression modeling. All calculations based on the estimation procedures have been obtained by optimization functions in the R software.

Empirical results for the MLE methods of the ULL distribution
In this section, we perform the graphical simulation studies to see the performance of the MLEs of the ULL distribution with respect to varying sample size n. We generate N = 1000 samples of size n = 20, 25, . . . , 1000 from the ULL distribution based on the actual parameter values α = 1 and β = 2. The random numbers from the ULL model are obtained by its quantile function. All estimations based on the estimation methods have been obtained by using constrOptim routine in the R program. Further, we calculate the empirical mean, bias and mean square error (MSE) of the estimations for comparisons between estimation methods. The bias and MSE are calculated by (for = α, β) respectively. We expect that the empirical means are close to true values when the MSEs and biases are near zero. The results of this simulation study are shown in Figure 2. This figure indicates that MLEs are consistent since the MSE and biasedness decrease to zero with increasing sample size as expected. Hence, MLEs are asymptotic unbiased.
Furthermore, based on the above MLEs, the simulation study has been given to see the performance efficiency of the 95% confidence intervals. In order to see this, the empirical coverage probability (CP) and coverage length (CL) have been used as criteria. The estimated CPs and CLs are respectively given by and  where the I(·, ·) is the indicator function and sˆ is the standard errors of the MLEs, for i = 1, . . . , N, are evaluated by inverting the observed information matrix I. Figure 3 shows the performance of this simulation results. As seen from these Figures, the CPs are near to the nominal value as well as the CL decreases for each parameter when the sample size increases for each parameter. The simulation results verify the consistency property of MLEs. Table 1. The empirical biases, MSE, 95% CP, and 95% CL for the QULL regression model with α = 0.5, δ 0 = −0.5 and δ 1 = 1.  Table 2. The empirical biases, MSE, 95% CP, and 95% CL for the QULL regression model with α = 2, δ 0 = 0.5 and δ 1 = −1.

Simulation studies for the proposed regression model
In this subsection, it has been given the simulation studies in order to see behaviors of the MLEs belonging to the QULL regression model based on the estimated bias, MSE, CP and CL measurements. For the varying sample size n, known u = 0.5, true α, and generated covariates values, the values of the unit response variable have been obtained with For different three simulation scenarios, it has been considered as the true parameters values are (α = 0.5, δ 0 = 0.5, δ 1 = 1), (α = 0.5, δ 0 = 0.5, δ 1 = 1) and (α = 0.5, δ 0 = 0.5, δ 1 = 1) with the following regression structure N(0, 1). In all simulation studies, the replication number is N = 1000, sample sizes are n = 25, 50, 100, 250, 500, 1000 and known quantile level is u = 0.5.
The simulation results for the QULL regression model have been shown in Tables 1-3. As seen from these Tables, the empirical CLs decrease while the sample size increases as well as the empirical CPs are around the 0.95 value. All biases are close to zero value as well as all MSEs tend to zero value at the same time as expected.

Data analysis
In this section, three data applications have been given to see the applicability of the proposed distribution model. The first application is about univariate modeling and others applications are about regression modeling for the ULL model. We obtain all data sets Table 3. The empirical biases, MSE, 95% CP, and 95% CL for the QULL regression model with α = 1.5, δ 0 = −1 and δ 1 = −0.5. Each theme is divided into topics. All used data sets are related to the educational measurements of the above countries. We also compare performances of the real data fittings of the proposed distribution under the MLE method with well know unit distributions in the literature for the first application. These distributions are the beta, Kumaraswamy (Kw) [35], exponentiated Topp Leone (ETL) [51], unit Gompertz (UGom) [41], log-weighted exponential (LWE) [2], unit gamma (UG) [15], and unit Birnbaum-Saunders (UBS) [40] distributions. Their pdfs have been given by Supplemental File. The second and third applications have been given in order to see the applicability of the newly defined regression model. Two competitor regression models, which are wellknown literature, also have been considered. They are beta regression [24] models as well as Kumaraswamy quantile regression [46] models. The used pdfs of the beta regression and Kumaraswamy quantile regression are where μ ∈ (0, 1) is the mean and α > 0 and where μ ∈ (0, 1) is the median and α > 0 respectively. For univariate data modeling application, theˆ values, Akaike Information Criteria (AIC), Bayesian information criterion (BIC), Kolmogorov-Smirnov (KS), Cramer-von-Mises, (W * ) and Anderson-Darling (A * ) goodness-of-fit statistics have been obtained based on all distribution models to determine the optimum model. In general, it can be chosen as the optimum model the one which has the smaller the values of the AIC, BIC, KS, W * and A * statistics and the larger the values ofˆ and p-value of the goodnessof-statistics. For regression modeling applications, the Vuong likelihood ratio test (see [60]) has been used for comparison of the non-nested two regression models whether there is any significant difference in the fit models. The Vuong statistic is defined as where λ))} 2 , g(y, θ) and f (y, λ) are the corresponding competitor pdfs calculated at the MLEs. It is noticed that, while n → ∞, the asymptotic distribution behind V is the standard normal distribution. Therefore, at a significance level of υ% distribution equivalence is rejected if |V| < z υ/2 . The calculations have been obtained by the maxLik function [31] of the R software. In addition, the betareg function of the R software for results of the beta regression model has been used. The details are the following.

Univariate real data modeling
This subsection introduces analyzing of the real data set to show the modeling ability of the proposed ULL model. We use the variable novice teachers who have an assigned a mentor at the current school of the Teachers' in the initial and induction training indicator of the TALIS (Teaching and Learning International Survey) in the Education and Training theme of the OECD.Stat as data set. This variable is defined by proportions of the novice teachers with mentor at the school and it has been given as percentage value for 48 countries which are the OECD countries and selected non-member of OECD economies. The reference year is 2018 for this data set. Some summary statistics of the data set have been given in Table 4. The data is rightskewed and has normal kurtosis.
We compare performance of the real data fitting of the proposed distribution under the MLE method with well know unit distribution in the literature. We give the data analysis results belong to other competitor models in Table 5. Table 5 shows that the ULL distribution has the lowest values of AIC and BIC statistics. Also, the ULL distribution has the lowest values of A * and W * and K-S statistics with a higher p-value. These results show that the ULL distribution can be chosen as the best model for the data set.  The ULL distribution has captured skewness and kurtosis of the data set successfully as well as the estimated cdf is near to empirical cdf. The plotted lines of the probability-probability (PP) are close to the diagonal line which indicates that the performance of the ULL distribution is acceptable for the modeled data. We note that to show the likelihood equations have a unique solution, we plot the profile log-likelihood (PLL) functions of the α and β parameters for the data set in Figure 5. From this figure, we see that the likelihood equations have a unique solution for the MLEs.

Quantile regression modeling for the educational attainment data set
In this subsection, we use the educational attainment, homicide rate and life satisfaction variables of the Better Life Index indicator in the Social Protection and Well-being theme of the OECD.Stat as data set. The reference year is 2017 for this data set. The data set can be found in https://stats.oecd.org/index.aspx?DataSetCode=BLI. In this application, the data set deals with the educational attainment values' percentage of the 35 OECD countries and those of Brazil, Russia, and South Africa. Educational attainment is defined as the highest grade completed within the most advanced level attended in the educational system of the country where the education was received. Educational attainment considers the number of adults aged 25-64 holding at least an upper secondary degree over the population of the same age.
The aim is to associate these educational attainment values (y) with covariates. The covariates associated with this response variable are: • x 1 : homicide rate (ratio); • x 2 : life satisfaction (average score of the Cantril Ladder known also as the Self-Anchoring Striving Scale).
The regression model based on μ i is given by where the μ i is the mean for the beta model whereas it denotes the median for the Kumaraswamy and QULL models. We give the results of the regression analysis in Table 6.   Table, the δ 1 and δ 2 parameters have been seen statistically significant at usual level for the QULL regression model. The δ 0 is the positive value for the beta and Kumaraswamy models however, it is negative for the QULL model. The δ 1 parameter has been found statistically significant at the usual level and has affected negatively for all regression models. Whereas, the δ 2 parameter has not been found statistically significant at the usual level for the beta and Kumaraswamy regression models. It is obvious that all covariates have affected educational attainment based on the QULL model and they are significant statistically. It is also noticed that there is a positive relationship between the median response (educational attainment) and the life satisfaction index of the countries whereas there is a negative relationship between the median response and the homicide rate index of the countries. This means that big values of life satisfaction tend to have more percentage of the educational attainment and its opposite is true for the homicide rate. It is interesting that the regression coefficient δ 2 of the median models has been affected by the distribution of the response variables. It has been obtained as positive signed for QULL and beta models whereas it has been obtained as negative signed for the Kumaraswamy model. This situation can emphasize better goodness of fit because the related coefficient is not statistically significant at any usual significance level for the Kumaraswamy and beta regression models. The fit of the errors also supports better goodness of fit for the QULL model. It has been pointed out fit of the errors following plots. One may see [42] for a similar situation. Further, the Vuong test shows that the QULL regression model is not equivalent to either the Kumaraswamy or the Beta regression models, considering a level of 6.5% of significance. Based on the estimated log-likelihood value and results of the Vuong test, the QULL regression model can be chosen as the best model among application models in terms of better modeling ability than other regression models. Figures 6 and 7 respectively display the Quantile-Quantile (QQ) plot of the randomized quantile residuals and PP plot of the Cox-Snell residuals for all regression models. We can say that the fit of the errors of the QULL regression model to related distributions are better than those of the beta and Kumaraswamy regression models. In addition, these figures support the above results and are a good fit of the QULL regression model to the educational attainment data set.

Quantile regression modeling for the adolescents, who want top grades at school, data set
In this subsection, we use the variables, which are the proportion of the Adolescents (15-year-olds) who want top grades at school, sciences performance at age 15 (PISA), proportion of the Adolescents (15-year-olds) with a desk and a quiet place to study at home, and proportion Adolescents (15-year-olds) who do paid work before or after school, of the education and school life sub-indicator of the Child Well-Being indicator in the Social Protection and Well-being theme of the OECD.Stat as data set. The reference year is 2015 for all variables. In this application, we use the proportion of the Adolescents (15-year-olds) who want top grades at school percentage of the 37 OECD countries as the response variable. The response variable is that the percent of 15-year-old students who report that they agree or strongly agree with the statement I want top grades in most or all of my courses. The aim is to associate these response values (y) with covariates. The covariates associated with this response variable are: • x 1 : sciences performance at age 15 (PISA); • x 2 : Adolescents (15-year-olds) with a desk and a quiet place to study at home; • x 3 : Adolescents (15-year-olds) who do paid work, before or after school.
The regression model based on μ i is given by where the μ i is the mean for the beta model whereas it denotes the median for the unit-Weibull, Kumaraswamy and QULL models. A similar data set has been analyzed by [14].
We give the results of the regression analysis in Table 7. From this Table, the δ 1 parameter has not been obtained statistically significant at the usual level for all models. The δ 2 and δ 3 parameters have been seen statistically significant at the usual level only for the QULL regression model whereas these parameters are statistically significant at 10% level for the beta regression model. The δ 0 is the positive value for all models. The δ 2 parameter has affected negatively to the QULL regression model whereas, the δ 3 parameter has affected positively to this model. It is also noticed that there is a positive relationship between the median response (percentage of the desire of the top grades) and the adolescents, who do paid work before or after school. There is a negative relationship between the median response and the adolescents, who have a desk and a quiet place for studying at home. This means that the percentage of the adolescents, who want the top grades, increases when percentage of the adolescents, who do paid work before or after school, percentage increases. However, it decreases when the percentage of the adolescents, who have a desk and a quiet place for studying at home, increases. The result obtained from all regression models shows that x 1 covariate (sciences performance) is not statistically significant to explain the median response variable. Further, the Vuong test shows that the QULL regression model is not  equivalent to either the Kumaraswamy or the Beta regression models. Based on the estimated log-likelihood value and results of the Vuong test, the QULL regression model can be chosen as the best model among application models in terms of better modeling ability than other regression models. Figure 8 displays the QQ plot of the randomized quantile residuals and PP plot of the Cox-Snell residuals for the QULL regression model. We can say that the fit of the errors of the QULL regression model to related distributions has a good fit.

Conclusion
We define a new unit model, called unit log-log distribution, in order to model the proportion of educational measurements as well as other data sets defined on the unit interval. We investigate the general structural properties of the new distribution. The model parameters are estimated by the six different methods. The simulation study is performed to see the performances of these estimates. The data sets with proportions of some educational measurements belonging to OECD countries are analyzed to point out its applicability, flexibility and comparability.
For the univariate real data modeling, we used to model the proportions of the novice teachers with a mentor at the school for 48 countries according to the data set of TALIS.
According to the results, the proposed distribution has been chosen as the best model under model selection criteria.
For applications of the proposed quantile regression model, we have targeted to explain linear relation between educational measurements of both OECD countries and some nonmembers of OECD countries, and their well-being index. The unit response variable has been considered as the conditional median of the covariates in the proposed regression models. According to the first regression application, it has been seen that there is a positive relationship between the median response (educational attainment) and the life satisfaction index of the countries whereas there is a negative relationship between the median response and the homicide rate index of the countries. This means that big values of life satisfaction tend to have more percentage of the educational attainment and its opposite is true for the homicide rate. Via a unit mean response modeling, Altun [2] has related the educational attainment values of the OECD countries with Labor market insecurity and homicide rate variables and has concluded that the Labor market insecurity and homicide rate affect the educational attainment negatively. Similar results have been also supported by [4]. According to Salinas-Jimenez et al. [19], a positive relationship between education level and life satisfaction has been observed based on data from some OECD countries. Salinas-Jimenez et al. [20] have reported that a positive correlation between education and life satisfaction is found according to data which comes from the World Values Survey of more than 50 countries. Botha [8] has explored various dynamics in the relationship between life satisfaction and education in South Africa using the 2008 National Income Dynamics Survey and his report has indicated a strong positive association between educational attainment and individual satisfaction with life. Nadanovsky and Cunha-Cruz [47] have shown that countries with higher education levels have lower homicide rates, based on a new index called impunity.
For the second regression application, it has been seen that there is a positive relationship between the median response (percentage of the desire of the top grades) and the adolescents, who do paid work before or after school. There is a negative relationship between the median response and the adolescents, who have a desk and a quiet place for studying at home. This means that percentage of the adolescents, who want the top grades, increases in the country when the percentage of the adolescents, who do paid work before or after school, percentage increases in the country. By using the Career Maturity Inventory data, based on the high school students, Creed and Pattan [17] have concluded that students with paid work experience have reported higher levels of Career Development Attitude than those with no paid work experience. Further, they have reported that gender differences have also occurred, with females with paid work experience generally reporting higher levels of career maturity than males with paid work experience. However, the unit median response variable decreases when the percentage of the adolescents, who have a desk and a quiet place for studying at home, increases in the country. The sciences performance covariate has not been found statistically significant to explain the median response variable. One may see [36,53] for some results about socio-economic status and student performance. The existence of the significant covariates has been seen on both real data applications for the unit median regression modeling. Based on another quantile regression modeling, similar results have been also observed by [14].
All empirical findings, which are about modeling of the educational measurements, show that the proposed model provides better fits than the well-known unit probability distribution in the literature for both its univariate data modeling and its regression modelings. It is hoped that the new distribution will attract attention in the other disciplines.

Disclosure statement
No potential conflict of interest was reported by the author(s).