# How to predict parameters of time-variant distributions?

I am trying to create a simulation, and I have my probability distribution function changing over time (getting more skewed, etc.) We can have up to 5 consecutive years. Can you please explain where to look at. What I do is that I first use EasyFit link then I see the changes in the parameters per year.

I am trying to predict the fitting distribution of literacy rate in all world countries. Let's assume for the first year, parameters were Alpha1, Beta1, Gamma1, and second year, the parameters were Alpha2, Beta2, Gamma2. For example, EasyFit chose for the cellular penetration rate in 2014, the Dagum distribution suggested had the following parameters: k= 0.17555, alpha=9.9954, beta= 147.31 and gamma=0 Then for 2016, the distribution had the following parameters k= 0.20865, alpha=9.9207, beta= 146.58 and gamma=0

Would it be possible to predict the shape and location of the distribution at year 3 by taking the annual growth rate from Alpha1 and Alpha2 to expect Aplha3 and so on?

If this is non-sense, can you suggest an easy method to know how the distribution will most likely look like in year 3?

• How do you obtain the distributions in each year? What distribution are you using?
– Carl
Commented Feb 2, 2017 at 5:29
• I am have a variety of indicators, therefore what I do is that I first use EasyFit (www.mathwave.com/easyfitxl-distribution-fitting-excel.html) then I see the changes in the parameters per year. Commented Feb 2, 2017 at 10:16

To determine what the distribution of the individual parameters look like, bootstrapping can be used within each time group to better explore what the basis is for extrapolation for prediction. If that is done then two years is enough data to test for a significance of difference, but not enough to establish a trend, because we lack a third value to test accuracy of forward and backward extrapolation (error of 1,2 for predicting 3, and error of 3,2 for predicting 1) or for interpolation (error of 1,3 for predicting 2). The absolute range of the parameters is needed information. The distributions of the parameters in each category is needed. Once that information is supplied, suitable transformations can be explored for transformation of parameters for linear model or other extrapolation model of parameter prediction.

Update from further information: The cumulative distribution function of the Dagum distribution (Type I) is given by $F(x;a,b,p)= {\left( 1+{\left(\frac{x}{b}\right)}^{-a} \right)}^{-p}$ for $x > 0$ and where $a, b, p > 0$.

The cumulative distribution function of the Dagum (Type II) distribution adds a point mass at the origin and then follows a Dagum (Type I) distribution over the rest of the support (i.e. over the positive halfline)

$F(x;a,b,p,\delta)= \delta + (1-\delta) {\left( 1+{\left(\frac{x}{b}\right)}^{-a} \right)}^{-p} .$

The results you list, since the last parameter is zero, seem to correspond to a Type I Dagum distribution. You need at least one more data point to test any trend assumption. Also, the Dagum parameter support for $x,a,b,p>0$, which may be problematic, as follows.

Next update: With the third set of parameters, provided that they correspond in the order listed, to those above, we now have

 year   k       a       b
2012   0.12941 10.504  146.90
2014   0.17555 9.9954  147.31
2016   0.20865 9.9207  146.58


Now, if we just do ordinary least squares linear regression on this we obtain

$k = -39.73 + 0.01981year$ with $p=0.0603$ (ANOVA F-test)

$a = 303.8 - 0.1458year$ with $p=0.2562$

$b = 308.1 - 0.08000year$ with $p=0.7119$

Only the regression line for $k$ has a borderline significant probability, and the others are not significant. Upon further testing we cannot be assured that the line slopes are in the directions indicated or the $y$-axis intercepts useful. Use of power functions and potentially more motivated fit models have the problem that the date is not a start time for literacy, whereas recorded history, our defacto marker of literacy, began ~5000 years ago, with the exact date being unknown.

Thus, if we use non-linear functions to fit the 3 points above with semi-infinite support having $0\leq t<\infty$, like power functions, we will need dates something like $t=5012,5014,5016$. Now, it is possible to float the date, by using $year-year_{\text{start of history}}$ in place of $year$ in the regressions above, by solving for a plausible start time of history (where BC values are negative), or we use only distributions with support on the whole interval $-\infty<t<\infty$ like the linear functions used above.

• We can have up to 5 consecutive years. Can you please explain where to look at. What I do is that I first use EasyFit [link](www.mathwave.com/easyfitxl-distribution-fitting-excel.html) then I see the changes in the parameters per year. Can you please elaborate on your suggestion. How can I get the distributions of the parameters in each category? And what should I learn to do the suitable transformations? Commented Feb 2, 2017 at 10:38
• EasyFit chooses distributions. which distribution is used for each year and what are the parameter values?
– Carl
Commented Feb 2, 2017 at 15:15
• For example, EasyFit chose for the cellular penetration rate in 2014, the Dagum distribution suggested had the following parameters: k= 0.17555, alpha=9.9954, beta= 147.31 and gamma=0 Then for 2016, the distribution had the following parameters k= 0.20865, alpha=9.9207, beta= 146.58 and gamma=0 Commented Feb 2, 2017 at 22:26
• Thanks Carl... Here is the parameters for the Dagnum for the distribution in 2012: k= 0.12941, alpha=10.504, beta= 146.9. Please advise what to do. Also let me know if your suggestion is valid for any other type of distribution? Or I should take into consideration certain points. Many many thanks for your support! Commented Feb 3, 2017 at 11:54
• Traveling today, more tomorrow.
– Carl
Commented Feb 3, 2017 at 13:55