Introduction to Splicing Model

There are two ways of splicing a model:

One approach to constructing a spliced model is by using known break points. In this method, the break points $b_0, b_1, ...b _n$ are predetermined and specified in advance. The spliced model is then created by selecting appropriate density functions for each interval between the break points.

Another way to construct a spliced model is by utilizing standard distributions over a specific range. This method involves selecting standard distributions that cover different intervals within the desired range. The density functions associated with these standard distributions are combined to form the spliced model

Let's look at one example of using the second method. Let us consider a model with uniform distribution from $(0, a)$ and Pareto distribution of type Lomax with parameters $\alpha$ and $\theta$. For the uniform distribution, we will have the pdf as $$f(x) = \frac{1}{a} \, \text{ for } x \in (0, a)$$

and for Pareto distribution part, we will have

$$S(x) = \left(1 + \frac{x}{\theta} \right)^{-\alpha} = \theta^\alpha(\theta +x)^{-\alpha}$$

and

$$f(x) = \frac{\alpha}{\theta}\left(1+\frac{x}{\theta}\right)^{-\alpha-1} = \alpha \theta^{\alpha}\left( \theta+x \right)^{-\alpha-1}$$

So the mixture distribution regarding splicing will be

$$f_X(x)= \begin{cases} b_1\frac{1}{a} & 0<x<a \\ (1-b_1) \frac{\alpha\left( \theta+x \right)^{-\alpha-1}}{(\theta +\alpha)^{-\alpha}} & x \ge a\end{cases}$$

As we can see from the following visualization that the spliced distribution are not continuous.