Let X have a Pareto distribution with parameters Λ and θ. Let Λ have a gamma distribution with parameters α and 1 (i.e., scale parameter = 1). Find the unconditional pdf of X.

I tried finding the unconditional pdf of X by:
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I'm stuck at the integral evaluation. Can I and how do I modify the integrand on the RHS to a density function such that direct integration is avoided in this case?


At first glance it looks like a gamma kernel; it's all powers of $\lambda$ or things to the power of $\pm\lambda$. So lets try that:

$$\frac{\theta^\lambda \lambda^\alpha e^{-\lambda}}{(x+\theta)^{\lambda+1}}=\frac{1}{x+\theta}\cdot \lambda^\alpha e^{-\lambda}(1+\frac{x}{\theta})^{-\lambda}$$

Now $\:-\lambda-\lambda\log(1+\frac{x}{\theta})=-k\lambda$, so we can write

$$\lambda^\alpha e^{-\lambda}(1+\frac{x}{\theta})^{-\lambda}=\lambda^{(\alpha+1)-1}e^{-k\lambda}$$

which is a shape-rate gamma kernel.

You will need to check that your parameter ranges and such will work with this.

After you integrate out you'll be left with the reciprocal of the normalizing constants you needed, which will include k - take such terms back to the original parameters.


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