这是一份manchester曼切斯特大学ECON10182T作业代写的成功案例

where $S_{M S K}=\left[1-\gamma \rho \alpha^{\beta}\right]^{-1}$ is the corresponding survivor function, the location parameter $\gamma$ is real, and the sample space for $x$ is $(0, \infty)$ for $\gamma<0$ and $\left(0,(\alpha \gamma)^{-(\beta-1)^{-1}}\right)$ for $\gamma>0$. Besides correcting for unobserved heterogeneity, the additional parameter $\gamma$ allows the hazard function to be nonlinearly monotonic increasing $(\beta>1, \gamma>0)$, nonlinearly monotonic decreasing $(\beta<1, \gamma<0)$, bathtub shaped $(\beta<1, \gamma>0)$, unimodal $(\beta>1, \gamma<0)$ or constant $(\beta=1, \gamma=0$ ). Finally, when $\beta>0$ and $\gamma \leq 0$ the generalized Weibull corresponds to the Burr type XII distribution.

For both models parameters’ estimation has been conducted by means of Maximum Likelihood. The log-likelihood function for a series of expansions (contractions) with observed magnitude $x_{i}$ is:

ECON10182T COURSE NOTES ：

Let the random variable $S$ to follow a Pareto distribution with parameter $\alpha$. Thus, the probability distribution of $\log (S)$ is:
$$
\operatorname{Pr}(\log (S) \geq k)=\operatorname{Pr}(S \geq \exp (k)) \propto(\exp (k))^{-\alpha}=\exp (-\alpha \mathrm{k})
$$
that is, an exponential distribution with parameter $\alpha$. In other terms, $\log (S)$ follows an exponential distribution with probability density function equal to:
$$
E\left(\log (S), \alpha^{-1}\right)=\frac{1}{\alpha} \exp \left(-\frac{\log (S)}{\alpha}\right)
$$
In the case of independent exponential variables, it is simple to prove that a Laplace distribution regarding growth rates emerges by making use of the convolution theorem and its relation with the characteristic function. In fact, the characteristic function of two independent exponential distributions $z_{j}, j=1,2$, with parameter $\alpha^{-1}$ is:
$$
C_{z j}(\gamma)=(1-i \alpha \gamma)^{-1}
$$