# Warning

The question is the sequel to this question and it was divided into three parts.

The first is this, while the second part is found here.

# Exercise

Let $$X \thicksim Pa(\lambda, \theta$$) with density function $$f(x; \theta, \lambda) = \frac{\lambda \theta^{\lambda}}{x^{\lambda+1}}$$ where $$x \geq \theta$$, $$\lambda > 0$$ and $$\theta > 0$$.

The CDF is: $$F(x) = 1 - \left( \frac{\theta}{x} \right)^{\lambda}$$

Let $$\lambda$$ known. Show if the statistic $$S = \min\{X_1, ..., X_n\}$$ is sufficient for $$\theta$$ and valuate if it's minimal.

# Try

First, I determine the CDF and the density of $$S=X_{(1)}$$:

$$F_S(x) = 1 - \left[1 - F(x)\right]^n = 1 - \left[ 1 - 1 + \left( \frac{\theta}{x} \right)^{\lambda} \right]^n = 1 - \left( \frac{\theta}{x} \right)^{n\lambda}$$

$$f_S(x) = n\left[ 1 - F(x) \right]^{n-1} f(x) = \frac{n\lambda \theta^{\lambda}}{x^{\lambda+1}} \left( \frac{\theta}{x} \right)^{\lambda(n-1)} = \frac{n\lambda \theta^{n\lambda}}{x^{n\lambda+1}}$$

The statistic $$S \thicksim Pa(n\lambda, \theta)$$.

I apply the factorization theorem for sufficiency: $$f(x;\theta) = \prod^{n}_{i=0} \frac{n\lambda \theta^{n\lambda}}{x^{n\lambda+1}_i} = (n\lambda)^n \theta^{n\lambda} \theta^{n^2 \lambda} \prod^{n}_{i=0} x^{-(n\lambda+1)} = g(x; \theta) \cdot h(x)$$

$$g(x; \theta) = (n\lambda)^n \theta^{n\lambda} \theta^{n^2 \lambda} \qquad ; \qquad h(x) = \prod^{n}_{i=0} x^{-(n\lambda+1)}$$

It's sufficient. I verify the status of minimal using Lehmann-Scheffé theorem:

• $$f(x;\theta) = (n\lambda)^n \theta^{n\lambda} \theta^{n^2 \lambda} \prod^{n}_{i=0} x^{-(n\lambda+1)}$$
• $$f(y; \theta) = (n\lambda)^n \theta^{n\lambda} \theta^{n^2 \lambda} \prod^{n}_{i=0} y^{-(n\lambda+1)}$$

Then: $$\frac{L(\theta; x)}{L(\theta; y)} \propto \frac{(n\lambda)^n \theta^{n\lambda} \theta^{n^2 \lambda} \prod^{n}_{i=0} x^{-(n\lambda+1)}}{(n\lambda)^n \theta^{n\lambda} \theta^{n^2 \lambda} \prod^{n}_{i=0} y^{-(n\lambda+1)}} = \frac{\prod^{n}_{i=0} x^{-(n\lambda+1)}}{\prod^{n}_{i=0} y^{-(n\lambda+1)}} = c(X;Y)$$

It's also minimal.

If I have missed anything else, please report it to me immediately.

• In your step 2, you calculate $\mathbb{E}\hat{\theta}$ using the density function of $x$, but $\hat{\theta} = x_{(1)}$, not $x$. Jan 23 at 19:34
• Actually, returning to this question, you did the same thing in step 3. Jan 23 at 19:51
• @jbowman Hi. Should I have used the density function of minimum $f_S(x)$? By virtue of that, it affects step 3, right? Jan 23 at 19:58
• And step 2; you should be able to see by inspection that the distribution of $x_{(1)}$ is Pareto with parameters $(\theta, n\lambda)$, and as a consequence see what the bias and $c$ are. Note that your bias term implies that $\hat{\theta}$ is inconsistent, but the correct bias term does not. Jan 23 at 20:02
• Yes, it is true for the general case. Jan 23 at 22:25