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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.

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  • $\begingroup$ In your step 2, you calculate $\mathbb{E}\hat{\theta}$ using the density function of $x$, but $\hat{\theta} = x_{(1)}$, not $x$. $\endgroup$
    – jbowman
    Commented Jan 23, 2023 at 19:34
  • $\begingroup$ Actually, returning to this question, you did the same thing in step 3. $\endgroup$
    – jbowman
    Commented Jan 23, 2023 at 19:51
  • $\begingroup$ @jbowman Hi. Should I have used the density function of minimum $f_S(x)$? By virtue of that, it affects step 3, right? $\endgroup$
    – iStats7238
    Commented Jan 23, 2023 at 19:58
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    $\begingroup$ 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. $\endgroup$
    – jbowman
    Commented Jan 23, 2023 at 20:02
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    $\begingroup$ Yes, it is true for the general case. $\endgroup$
    – jbowman
    Commented Jan 23, 2023 at 22:25

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