Suppose that $X_1,\cdots ,X_n$ is a random sample from the following Pareto density with parameters $\left(\theta ,\eta \right)$

$f_X\left(x_i;\theta ,\eta \right)=\frac{\theta {\eta }^{\theta }}{x_i^{\theta +1}}{I}_{\left[\eta ,\infty \right)}\left(x_i\right),\theta >0,\eta >0,$(5)

where $\theta$ is the shape parameter, often denoted as Pareto index, and $\eta$ is the scale parameter. In the following, when considering the parameters $\theta$ and $\eta$ random, they will be denoted as Θ and H.

In Sections 3.1 and 3.2, we derive closed-form expressions for $r_{{\pi }_D}$ and g for point estimators and interval estimators of $\theta$ and $\eta$. Specifically, for point estimation, we consider maximum likelihood estimators and quadratic loss function, whereas for interval estimation, we consider the length of $1-\gamma$ confidence intervals as loss function. We show that all the corresponding random risk functions are scale transformations either of the random parameters or of their square. We also show that, as a consequence, by adopting conjugate densities as design priors (generalized Gamma and Pareto respectively for Θ and H⁻¹), each resulting density g belongs to the same family of the design prior.

We consider independent conjugate design prior distributions for the two parameters. In particular, we adopt a Gamma density for the Pareto index Θ and a Pareto density for the reciprocal of the scale parameter H. Prior parameters are usually elicited using moments (such as mean and variance) or relevant quantiles of historical data.

Prior for Θ. For a given pair $\left(\nu ,\lambda \right)$ (both positive), the prior for the Pareto index $\theta$ is the Gamma density

${\pi }_D\left(\theta \right)=\frac{{\lambda }^{\nu }}{\text{Γ}\left(\nu \right)}{\theta }^{\nu -1}{\text{e}}^{-\lambda \theta },\theta >0.$(6)

If prior expectation m and variance v are elicited for the shape parameter, for instance, on the basis of historical data, the corresponding values for the prior parameters $\nu$ and $\lambda$ are

$\nu =\frac{m^2}{v}\text{and}\lambda =\frac{m}{v}.$(7)

Prior for H. Adopting a Pareto density for H⁻¹ implies that the prior density of the scale parameter H is

${\pi }_D\left(\eta \right)=\alpha {\beta }^{\alpha }{\eta }^{\alpha -1},\eta \in \left[0,\frac{1}{\beta }\right].$(8)

This density is often referred to as Inverse Pareto with parameters $\left(\alpha ,\beta \right)$, but the denomination is not unique in the literature (see, for instance, [6] ).

The two parameters $\alpha$ and $\beta$ are usually chosen by eliciting: (i) expectation m and variance v; (ii) some quantiles. In case (i), the resulting pair of hyperparameters is

$\alpha =\sqrt{1+\frac{m^2}{v}}-1\text{and}\beta =\frac{\sqrt{1+\frac{m^2}{v}}-1}{m\sqrt{1+\frac{m^2}{v}}}.$(9)

In case (ii) two quantiles are elicited (for example $q_{{\gamma }_1}$ and $q_{{\gamma }_2}$ are the quantiles at level ${\gamma }_1$ and ${\gamma }_2$ respectively, with ${\gamma }_1\ne {\gamma }_2$), then the prior parameters are

$\alpha =\frac{\text{ln}\left(\frac{{\gamma }_1}{{\gamma }_2}\right)}{\text{ln}\left(\frac{q_{{\gamma }_1}}{q_{{\gamma }_2}}\right)}\text{and}\beta =\frac{{\gamma }_1^{\frac{1}{\alpha }}}{q_{{\gamma }_1}}.$(10)

We now show the elicitation procedure to obtain distributions that will be used for implementing our proposed methodology in the examples of Sections 3.1 and 3.2. In this regard, we exploit data from a numerical example in Section 5 of [7] . The authors consider a dataset previously analyzed by [8] on annual wage (in multiples of 100 US dollars) relative to a random sample of 30 production-line workers in a large industrial firm. Specifically, income data are analyzed by adopting a Pareto model for the observations and independent conjugate densities for the two parameters. In our examples, we use the same prior assumptions as in [7] , that are:

(i) expectation $m=1.9$ and variance $v=0.6$ for the shape parameter Θ;

(ii) median and 5th percentile equal to 100 and 85 respectively, for the scale parameter H.

Then, according to (7) and (10), we elicit the prior hyperparameters as follows:

(i) prior hyperparameters for Θ

$\nu =\frac{{1.9}^2}{0.6}=6.02\text{and}\lambda =\frac{1.9}{0.6}=3.17;$(11)

(ii) prior hyperparameters for H

$\alpha =\frac{\text{ln}\left(\frac{0.50}{0.05}\right)}{\text{ln}\left(\frac{100}{85}\right)}=14.17\text{and}\beta =\frac{{0.50}^{\frac{1}{\alpha }}}{100}=\mathrm{0.009.}$(12)

These values are employed as hyperparameters in the design priors for Θ and H in Sections 3.1 and 3.2, where we continue this example by deriving the density of the random risk for point and interval estimators.

It is easy to check that (see [9] ) the maximum likelihood (ML) estimator of $\theta$ is

$\hat{\theta }=\frac{n}{{\displaystyle \sum \ln \frac{x_i}{\eta }}}$

that is distributed as an Inverse Gamma with shape n and scale $n\theta$ [10] . Noting that $\mathbb{E}\left[\hat{\theta }\right]=\frac{n\theta }{n-1}$ and that $\mathbb{V}\left[\hat{\theta }\right]=\frac{n^2{\theta }^2}{{\left(n-1\right)}^2\left(n-2\right)}$, using the quadratic loss function we obtain

$R\left(\theta ,\hat{\theta }\right)=\frac{n^2+n-2}{{\left(n-1\right)}^2\left(n-2\right)}{\theta }^2=\frac{n+2}{\left(n-1\right)\left(n-2\right)}{\theta }^2=k{\theta }^2.$

Let us now consider the random risk of $\hat{\theta }$, that is $Y=R\left(\Theta ,\hat{\theta }\right)=k{\Theta }^2$. For any design prior density ${\pi }_D\left(\theta \right)$ on Θ, we have that

$r_{{\pi }_D}\left(\hat{\theta }\right)=k{\mathbb{E}}_{{\pi }_D}\left[{\Theta }^2\right]$(13)

$g\left(y\right)=\frac{1}{2}\sqrt{\frac{1}{ky}}{\pi }_D\left(\sqrt{\frac{y}{k}}\right),y\ge 0.$(14)

Assuming for Θ a conjugate Gamma prior with shape parameter $\nu$ and rate parameter $\lambda$, from Equations (13) and (14) it follows that

$r_{{\pi }_D}\left(\hat{\theta }\right)=\frac{k\nu \left(\nu +1\right)}{{\lambda }^2}$(15)

$g\left(y\right)=\frac{{\lambda }^{\nu }}{2k^{\frac{\nu }{2}}\Gamma \left(\nu \right)}{y}^{\frac{\nu }{2}-1}{\text{e}}^{-\lambda \sqrt{\frac{y}{k}}},y\ge 0$(16)

that is a Generalized Gamma density [10] with parameters ( $c_1=\nu$, $c_2=\frac{1}{2}$, $c_3=\frac{{\lambda }^2}{k}$) in the parameterization of [11] . This result is a consequence of the closure of the Generalized Gamma distribution under both scale and power transformations.

Since $\hat{\theta }$ follows an Inverse Gamma distribution, it is easy to check that (see [9] ) $n\theta /\hat{\theta }$ is a pivotal quantity for $\theta$ with distribution $\text{Gamma}\left(n,1\right)$. Therefore, the $1-\gamma$ equal-tails confidence interval is

$C_{\theta }=\left[\frac{q_{\frac{\gamma }{2}}}{n}\hat{\theta },\frac{q_{1-\frac{\gamma }{2}}}{n}\hat{\theta }\right],$

where $q_{ϵ}$ denotes the $ϵ$-level quantile of the $\text{Gamma}\left(n,1\right)$. Using the length of the interval as loss function, we have

$\mathbb{L}\left(\theta ,C_{\theta }\right)=\frac{q_{1-\frac{\gamma }{2}}-q_{\frac{\gamma }{2}}}{n}\hat{\theta }$

and the resulting risk function is

$R\left(\theta ,C_{\theta }\right)=\frac{q_{1-\frac{\gamma }{2}}-q_{\frac{\gamma }{2}}}{n}{\mathbb{E}}_{\theta }\left[\hat{\theta }\right]=h\theta ,$

where $h=\left(q_{1-\frac{\gamma }{2}}-q_{\frac{\gamma }{2}}\right)/n$. Note that, for any $\theta$, $R\left(\theta ,C_{\theta }\right)$ tends to 0 as n diverges.

In this case, the random risk is $Y=R\left(\Theta ,C_{\theta }\right)=h\Theta$ and it is straightforward to check that

$r_{{\pi }_D}\left(C_{\theta }\right)=h{\mathbb{E}}_{{\pi }_D}\left(\text{Θ}\right),$

$g\left(y\right)=\frac{1}{h}{\pi }_D\left(\frac{y}{h}\right),y\ge 0.$

Using again for Θ a Gamma prior density of parameters $\left(\nu ,\lambda \right)$, we have that

$r_{{\pi }_D}\left(C_{\theta }\right)=\frac{\nu h}{\lambda }$

$g\left(y\right)=\frac{{\left(\frac{\lambda }{h}\right)}^{\nu }}{\text{Γ}\left(\nu \right)}{y}^{\nu -1}{\text{e}}^{-\frac{\lambda }{h}y},y\ge 0.$(17)

As a consequence of the above-mentioned closure under scale transformation, g is still a Gamma density with parameters $\left(\nu ,\frac{\lambda }{h}\right)$.

Let us consider the income data example in Section 2.2 again. Assume now that the scale parameter $\eta$ is known while the Pareto index $\theta$ needs to be estimated. Values of prior hyperparameters are given by (11).

Figure 1 shows: risk density (16) relative to the ML estimator (left panel); risk density (17) relative to the equal-tails 95%-confidence interval (right panel). In both cases, as an example, we consider sample sizes of $n=30$ and $n=50$. Note that the density of the risk function shrinks towards 0 as the sample size increases, as a consequence of the consistency of the estimators. Moreover, for larger and larger values of n, the skewness of the distribution reduces, which results in closer and closer values of the main summaries of g. Expectation, median and mode of g are reported in Table 1 for $n=30$ and $n=50$ for point estimation.

Since g is remarkably right-skewed, the mean is larger than the median and the mode; discrepancies between the three summaries are smaller when the sample size increases.

The maximum likelihood estimator for the parameter $\eta$ is the sample minimum $\hat{\eta }=X_{\left(1\right)}$ that is a Pareto random variable with parameters $\left(n\theta ,\eta \right)$. Therefore, under the quadratic loss function,

$R\left(\eta ,\hat{\eta }\right)=\frac{2}{\left(n\theta -1\right)\left(n\theta -2\right)}{\eta }^2=k{\eta }^2.$

Then, the random risk for $\hat{\eta }$ is $Y=R\left(\text{H},\hat{\eta }\right)=k{\text{H}}^2$ and, similarly to the previous case,

$r_{{\pi }_D}\left(\hat{\eta }\right)=k{\mathbb{E}}_{{\pi }_D}\left[{\text{H}}^2\right]$

$g\left(y\right)=\frac{1}{2}\sqrt{\frac{1}{ky}}{\pi }_D\left(\sqrt{\frac{y}{k}}\right),y\ge 0,$

where ${\pi }_D\left(\eta \right)$ is the design prior density for H and $k=2/\left[\left(n\theta -1\right)\left(n\theta -2\right)\right]$.

When we assume a conjugate prior (8) for H, i.e. an inverse Pareto density with parameters $\left(\alpha ,\beta \right)$, we obtain

$r_{{\pi }_D}\left(\hat{\eta }\right)=\frac{\alpha k}{{\beta }^2\left(\alpha +2\right)}$(18)

$g\left(y\right)=\frac{\alpha }{2}{\left(\frac{{\beta }^2}{k}\right)}^{\frac{\alpha }{2}}{y}^{\frac{\alpha }{2}-1},y\in \left[0,\frac{k}{{\beta }^2}\right],$(19)

that is an inverse Pareto density with parameters $\left(\alpha /2,{\beta }^2/k\right)$. Note that closure with respect to scale and power transformations also holds for the distribution of the reciprocal of a Pareto random variable.

A pivotal quantity for determining an interval estimator for $\eta$ is $\hat{\eta }/\eta$ that is a Pareto random variable with parameters $\left(n\theta ,1\right)$. Recalling that the $1-\gamma$ quantile of this distribution is ${\gamma }^{-\frac{1}{n\theta }}$, the $1-\gamma$ confidence interval for $\eta$ is

$C_{\eta }=\left[{\gamma }^{\frac{1}{n\theta }}\hat{\eta },\hat{\eta }\right].$

The length of the interval is $\left(1-{\gamma }^{\frac{1}{n\theta }}\right)\hat{\eta }$ and the risk function is

$R\left(\eta ,C_{\eta }\right)=\left(1-{\gamma }^{\frac{1}{n\theta }}\right){\mathbb{E}}_{\eta }\left[\hat{\eta }\right]=\frac{n\theta \left(1-{\gamma }^{\frac{1}{n\theta }}\right)}{n\theta -1}\eta =h\eta$

with h decreasing to 0 with n. The random risk of $C_{\eta }$ is $Y=R\left(\text{H},C_{\eta }\right)=h\text{H}$. Assuming again the conjugate prior density (8) for $\eta$ we obtain

$r_{{\pi }_D}\left(C_{\eta }\right)=\frac{\alpha h}{\beta \left(\alpha +1\right)}$(20)

and the resulting g is again a density of the form (8) with parameters $\left(\alpha ,\beta /h\right)$

$g\left(y\right)=\alpha {\left(\frac{\beta }{h}\right)}^{\alpha }{y}^{\alpha -1},y\in \left[0,\frac{h}{\beta }\right].$(21)

We notice that the forms $k{\text{H}}^2$ and $h\text{H}$ of the risk functions are not characteristic of the estimation method considered in the previous section. As an example, the moment estimator for $\eta$ is ${\hat{\eta }}_m={\theta }^{-1}\left(\theta -1\right){\bar{X}}_n$, with random risk function $Y=R\left(\text{H},{\hat{\eta }}_m\right)=k_m{\text{H}}^2$, where $k_m={\left[n\theta \left(\theta -2\right)\right]}^{-1}$.

As regards interval estimation, recalling that the asymptotic distribution of ${\hat{\eta }}_m$ is normal with mean $\eta$ and variance $k_m{\eta }^2$, the asymptotic $1-\gamma$ confidence interval is ${\hat{\eta }}_m\pm z_{1-\frac{\gamma }{2}}{\hat{\eta }}_m\sqrt{k_m}$, with length $2z_{1-\gamma /2}{\hat{\eta }}_m\sqrt{k_m}$, where $z_{ϵ}$ is the $ϵ$ quantile of a standard normal. Then the random risk is given by $h_m\text{H}$, where $h_m=2z_{1-\frac{\gamma }{2}}{\left[n\theta \left(\theta -2\right)\right]}^{-\frac{1}{2}}$ that decreases to 0 with n.

Let us consider the income data example in Section 2.2 again. Assume now that the scale parameter $\eta$ needs to be estimated while the Pareto index is known to be equal to $\theta =1.584$ (that corresponds to the mode of the Gamma prior density elicited in 0). Values of prior hyperparameters for H are given by (12).

Figure 2 shows: risk density (19) relative to the ML estimator (left panel); risk density (21) relative to the equal-tails 95%-confidence interval (right panel). In both cases, as an example, sample sizes of $n=30$ and $n=50$ have been considered. Again, a larger value of the sample size induces higher densities for lower values of the risk function of $\eta$. Note however that in this case, g is always an increasing function (which is remarkably left-skewed) but, again, the values of the main summaries of g become closer and closer as n increases. Numerical values of expectation, median and mode of g are given in Table 2 for $n=30$ and $n=50$ for point estimation.

In many relevant applications that involve the Pareto model, the quantity of inferential interest is often represented by a specific quantile of the distribution, for instance, Value at Risk [12] . In this section, we extend the approach of the previous sections to this problem. Specifically, for a Pareto distribution with parameters $\left(\theta ,\eta \right)$, the γ-quantile is $q_{\gamma }=\eta {\left(1-\gamma \right)}^{-\frac{1}{\theta }}$, which is a function of both parameters. The maximum likelihood estimator for this quantity is

${\hat{q}}_{\gamma }=\hat{\eta }{\left(1-\gamma \right)}^{-\frac{1}{\hat{\theta }}}.$(22)

For the sake of analytical tractability, let us consider the following loss function

$\mathbb{L}\left(q_{\gamma },{\hat{q}}_{\gamma }\right)={\left(\ln {\hat{q}}_{\gamma }-\ln q_{\gamma }\right)}^2,$

i.e. the quadratic loss on the logarithmic scale. The corresponding risk function is

$R\left(q_{\gamma },{\hat{q}}_{\gamma }\right)={\mathbb{E}}_{\theta ,\eta }\left[\mathbb{L}\left(q_{\gamma },{\hat{q}}_{\gamma }\right)\right]=\frac{2+\ln \left(1-\gamma \right)+n\ln {\left(1-\gamma \right)}^2}{n^2{\theta }^2}=\frac{k}{{\theta }^2}$

with $k=n^{-2}\left[2+\ln \left(1-\gamma \right)+n\ln {\left(1-\gamma \right)}^2\right]$.

It follows that $Y=k{\Theta }^{-2}$ and its density is

$g\left(y\right)=\frac{\sqrt{k}}{2}{y}^{-\frac{3}{2}}{\pi }_D\left(\sqrt{\frac{k}{y}}\right),y\ge 0.$

When a Gamma prior with parameters $\left(\nu ,\lambda \right)$ is adopted, the expression of g becomes

$g\left(y\right)=\frac{{\left({\lambda }^{2}k\right)}^{\frac{\nu }{2}}}{2\Gamma \left(\nu \right)}{y}^{-\frac{\nu }{2}-1}{\text{e}}^{-\frac{\lambda \sqrt{k}}{\sqrt{y}}},y\ge 0,$(23)

that is the density function of a Generalized Inverse Gamma of parameters ( $c_1=\nu$, $c_2=1/2$, $c_3={\lambda }^{2}k$) in the parametrization of [11] .

Risk density can be a useful tool when comparing different estimators. Using the same prior information as in the income data example of Section 2.2, i.e. (11) and (12), we consider the maximum likelihood estimator (22) of the 95th percentile of a Pareto population and the nonparametric estimator obtained as the 95th sample percentile. Since we do not have an analytic expression for the distribution of the risk relative to the nonparametric estimator, we have simulated the distribution of the risk, whose histogram is reported in Figure 3 and Figure 4 . The red curves in the same figure are the plot of the density (23) of the risk relative to the maximum likelihood estimator. The advantage, in terms of risk, of the maximum likelihood estimator seems to be quite evident from Figure 3 and Figure 4 . Table 3 reports the mean and median of the risk associated with the two estimators for moderate ( $n=30$) and large ( $n=100$) sample sizes.