1 Approximating lower bounds on S_{n}.

(This first paragraph below is to help keep notation straight, but probably should be removed ultimately.)

We suppose as before that X_{i} are iid Pareto random variables with distribution function F (x) = 1 − x^{−α} for x ≥ 1, that S_{n} = ∑_{i=1}^{n} X_{i}, and that M_{n} = max{X_{1}, X_{2}, ..., X_{n}}.

An approximation for the qth quantile of S_{n}, for small q (e.g. q = 2.5%), may be constructed based on the following idea. One seeks c_{n}, where

P (S_{n} < c_{n}) ≈ q.  (1)

For any value y_{n},

P (S_{n} < c_{n}) = P (S_{n} < c_{n}|M_{n} ≤ y_{n})P (M_{n} ≤ y_{n})
+ P (S_{n} < c_{n}|M_{n} > y_{n})P (M_{n} > y_{n}).  (2)

The term P (S_{n} < c_{n}|M_{n} ≤ y_{n}) in (??) can be approximated using the central limit theorem, since the variables being summed are now truncated and hence have finite moments. The resulting approximation will be close for sufficiently small values of y_{n} and sufficiently large n. At the same time, for any reasonably large value of y_{n}, and for c_{n} satisfying (??) for small q, the quantity P (S_{n} < c_{n}|M_{n} > y_{n}) in (??) will be infinitessimal; hence the entire final term in (??) may be considered negligeable. Thus, in approximating c_{n}, we suggest choosing an appropriate value of y_{n}, and considering the approximation

P (S_{n} < c_{n}) ≈ P (S_{n} < c_{n}|M_{n} ≤ y_{n})P (M_{n} ≤ y_{n})  (3)
≈ Φ\{(c_{n} − nμ_{y_{n}})/(σ_{y_{n}} \sqrt{n})\}P (M_{n} ≤ y_{n}),  (4)

where Φ is the standard normal distribution function, and

μ_{y} := E[X_{1}|X_{1} ≤ y] = \frac{αy^{1−α} − α}{(1 − α)(1 − y^{−α})},  α ≠ 1,  (5)

= \frac{ln(y)}{1 − y^{−1}} , α = 1,

σ_{y}^{2} := V [X_{1}|X_{1} ≤ y] = \frac{αy^{2−α} − α}{(2 − α)(1 − y^{−α})} − μ_{y}^{2}, α ≠ 2,  (6)

= \frac{2ln(y)}{1 − y^{−2}} − μ_{y}^{2}, α = 2.

Note that there is a tradeoff in choosing y_{n} in (??): if one selects too small a value of y_{n}, then the term P (S_{n} < c_{n}|M_{n} > y_{n}) in (??) is not negligeable, so the resulting approximation may not be satisfactory. On the other hand, if y_{n} is too large, then the approximation of P (S_{n} < c_{n}|M_{n} < y_{n}) using the central limit theorem may be unsatisfactory; this is particularly true for small n.

One option is to choose some value p^{∗} to represent the probability P (S_{n} < c_{n}|M_{n} ≤ y_{n}) in (??). From (??), one then has q/p^{∗} = P (M_{n} ≤ y_{n}) = \{1 − (\frac{1}{y_{n}})^{α}\}^{n}, and solving this for y_{n} one obtains

y_{n} = {1 − (q/p^{∗})^{1/n}}^{−1/α}.  (7)

One may then obtain an approximation of c_{n} by plugging this value of y_{n} from (??) into the first term in (??), yielding

ĉ_{n} = σ_{y_{n}} \sqrt{n}Φ^{−1}(p^{∗}) + nμ_{y_{n}},  (8)

where μ_{y_{n}} and σ_{y_{n}} are given by equations (??-??).

A naive choice for p^{∗} is \sqrt{q}; this seems to balance the aforementioned tradeoff, since then P (S_{n} < c_{n}|M_{n} ≤ y_{n}) = P (M_{n} ≤ y_{n}) = \sqrt{q}.

The values reported in column 3 of Table 1 reflect the approximation ĉ_{n} in (??), with p^{∗} = \sqrt{q} and y_{n}, μ_{y_{n}}, and σ_{y_{n}} given by (??-??). However, plots of ideal choices of p^{∗} versus q, n, and α suggest that instead, the choice

p^{∗} = 0.136 + 0.235q + q^{2} + 0.0066 min(n, 10) − 0.05 max(a, 1)  (9)

provides a better approximation. Performance of the approximation c̃_{n} resulting from the use of (??) with (??) for p^{∗} and y_{n}, μ_{y_{n}}, and σ_{y_{n}} given by (??-??) is shown in column 4 of Table 1.

Although the slightly simpler ĉ_{n} approximates the lower quantile quite well, in most cases there is substantial improvement from the use of the approximation c̃_{n} which employs (??). This is also displayed graphically in Figures 1 and 2 below. Figure 1 shows how the 0.02 quantile c_{n} and the two approximations ĉ_{n} and c̃_{n} vary with n, for α = 2/3. One sees that the two approximations match the true quantile quite closely. Figure 2 highlights the error rates for the two approximations ĉ_{n} and c̃_{n} as a function of n, again for α = 2/3. Table 1 and Figures 1 and 2 show the error in the approximations for q = 0.02, but results for other small values of q are rather similar.

Figure Captions:

Figure 1: Quantile c_{n} (solid curve), along with approximations ĉ_{n} (dotted) and c̃_{n} (dashed) as functions of n, for α = 2/3 and q = 0.02. For each n, the values of c_{n} shown is the empirical 0.02 quantile from 10 million simulations of S_{n}.

Figure 2: Error rates for the two approximations ĉ_{n} (dotted) and c̃_{n} (dashed) as functions of n, for α = 2/3 and q = 0.02. Percentage error rates are calculated as 100(ĉ_{n} − c_{n})/c_{n} and 100(c̃_{n} − c_{n})/c_{n}, where each value of c_{n} is the empirical 0.02 quantile taken from 10 million simulations of S_{n}.

The table below is for q = 0.02. Error rates are rounded to 4 decimal places.

\begin{matrix}α & n & (ĉ_{n} − c_{n})/c_{n} & (c̃_{n} − c_{n})/c_{n} \\ 0.5 & 2 & 0.0157 & -0.0023 \\ 0.5 & 5 & 0.0120 & 0.0029 \\ 0.5 & 10 & 0.0079 & 0.0041 \\ 0.5 & 20 & 0.0052 & -0.0018 \\ 0.5 & 50 & 0.0040 & -0.0080 \\ 0.5 & 100 & 0.0035 & -0.0021 \\ 0.67 & 2 & 0.0019 & -0.0005 \\ 0.67 & 5 & 0.0012 & -0.0030 \\ 0.67 & 10 & -0.0050 & 0.0045 \\ 0.67 & 20 & -0.0019 & 0.0027 \\ 0.67 & 50 & -0.0010 & 0.0013 \\ 0.67 & 100 & 0.0005 & -0.0004 \\ 1 & 2 & -0.0110 & 0.0007 \\ 1 & 5 & -0.0056 & 0.0000 \\ 1 & 10 & -0.0006 & 0.0023 \\ 1 & 20 & 0.0019 & 0.0007 \\ 1 & 50 & -0.0149 & -0.0015 \\ 1 & 100 & -0.0076 & -0.0014 \\ 1.5 & 2 & 0.0003 & 0.0035 \\ 1.5 & 5 & 0.0045 & 0.0032 \\ 1.5 & 10 & -0.0182 & -0.0043 \\ 1.5 & 20 & -0.0085 & -0.0022 \\ 1.5 & 50 & 0.0017 & 0.0049 \\ 1.5 & 100 & 0.0059 & 0.0046 \end{matrix}