PROOF OF THEOREM 2.. We prove the theorem together with the additional result (iii) on by induction on *n*. The proof proceeds as follows. First, we prove that (i) holds for *N*, (ii) holds for *N* and *N* + 1, and (iii) holds for *N*. Then, by inductively assuming that (i) and (iii) hold for *n* + 1 and (ii) holds for *n* + 1 and *n* + 2, we prove that (i), (ii) and (iii) hold for *n*.

We first prove that (i) and (iii) hold for *N* and (ii) holds for *N* and *N* + 1. By (5) and , we have . Thus, it follows that, when ,

This shows that (ii) holds for *N* + 1. When *n* = *N*, since , it follows from (6), (7), and (8) that

Note that is concave in *y* and is increasing in *y* on . Then, when . When , since is increasing in *y* on and it is decreasing in *y* on , it follows that . Thus, (i) holds when *n* = *N*. In addition, when , we have

Thus, (ii) also holds when *n* = *N*. Finally, for any , since is a concave function, we have . Thus, (iii) also holds for *N*. Now assume inductively that (i) and (iii) hold for *n* + 1 and (ii) holds for *n* + 1 and *n* + 2. In what follows, we prove that (i), (ii) and (iii) hold for *n*, which then completes the proof. First, we prove (i) holds for *n*. Suppose and . Then, by Lemma 2. By the inductive assumption on (ii) for *n* + 1, we have

- (13)

When , since , it follows from (13) that is increasing in *y* when , and it is decreasing in *y* when . Note that is concave in *y* when . Thus, when .

Now suppose ; and we shall prove that in this case. By the definition of , it suffices to prove that is increasing in *y* when , and . In what follows, we prove these two results sequentially.

We first prove is increasing in *y* on . When is given by (13) and the desired result is clearly true. Now suppose , and we prove that the desired result is also true in this case. Since is concave in *y*, we only need to prove that . When , by the inductive assumption on (ii) for *n* + 1, we have

- (14)

Taking the partial derivative with respect to *y* and letting , we obtain

- (15)

where is the indicator function; the first inequality is from the concavity of ; and the second one is from the inductive assumption on (iii), is increasing in *R*, and .

By (15), to prove , it suffices to show that

- (16)

Using the inductive assumptions on (ii) for *n* + 1 and *n* + 2, we obtain

and

Combining the above two inequalities, we obtain

Since and by the inductive assumption on (i), we have . Thus, (16) holds. Therefore, when is increasing in *y* on .

We next prove when . For , we define

Then, by Lemma 1, it is easy to verify that is concave in *y*. When , it follows from (6) that

Since is a convex function, when . Then, by Lemma 1 (i), it follows that when and . Therefore, to prove the desired result, it suffices to prove that is decreasing in *y* when , with *∊* being a sufficiently small positive number, or equivalently, . In what follows, we divide the analysis into two cases.

CASE 1. . In this case, when , by the inductive assumption on (ii),

Since , it is obviously that .

CASE 2. . In this case, by Lemma 2, . When , by the inductive assumption on (ii),

Taking the partial derivative with respect to *y* and letting , we obtain

where the first inequality is from the concavity of ; and the last one holds as and is an increasing function.

Summarizing the above arguments, we have proved that (i) holds for *n*.

Second, we prove (ii) holds for *n*. When , by part (i) for *n* and (13),

Thus, (ii) holds for *n*.

Finally, we prove (iii) for *n*. When , since (ii) holds for *n* and *n* + 1 and is a concave function, then .

Now suppose . In this case, by (ii) for *n* and (14), we obtain

while by the inductive assumptions on (ii) for *n* + 1 and *n* + 2, we have

where the inequality is from that is decreasing in *x*.

Since *D*_{n} and are identically distributed, it follows from the inductive assumption on (iii) that . Thus, (iii) holds for *n*. The proof is complete.