On measures of central location

ON MEASURES OF CENTRAL LOCATION

DEPARTMENT OF MATHEMATICS, BANDUNG INSTITUTE OF TECHNOLOGY

BANDUNG 40132, INDONESIA

Abstract.

Given some data consisting of a finite sequence of real numbers, we

study some measures of central location of the data which we define to be the best

constants that approximate the data in certain norms. We also discuss how one

measure of central location is related to another.

1. Introduction and Main Results

, . . . , x Given a finite sequence of real numbers, (x

1 N ), say, one usually computes the average or the (arithmetic) mean, among others, as the measures of central

1 location. Here the mean is given by µ = x . One property of the mean is i

P N

N i=1 P N that the sum of the deviations is equal to 0, that is, (x − µ) = 0. Another i i=1

P N

2 nice property of the mean is that it minimizes the function F (ξ) := (x − ξ) . i i=1

2 N If we denote by l the space of real sequences X = (x ) of length N , equipped i

N i=1 ¢ 1/2

¡P N

2 with the norm kXk := x , then the last property says that the average of

2 i i=1

2 a sequence X is the best constant that approximates X in l . N p

N Now, for each 1 ≤ p ≤ ∞, let l denote the space of real sequences X = (x ) of i

N i=1

¢ 1/p ¡P N p length N , equipped with the norm kXk := |x | . (Note that, for p = ∞, p i i=1

∞ , i kXk takes the maximum value of x i = 1, . . . , N .) Then, given a sequence X = p

N (x i ) , one may view it as a member of l and ask: what is the best constant that i=1 N p approximates X in l , that is, what is the value of ξ = ξ p that will minimize the

N N function F (ξ) := kX − ξk , where X − ξ = (x − ξ) ? p p i i=1

The following theorem tells us that such a value of ξ at which F is minimum p p always exists and, in most cases, is unique.

2000 Mathematical Subject Classification : 26A06, 41A44, 41A50.

Keywords and phrases : measures of central location, approximation by constants.

£ Theorem 1. For each 1 ≤ p ≤ ∞, there exists ξ ∈ min x , max x ] at which p i i i=1,...,N i=1,...,N

F is minimum; and such a value of ξ is unique except for the case where p = 1 and p p N is even.

¡ Proof. x x

First observe that F (ξ) > F min ¢ for ξ < min and that F (ξ) > p p i i p i=1,...,N i=1,...,N

¡ F x x max ¢ for ξ > max . Hence our search for ξ at which F is minimum can p i i p p i=1,...,N i=1,...,N

£ x , x be restricted on the interval I := min i max i ¤. But F p is continous in ξ, and i=1,...,N i=1,...,N so by the Maximum-Minimum Theorem such a value of ξ p must exist.

To prove the uniqueness, we shall use the fact that minimizing F p amounts to p minimizing G := F , especially for 1 < p < ∞. For these values of p’s, G is p p p

P ′ N p−1 differentiable and G (ξ) = −p |x − ξ| sgn(x − ξ) for every ξ ∈ I. Since i i p i=1

¡ ¡ ′ ′

G min x ¢ < 0 and G max x ¢ > 0, G must achieve its minimum value at i i p p p i=1,...,N i=1,...,N

′ an interior point of I at which G is 0. But G is also twice differentiable and p p

′′ P N p−1 G (ξ) = p(p − 1) |x − ξ| > 0 for every ξ ∈ I, and so we conclude that such a i p i=1

′′ point must be unique (for otherwise we would have a point in I at which G is equal p to 0). This proves the uniqueness of ξ at which G , as well as F , is minimum for p p p

1 < p < ∞. The proofs of the two remaining cases are implicitly contained in the proofs of Theorems 2 and 3 below.

¤ For p = 2, we know that F achieves its minimum at ξ = µ, the mean. For

2 1 ≤ p ≤ ∞ in general, we may interpret the value of ξ at which F is minimum p p p as the measure of central location of X in l . In addition to the classical result for

N p = 2, we also have interesting results for p = 1 and for p = ∞. See [?] for analogous results in the continuous case. See also [?] for related work. is minimized at Theorem 2. If X is a monotone sequence, then F the median.

1 N Note

N +1

. If N is odd, the median of X = (x i ) is the middle term x . If N is even, i=1

1 N N the median of X is (x + x ).

Proof. Let us first consider the case where N is odd. Without loss of generality, we P N may assume that X is increasing. Our task is to show that F (ξ) = |x − ξ| is

1 i i=1

N +1

minimized at ξ = x , that is,

2 N N

X X

N +1

|x − x | ≤ |x − ξ| i i

i=1 i=1 , x N +1 for every ξ ∈ [x

1 N ]. To do so, observe that if ξ = x + δ for some δ > 0, then

2 N

X N +1 N +1 N +1 |x i − ξ| = (x + δ − x 1 ) + · · · + (x + δ − x )

i=1

N +3 N +1 N +1

|x − x − δ| + · · · + |x − x − δ|

2 N +1

2 N

1 − 1

X X N N

≥ |x i − x | + |x i − x | −

N +1 δ N +1 δ

2 N +3 i=1 i=

2 N

X N +1 = |x − x | + δ i

i=1 N

X N +1 > |x − x |. i

i=1

N +1

Similarly, if ξ = x − δ for some δ > 0, then

2 N N

X X

i i

N +1 |x − ξ| > |x − x |.

i=1 i=1 This proves the case where N is odd.

The proof of the case where N is even is similar, except that now F is minimized

1 N N at ξ = (x + x ), that is,

1 2 +1

2 N N

X X ¯ 1 ¯ ¯ N N ¯ x − (x + x ) ≤ |x − ξ| i i

2 i=1 i=1

N N

for every ξ ∈ [x , x ], and that the equality holds for every ξ ∈ [x , x ] (meaning

1 N

2 N , x N that F actually achieves its minimum at every point in [x ]).

1 ¤

∞ achieves its minimum value at Theorem 3. For any sequence X, the function F

1 ξ ∞ x x = min + max ¢. i i

2 i=1,...,N i=1,...,N Proof.

Obvious.

1 It may be suitable for us to call ξ ∞ = min x + max x ¢ the centre of the i i 2 i=1,...,N i=1,...,N sequence. Then, one can also think of the radius of the sequence as the maximum deviation about the centre, which is of course equal to half of the range:

1 ∞ x x ¢. radius := max |x i − ξ | = max i − min i i=1,...,N i=1,...,N i=1,...,N

2 While there is no obvious relation between the measures of central location ξ ’s at p which F ’s are minimized, the following theorem tells us that there is a nice relation p between the values of kX − ξ k ’s. p p

N Theorem 4. For any sequence X = (x ) , we have i i=1

N N ³ ´ ³ ´

1/p 1/q

X X

1 p q

|x − ξ | ≤ |x − ξ | , i p i q

N N i=1 i=1 whenever 1 ≤ p < q ≤ ∞.

Proof. By the fact that ξ minimizes kX − ξ k and H¨older’s inequality, we have p p p

N N

X X p p

|x − ξ | ≤ |x − ξ | i p i q i=1 i=1

N N ³ ´ p/q ³ ´ 1−p/q

X X q

≤ |x − ξ |

1 i q i=1 i=1

N ³ ´ p/q

X 1−p/q q

, = N |x − ξ | i q i=1 whence

N N ³ ´ p/q

X X

1 p q .

|x i − ξ p | ≤ |x i − ξ q | N N i=1 i=1 Taking the p-th root of both sides, we get the desired inequality.

¤ Remark

. As a consequence of the above theorem, we have in particular the inequalities N N

³ ´ 1/2

X X

2 |x − ξ | ≤ (x − ξ ) ≤ max |x − ξ ∞ |, i 1 i 2 i i=1,...,N

N N i=1 i=1 which relates the mean absolute deviation (about the median), the standard deviation (about the mean), and the maximum deviation (about the centre) of any

N sequence (x ) . i i=1

2. More about the Mean and the Centre

N Given a monotone sequence (x ) of real numbers, it is rather laborious to com- i i=1 pute the mean, especially when N is large. In contrast, it is very easy for us to compute the centre, which is just half of the sum of the first and the last terms.

Now, if one is curious to know how far the mean is from the centre, one may express the difference between the two values as N N −1

³ ´

1 N

1 N − x = i − ∆ i i

X X x + x

N N

2 i=1 i=1 where ∆ = x − x , i = 1, . . . , N − 1. (Here we assume that N ≥ 2.) Next, since i i+1 i

P N −1 N

¡i − ¢ = 0, we have i=1

2 N N −1 ³ ´

1 N x i

X X x N

− = − (∆ − ξ) i i

2 N N

2 i=1 i=1 for any ξ ∈ R. Taking the absolute values, we get

N N −1 ¯ ¯ ¯ ¯

1 N ¯ ¯ ¯ ¯ x i

X X x N

− − − ξ| i i

¯ ¯ ≤ ¯ ¯ |∆

2 N N

2 i=1 i=1 for any ξ ∈ R. By H¨older’s inequality, we have in particular the following two estimates

N N −1 ¯ ¯

X X x N 1 + x N 1 − 2

¯ ¯ x (1) − |∆ − ξ| i i

¯ ¯ ≤ N

2N i=1 i=1 and

N ¯ ¯

1 ¯ ¯ x S

X x 1 + x N

(2) − max |∆ − ξ| i N i

¯ ¯ ≤ i=1,...,N −1

2 i=1

(N −1) N −2 for any ξ ∈ R, where S = (if N is even) or (if N is odd). The problem

4N P N −1 then is to choose ξ = ξ in (1) at which |∆ − ξ| is minimum, and to choose

1 i i=1 ξ ∞ = ξ in (2) at which max |∆ − ξ| is minimum. i i=1,...,N −1

P N −1

From Theorem 2, we know that in order to minimize |∆ − ξ|, we must first i i=1

∗ ∗ N −1 N −1 rearrange (∆ ) , to get a monotone sequence (∆ ) , and then choose ξ = ∆ N i i=1 i i=1

∗ ∗

1 (if N is even) or (∆ N −1 + ∆ N +1 ) (if N is odd). Meanwhile, from Theorem 3, we

1 know that in order to minimize max

|∆ −ξ| we must choose ξ = min ∆

i i

2 i=1,...,N −1 i=1,...,N −1 max ∆ ¢. If, for instance, N is even, then the estimate (1) will become i i=1,...,N −1

N N −1 ¯ ¯

x 1 − 2 ∗ ∗

1 N ¯ ¯ x

X X x N

− |∆ − ∆ N |, i

¯ ¯ ≤ i

2 N

2N i=1 i=1 while the estimate (2) becomes

N ¯ ¯

1 N − 2 ¡

1 N ¯ ¯ − x max ∆ − min ∆ ¢. i i i

X x + x

¯ ¯ ≤ i=1,...,N −1 i=1,...,N −1

2 N

8 i=1

Both estimates tell us that the mean will not be very far from the centre if the ∆ ’s i are approximately constant, as we expect. Note, however, that the second estimate is easier to compute than the first one.

References

[1] S.S. Dragomir and A. Mcandrew, “On trapezoid inequality via a Gr¨ uss type result and appli-

cations”, Tamkang J. Math. 31 (2000), 193–201.

[2] H. Gunawan, “A note on Dragomir-Mcandrew’s trapezoid inequalities”, Tamkang J. math., to

appear.

E-mail address : hgunawan@dns.math.itb.ac.id

On measures of central location