Sınırlı veri kümesi için maksimum varyasyon katsayısı değeri


17

Standart sapmanın ortalamayı aşıp aşmayacağı ile ilgili son soruyu takip eden tartışmada , bir soru kısaca gündeme getirildi, ancak hiçbir zaman tam olarak cevaplanmadı. Ben de burada soruyorum.

Bir dizi göz önünde n negatif olmayan sayılar xi burada 0xic için 1in . xi farklı olması, yani kümenin bir çoklu-set olması gerekli değildir . Setin ortalaması ve varyansı

x¯=1ni=1nxi,  σx2=1ni=1n(xix¯)2=(1ni=1nxi2)x¯2
ve standart sapmaσx. Sayı kümesininbir popülasyondan örnekolmadığınıve bir popülasyon ortalamasını veya popülasyon varyansını tahmin etmediğimizi unutmayın. O zaman soru şu:

Maksimum değeri nedir σxx¯ , varyasyon katsayısı,xi's aralığınıntüm seçimlerinde[0,c]mı?

Σ x için bulabileceğim maksimum değerσxx¯ isimli n1 zaman elde edildiğin1arasındaxideğerine sahip0ve kalan (uç değerlerin)xi sahiptir değeric, verme

x¯=cn,  1nxi2=c2nσx=c2nc2n2=cnn1.
Ama bu bağımlı değildirchem hiç, ben büyük değerler merak ediyorum, muhtemelen bağımlın ve c elde edilip edilemeyeceğini .

Herhangi bir fikir? Bu sorunun daha önce istatistik literatüründe çalışıldığından eminim ve bu nedenle gerçek sonuçlar olmasa bile referanslar çok takdir edilecektir.


Bence bunun mümkün olan en büyük değer olduğu konusunda haklısınız ve önemli olmadığına da şaşırıyorum . Güzel. c
Peter Flom - Monica'yı eski durumuna döndürün

7
sonucu σ x olarak etkilememelidirc , tüm değerler pozitif bir sabitkile çarpılırsa değişmez. σxx¯k
Henry

Yanıtlar:


15

Geometry provides insight and classical inequalities afford easy access to rigor.

Geometric solution

We know, from the geometry of least squares, that x¯=(x¯,x¯,,x¯) is the orthogonal projection of the vector of data x=(x1,x2,,xn) onto the linear subspace generated by the constant vector (1,1,,1) and that σx is directly proportional to the (Euclidean) distance between x and x¯. The non-negativity constraints are linear and distance is a convex function, whence the extremes of distance must be attained at the edges of the cone determined by the constraints. This cone is the positive orthant in Rn and its edges are the coordinate axes, whence it immediately follows that all but one of the xi must be zero at the maximum distances. For such a set of data, a direct (simple) calculation shows σx/x¯=n.

Solution exploiting classical inequalities

σx/x¯ is optimized simultaneously with any monotonic transformation thereof. In light of this, let's maximize

x12+x22++xn2(x1+x2++xn)2=1n(n1n(σxx¯)2+1)=f(σxx¯).

(The formula for f may look mysterious until you realize it just records the steps one would take in algebraically manipulating σx/x¯ to get it into a simple looking form, which is the left hand side.)

An easy way begins with Holder's Inequality,

x12+x22++xn2(x1+x2++xn)max({xi}).

(This needs no special proof in this simple context: merely replace one factor of each term xi2=xi×xi by the maximum component max({xi}): obviously the sum of squares will not decrease. Factoring out the common term max({xi}) yields the right hand side of the inequality.)

Because the xi are not all 0 (that would leave σx/x¯ undefined), division by the square of their sum is valid and gives the equivalent inequality

x12+x22++xn2(x1+x2++xn)2max({xi})x1+x2++xn.

Because the denominator cannot be less than the numerator (which itself is just one of the terms in the denominator), the right hand side is dominated by the value 1, which is achieved only when all but one of the xi equal 0. Whence

σxx¯f1(1)=(1×(n1))nn1=n.

Alternative approach

Because the xi are nonnegative and cannot sum to 0, the values p(i)=xi/(x1+x2++xn) determine a probability distribution F on {1,2,,n}. Writing s for the sum of the xi, we recognize

x12+x22++xn2(x1+x2++xn)2=x12+x22++xn2s2=(x1s)(x1s)+(x2s)(x2s)++(xns)(xns)=p1p1+p2p2++pnpn=EF[p].

The axiomatic fact that no probability can exceed 1 implies this expectation cannot exceed 1, either, but it's easy to make it equal to 1 by setting all but one of the pi equal to 0 and therefore exactly one of the xi is nonzero. Compute the coefficient of variation as in the last line of the geometric solution above.


Thanks for a detailed answer from which I have learned a lot! I assume that the difference between the n in your answer and the n1 that I obtained (and Henry confirmed) is due to the fact that you are using
σx=1n1i=1n(xix¯)2
as the definition of σx while I used
σx=1ni=1n(xix¯)2?
Dilip Sarwate

1
Yes Dilip, that's right. Sorry about the discrepancy with the question; I should have checked first and I should have defined σx (which I intended to do but forgot).
whuber

10

Some references, as small candles on the cakes of others:

Katsnelson and Kotz (1957) proved that so long as all xi0, then the coefficient of variation cannot exceed n1. This result was mentioned earlier by Longley (1952). Cramér (1946, p.357) proved a less sharp result, and Kirby (1974) proved a less general result.

Cramér, H. 1946. Mathematical methods of statistics. Princeton, NJ: Princeton University Press.

Katsnelson, J., and S. Kotz. 1957. On the upper limits of some measures of variability. Archiv für Meteorologie, Geophysik und Bioklimatologie, Series B 8: 103–107.

Kirby, W. 1974. Algebraic boundedness of sample statistics. Water Resources Research 10: 220–222.

Longley, R. W. 1952. Measures of the variability of precipitation. Monthly Weather Review 80: 111–117.

I came across these papers in working on

Cox, N.J. 2010. The limits of sample skewness and kurtosis. Stata Journal 10: 482-495.

which discusses broadly similar bounds on moment-based skewness and kurtosis.


8

With two numbers xixj, some δ>0 and any μ:

(xi+δμ)2+(xjδμ)2(xiμ)2(xjμ)2=2δ(xixj+δ)>0.

Applying this to n non-negative datapoints, this means that unless all but one of the n numbers are zero and so cannot be reduced further, it is possible to increase the variance and standard deviation by widening the gap between any pair of the data points while retaining the same mean, thus increasing the coefficient of variation. So the maximum coefficient of variation for the data set is as you suggest: n1.

c should not affect the result as σxx¯ does not change if all the values are multiplied by any positive constant k (as I said in my comment).

Sitemizi kullandığınızda şunları okuyup anladığınızı kabul etmiş olursunuz: Çerez Politikası ve Gizlilik Politikası.
Licensed under cc by-sa 3.0 with attribution required.