Czy istnieje przykładowa wersja jednostronnej nierówności Czebyszewa?

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Interesuje mnie następująca jednostronna wersja nierówności Czebyszewa Cantellego :

P(XE(X)t)Var(X)Var(X)+t2.

Zasadniczo, jeśli znasz średnią populacji i wariancję, możesz obliczyć górną granicę prawdopodobieństwa zaobserwowania określonej wartości. (Tak przynajmniej rozumiałem.)

Chciałbym jednak użyć średniej próby i wariancji próbki zamiast rzeczywistej średniej populacji i wariancji.

Domyślam się, że skoro wprowadziłoby to więcej niepewności, górna granica wzrósłaby.

Czy istnieje nierówność analogiczna do powyższej, ale która wykorzystuje średnią próbki i wariancję?

Edit: The "sample" analog of the Chebyshev Inequality (not one sided), has been worked out. The Wikipedia page has some details. However, I am not sure how it would translate to the one sided case I have above.

casandra
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Thanks Glen_b. It is quite an interesting problem. I always thought that the Chebyshev inequality was powerful (since it let's you do statistical inference without requiring a probability distribution); so being able to use it with the sample mean and variance would be pretty awesome.
casandra

Odpowiedzi:

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Yes, we can get an analogous result using the sample mean and variance, with perhaps, a couple slight surprises emerging in the process.

First, we need to refine the question statement just a little bit and set out a few assumptions. Importantly, it should be clear that we cannot hope to replace the population variance with the sample variance on the right hand side since the latter is random! So, we refocus our attention on the equivalent inequality

P(XEXtσ)11+t2.
In case it is not clear that these are equivalent, note that we've simply replaced t with tσ in the original inequality without any loss in generality.

Second, we assume that we have a random sample X1,,Xn and we are interested in an upper bound for the analogous quantity P(X1X¯tS), where X¯ is the sample mean and S is the sample standard deviation.

A half-step forward

Note that already by applying the original one-sided Chebyshev inequality to X1X¯, we get that

P(X1X¯tσ)11+nn1t2
where σ2=Var(X1), which is smaller than the right-hand side of the original version. This makes sense! Any particular realization of a random variable from a sample will tend to be (slightly) closer to the sample mean to which it contributes than to the population mean. As we shall see below, we'll get to replace σ by S under even more general assumptions.

A sample version of one-sided Chebyshev

Claim: Let X1,,Xn be a random sample such that P(S=0)=0. Then,

P(X1X¯tS)11+nn1t2.
In particular, the sample version of the bound is tighter than the original population version.

Note: We do not assume that the Xi have either finite mean or variance!

Proof. The idea is to adapt the proof of the original one-sided Chebyshev inequality and employ symmetry in the process. First, set Yi=XiX¯ for notational convenience. Then, observe that

P(Y1tS)=1ni=1nP(YitS)=E1ni=1n1(YitS).

Now, for any c>0, on {S>0},

1(YitS)=1(Yi+tcStS(1+c))1((Yi+tcS)2t2(1+c)2S2)(Yi+tcS)2t2(1+c)2S2.

Then,

1ni1(YitS)1ni(Yi+tcS)2t2(1+c)2S2=(n1)S2+nt2c2S2nt2(1+c)2S2=(n1)+nt2c2nt2(1+c)2,
since Y¯=0 and iYi2=(n1)S2.

The right-hand side is a constant (!), so taking expectations on both sides yields,

P(X1X¯tS)(n1)+nt2c2nt2(1+c)2.
Finally, minimizing over c, yields c=n1nt2, which after a little algebra establishes the result.

That pesky technical condition

Note that we had to assume P(S=0)=0 in order to be able to divide by S2 in the analysis. This is no problem for absolutely continuous distributions, but poses an inconvenience for discrete ones. For a discrete distribution, there is some probability that all observations are equal, in which case 0=Yi=tS=0 for all i and t>0.

We can wiggle our way out by setting q=P(S=0). Then, a careful accounting of the argument shows that everything goes through virtually unchanged and we get

Corollary 1. For the case q=P(S=0)>0, we have

P(X1X¯tS)(1q)11+nn1t2+q.

Proof. Split on the events {S>0} and {S=0}. The previous proof goes through for {S>0} and the case {S=0} is trivial.

A slightly cleaner inequality results if we replace the nonstrict inequality in the probability statement with a strict version.

Corollary 2. Let q=P(S=0) (possibly zero). Then,

P(X1X¯>tS)(1q)11+nn1t2.

Final remark: The sample version of the inequality required no assumptions on X (other than that it not be almost-surely constant in the nonstrict inequality case, which the original version also tacitly assumes), in essence, because the sample mean and sample variance always exist whether or not their population analogs do.

cardinal
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15

This is just a complement to @cardinal 's ingenious answer. Samuelson Inequality, states that, for a sample of size n, when we have at least three distinct values of the realized xi's, it holds that

xix¯<sn1,i=1,...n
where s is calculated without the bias correction, s=(1ni=1n(xix¯)2)1/2.

Then, using the notation of Cardinal's answer we can state that

P(X1X¯Sn1)=0a.s.[1]

Since we require, three distinct values, we will have S0 by assumption. So setting t=n1 in Cardinal's Inequality (the initial version) we obtain

P(X1X¯Sn1)11+n,[2]

Eq. [2] is of course compatible with eq. [1]. The combination of the two tells us that Cardinal's Inequality is useful as a probabilistic statement for 0<t<n1.

If Cardinal's Inequality requires S to be calculated bias-corrected (call this S~) then the equations become

P(X1X¯S~n1n)=0a.s.[1a]

and we choose t=n1n to obtain through Cardinal's Inequality

P(X1X¯S~n1n)1n,[2a]
and the probabilistically meaningful interval for t is 0<t<n1n.
Alecos Papadopoulos
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2
(+1) Incidentally, as I was first considering this problem, the fact that maxi|XiX¯|Sn1 was actually the initial clue that the sample inequality should be tighter than the original. I wanted to squeeze that into my post, but couldn't find a (comfortable) place for it. I'm glad to see you mention it (actually a very slight improvement on it) here along with your very nice additional elaboration. Cheers.
cardinal
Cheers @Cardinal, great answer -just clarify for me -does it matter for your Inequality how one defines the sample variance (bias-corrected or not)?
Alecos Papadopoulos
Only ever so slightly. I used the bias-corrected sample variance. If you use n instead of n1 to normalize, then you'll end up with
1+t2c2t2(1+c)2
instead of
(n1)+nt2c2nt2(1+c)2,
which means the n/(n1) term in the final inequality will disappear. Thus, you'll get the same bound as in the original one-sided Chebyshev inequality in that case. (Assuming I've done the algebra correctly.) :-)
cardinal
@Cardinal ...which means that the relevant equations in my answer are 1a and 2a, which means that your inequality tells us that for t chosen to activate Samuelson Inequality, the probability of the event we are examining, cannot be greater than 1/n, i.e. not greater than randomly choosing any one realized value from the sample... which somehow makes some hazy intuitive sense: what is proven certainly impossible in deterministic terms, when approached probabilistically its probability bound does not exceed equiprobability... not clear in my mind yet.
Alecos Papadopoulos