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| In [[statistics]], '''[[cumulative distribution function]] (CDF)-based nonparametric confidence intervals''' are a general class of [[confidence interval]]s around [[V-statistic#Statistical_functions|statistical functionals]] of a distribution. To calculate these confidence intervals, all that is required is an
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| [[Independent_and_identically_distributed_random_variables|independently and identically distributed]] (iid) sample from the distribution and known bounds on the support of the distribution. The latter requirement simply means that all the nonzero probability mass of the distribution must be contained in some known interval <math>[a,b]</math>.
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| ==Intuition==
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| The intuition behind the CDF-based approach is that bounds on the CDF of a distribution can be translated into bounds on statistical functionals of that distribution. Given an upper and lower bound on the CDF, the approach involves finding the CDFs within the bounds that maximize and minimize the statistical functional of interest.
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| ==Properties of the bounds==
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| Unlike approaches that make asymptotic assumptions, including [[Bootstrapping (statistics)|bootstrap approaches]] and those that rely on the [[central limit theorem]], CDF-based bounds are valid for finite sample sizes. And unlike bounds based on inequalities such as [[Hoeffding's_inequality|Hoeffding's]] and [[Doob martingale#McDiarmid.27s_inequality|McDiarmid's]] inequalities, CDF-based bounds use properties of the entire sample and thus often produce significantly tighter bounds.
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| ==CDF bounds==
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| CDF-based confidence intervals require a probabilistic bound on the CDF of the distribution from which the sample were generated. A variety of methods exist for generating confidence intervals for the CDF of a distribution, <math>F</math>, given an i.i.d. sample drawn from the distribution. These methods are all based on the [[empirical distribution function]] (empirical CDF). Given an i.i.d. sample of size ''n'', <math>x_1,\ldots,x_n\sim F</math>, the empirical CDF is defined to be
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| : <math> | |
| \hat{F}_n(t) = \frac{1}{n}\sum_{i=1}^n1\{x_i\le t\},
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| </math> | |
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| where <math>1\{A\}</math> is the indicator of event A. The [[Dvoretzky–Kiefer–Wolfowitz inequality]],<ref name=dvoretzky>{{cite journal|last=A.|first=Dvoretzky|coauthors=Kiefer, J.; Wolfowitz, J.|title=Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator|journal=The Annals of Mathematical Statistics|year=1956|volume=27|issue=3|pages=642–669}}</ref> whose tight constant was determined by Massart,<ref name=massart>{{cite journal|last=Massart|first=P.|title=The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality|journal=The Annals of Probability|year=1990|pages=1269–1283}}</ref> places a confidence interval around the [[Kolmogorov–Smirnov_test#Kolmogorov.E2.80.93Smirnov_statistic|Kolmogorov–Smirrnov statistic]] between the CDF and the empirical CDF. Given an i.i.d. sample of size ''n'' from <math>F</math>, the bound states
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| : <math> | |
| P(\sup_x|F(x)-F_n(x)|>\varepsilon)\le2e^{-2n\varepsilon^2}.
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| </math>
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| This can be viewed as a confidence envelope that runs parallel to, and is equally above and below, the empirical CDF.
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| [[File:MassartBound.png|thumb|300px|Illustration of the bound on the empirical CDF that is obtained using the Dvoretzky–Kiefer–Wolfowitz inequality. The notation <math>X_{(j)}</math> indicates the <math>j^\text{th}</math> [[order statistic]].]] | |
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| The equally spaced confidence interval around the empirical CDF allows for different rates of violations across the support of the distribution. In particular, it is more common for a CDF to be outside of the CDF bound estimated using the Dvoretzky–Kiefer–Wolfowitz inequality near the
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| median of the distribution than near the endpoints of the distribution. In contrast, the order statistics-based bound introduced by Learned-Miller and DeStefano<ref name=entropy>{{cite journal|last=Learned-Miller|first=E.|coauthors=DeStefano, J.|title=A probabilistic upper bound on differential entropy|journal=IEEE Transactions on Information Theory|year=2008|volume=54|issue=11|pages=5223–5230}}</ref> allows for an equal rate
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| of violation across all of the order statistics. This in turn results in a bound that is tighter near the ends of the support of the distribution and looser in the middle of the support. Other types of bounds can be generated by varying the rate of violation for the order statistics. For example, if a tighter bound on the distribution is desired on the upper portion of the support, a higher rate of violation can be allowed at the upper portion of the support at the expense of having a lower rate of violation, and thus a looser bound, for the lower portion of the support.
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| ==A nonparametric bound on the mean==
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| Assume without loss of generality that the support of the distribution is contained in <math>[0,1].</math> Given a confidence envelope for the CDF of <math>F</math> it is easy to derive a corresponding confidence interval for the mean of <math>F</math>. It can be shown<ref name=anderson>{{cite journal|last=Anderson|first=T.W.|title=Confidence limits for the value of an arbitrary bounded random variable with a continuous distribution function|journal=Bulletin of The International and Statistical Institute|year=1969|volume=43|pages=249–251}}</ref> that the CDF that maximizes
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| the mean is the one that runs along the lower confidence envelope, <math>L(x)</math>, and the CDF that minimizes the mean is the one that runs along the upper envelope, <math>U(x)</math>. Using the identity
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| : <math>
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| E(X) = \int_0^1(1-F(x))\,dx,
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| </math>
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| the confidence interval for the mean can be computed as
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| : <math>
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| \left[\int_0^1(1-U(x))\,dx, \int_0^1(1-L(x))\,dx \right].
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| </math>
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| ==A nonparametric bound on the variance==
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| Assume without loss of generality that the support of the distribution of interest, <math>F</math>, is contained in <math>[0,1]</math>. Given a confidence envelope for <math>F</math>, it can be shown<ref name=romano2002explicit>{{cite journal|last=Romano|first=J.P.|coauthors=M., Wolf|title=Explicit nonparametric confidence intervals for the variance with guaranteed coverage|journal=Communications in Statistics - Theory and Methods|year=2002|volume=31|issue=8|pages=1231–1250}}</ref> that the CDF within the envelope that minimizes the variance begins on the lower envelope, has a jump continuity to the upper envelope, and then continues along the upper envelope. Further, it can be shown that this variance-minimizing CDF, F', must satisfy the constraint that the jump discontinuity occurs at <math>E[F']</math>. The variance maximizing CDF begins on the upper envelope, horizontally transitions to the lower envelope, then continues along the lower envelope. Explicit algorithms for calculating these variance-maximizing and minimizing CDFs are given by Romano and Wolf.<ref name=romano2002explicit />
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| ==Bounds on other statistical functionals==
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| The CDF-based framework for generating confidence intervals is very general and can be applied to a variety of other statistical functionals including
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| *Entropy<ref name=entropy />
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| *Mutual Information<ref name=mutualInformation>{{cite journal|last=VanderKraats|first=N.D.|coauthors=Banerjee, A.|title=A finite-sample, distribution-free, probabilistic lower bound on mutual information|journal=Neural Computation|year=2011|volume=23|issue=7|pages=1862–1898}}</ref>
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| *Arbitrary percentiles
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| ==See also==
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| * [[Bootstrapping (statistics)]]
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| * [[Non-parametric statistics]]
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| * [[Confidence interval]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://mathworld.wolfram.com/ConfidenceInterval.html Confidence Interval] – An explanation of confidence intervals.
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| * [http://www.stat.rutgers.edu/~mxie/RCPapers/bootstrap.pdf Bootstrap: A Statistical Method] – An overview of bootstrap methods
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| [[Category:Non-parametric statistics]]
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| [[Category:Statistical inference]]
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| [[Category:Robust statistics]]
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| [[Category:Empirical process]]
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If you are looking for the best Optometrists in Brampton, to get an eye examination, just because you are having some difficulty with your vision, you might be thinking about whether to choose between contact lenses or eyeglasses in Brampton, if such a prescription is given by the doctor. Generally, choosing between the two purely depends uggs on sale your personal preference, some of the factors like aesthetics, budget, convenience, comfort and lifestyle should be taken into consideration in the process of selection between the two. The thing to remember here is that one is not better than the other as each of these two options has their own pros and cons with respect to the health of your eyes, easiness to use and in terms of vision.
Contact lenses: When you opt for contact lenses in Brampton, it is better to understand the following pros and cons associated with this option:
Pros:
1. They conform to the curvature of your eyes, thereby causing lesser vision distortions and providing a wider field of view as compared to the other alternative.
2. They will not create any disturbance when you engage in exercising and sporting activities
3. They do not fog up because of cold weather like glasses.
4. You can try out different contact lenses in Brampton in different colors ugg boots to match your costume
Cons:
1. If you frequently work in computer, ugg boots sale using lenses can cheap ugg boots cause a problem called as computer vision syndrome
2. Some people will have a problem in applying lenses in their eyes, but this problem can http://tinyurl.com/pch83be be rectified with practice.
3. This alternative can cause dry eye syndrome in some people.
Eyeglasses: When it comes to Eyeglasses in Brampton, you can enjoy the following pros:
Pros:
1. If you have dry eye problems, it will not worsen when you use eye glasses as against the contacts.
2. When you use this alternative, there will not be any need for touching your eyes as against the contacts. So, chances of irritation and infections can be avoided
3. You can get extra protection against external factors like debris, dust and wind when you go for this alternative as per the suggestion of optometrists in Brampton.
Cons:
1. When you do not choose the right frame, there are chances of pressure to your nose and behind the ears, which can lead to headaches and other issues.
2. If you are concerned about aesthetics, this cannot be the right option for you.
When you visit an optometrist for an eye exam in Brampton, you can consult the professional to suggest the right alternative for you. Also, regular eye exam in Brampton will be helpful regardless of the option you choose.