Probability & Statistics: Order Statistics

In summary: So, in summary, the difference between X and order statistics is that order statistics denote a specific observed value while X denotes a random variable. Order statistics can be denoted by x(1) or X(1). Furthermore, order statistics is a random variable when it is denoted by X(1). Lastly, when ordering a bunch of random variables, you can do so by considering the joint density as a function of a single variable x.
  • #1
kingwinner
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Homework Statement


Q1) About "order statistics", sometimes it's denoted x(1) and sometimes it's denoted X(1). What is the difference between the two?
Also, for X(1)=min{X1,X2,...,Xn}, it's a random variable. What does it mean to be the minimum of a bunch of random variables? If they are SPECIFIC observed values, then we can order them (e.g. if we have 6,3,8,7, then ordering them gives 3,6,7,8)...that I understand. But if they are random variables, HOW can we order them?



Q2) (more about order statistics)
http://www.geocities.com/asdfasdf23135/stat7.JPG
Here we have n random variables X1,X2...,Xn and we see FX(x) here. Why can we label it just based on one single varaible "x" instead of x1,x2,...,xn? Don't we have to treat them separately as x1,x2,...xn instead of just one "x"? Well, you may say it is because they're identically distributed, so we can just use a single "x" to represent each of x1,x2,...xn. But consider the following case:
Let X1,X2,...,Xn be iid random variables with density f(xi)=xi, 0<xi<sqrt2, then in this case the joint density must be f(x1,x2,...,xn)=x1x2...xn, and is definitely NOT (x1)n
So we've seen two different situations. In the first case, we can say x=x1=x2=...=xn, but not so in the second case. What is going on? Can someone please explain? I am always confused between these two cases. I am confused whenever they say X1,...Xn are iid with COMMON density fX(x). If this is the case, then the JOINT density [fX(x)]n would be a function of only a single variable "x" which doesn't make any sense to me (the joint density should be a function of n variables x1,x2,...,xn)


Homework Equations


Order Statistics

The Attempt at a Solution


As shown above.


Thank you for clearing my doubts! I appreciate your great help!
 
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  • #2
x usually denotes an observed value; whereas X denotes a random variable (i.e., a distribution). You cannot order random variables but using them you can derive the distribution that governs the value of the order statistic variable. As for Q2, you are confusing particular values with a distribution. If it makes it easier, you can try thinking of them not as probability distributions but frequency distributions (e.g., body height in a given population of N individuals).
 
  • #3
For Q1, maybe it would help you if you would recall that a random variable X is a function from a sample space S to the set of real numbers.

If s is an outcome in S, then X(s) is a real number.

To say [tex]X_{(1)}=\min\{X_1,X_2,\dots,X_n\}[/tex], this really means that for each outcome s in S, we have [tex]X_{(1)}(s)=\min\{X_1(s),X_2(s),\dots,X_n(s)\}[/tex].

For Q2, you are right that you can not simply say that the joint f is the product [tex][f_X(x)]^n[/tex]. There is a different reason for this in this example.

First, for emphasis and clarity, use the letter k instead of x, where k is a constant.

Now [tex]\{X_{(n)} \le k\}[/tex] is shorthand for [tex]\{ s\in S:X_{(n)}(s) \le k\}[/tex]. But [tex]X_{(n)}(s)=\max\{X_1(s),X_2(s),\dots,X_n(s)\}[/tex], so [tex]X_{(n)}(s) \le k[/tex] if and only [tex]X_i(s)\le k[/tex] for all i from 1 to n.

That is, [tex]\{ s\in S:X_{(n)}(s) \le k\}=\{s\in S:X_1(s)\le k\ \land\ X_2(s)\le k\ \land\ \dots\ \land\ X_n(s)\le k\}=\{s\in S:X_1(s)\le k\}\cap \dots\cap\{s\in S: X_n(s)\le k\}[/tex].

By independence, [tex]P(\{s\in S:X_1(s)\le k\}\cap \dots\cap\{s\in S: X_n(s)\le k\})=P(X_1\le k)P(X_2\le k)\dots P(X_n\le k)[/tex].
 

Related to Probability & Statistics: Order Statistics

1. What is order statistics in probability and statistics?

Order statistics refers to the study of the statistical properties of the sorted values in a random sample. This includes analyzing the minimum, maximum, and other percentiles of the data set.

2. How is order statistics related to probability distributions?

Order statistics can be used to determine the probability of certain events occurring in a data set, such as the probability of the smallest value being within a certain range. Order statistics can also be used to estimate the parameters of a probability distribution, such as the mean and variance.

3. What is the difference between order statistics and rank statistics?

Order statistics refer to the actual values in a data set, whereas rank statistics refer to the position of a value within the data set. For example, the minimum value in a data set is an order statistic, while the first value in a sorted list would be its rank.

4. How are order statistics used in practical applications?

Order statistics have various practical applications, such as in quality control to determine the minimum or maximum acceptable value for a product. They are also used in finance to analyze the extreme values of stock prices. Additionally, order statistics are used in survival analysis to estimate the probability of an event occurring within a certain time frame.

5. What are the limitations of order statistics?

Order statistics can be affected by outliers, as extreme values can significantly impact the results. Additionally, order statistics may not be accurate if the sample size is small. It is important to consider these limitations when using order statistics in data analysis.

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