The characteristic function of order statistics

In summary, the characteristic function of a sum of ordered random variables is the product of integrals over the ranges of the individual variables.
  • #1
EngWiPy
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Suppose that ##Y=\sum_{k=1}^KX_{(k)}##, where ##X_{(1)}\leq X_{(2)}\leq\cdots X_{(N)}## and (##N\geq K##). I want to find the characteristic function of ##Y## as

[tex]\phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right][/tex]

In the case where ##\{X\}## are i.i.d random variables, the above characteristic function will be

[tex]\phi(jvY)=\prod_{k=1}^K\phi(jvX_k)[/tex]

but when the random variables are ordered, they are no longer independent. How can the characteristic function be found in this case?
 
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  • #2
, Are the random variables independently chosen and then ordered? If so it doesn't matter, since the definition of Y doesn't depend on the order. . On the other hand if your question means choose a random variable, then a second variable greater than the first, etc., it is a different problem.
 
  • #3
mathman said:
, Are the random variables independently chosen and then ordered? If so it doesn't matter, since the definition of Y doesn't depend on the order. . On the other hand if your question means choose a random variable, then a second variable greater than the first, etc., it is a different problem.

The selection is like this: I have ##N## i.i.d random variables. Then ##Y## is the sum of the ##K## smallest random variables of the ##N## random variables.
 
  • #4
I found that the joint PDF

##f_{X_{(1)},\,X_{(2)},\ldots,\,X_{(K)}}(x_1,\,x_2,\,\dots,\,x_K)=K!\prod_{k=1}^Kf_X(x_k)##

for ##0<x_1<x_2<\cdots<x_K<\infty##, where ##f_X(x_k)## is the PDF of the original random variables before ordering. Then we can write the characteristic function as

[tex]\phi(jvY)=\int_{x_1=0}^{\infty}\int_{x_2>x_1}\cdots\int_{x_K>x_{K-1}}K!\prod_{k=1}^Ke^{jvx_k}f_X(x_k)\,dx_1\,dx_2\cdots\,dx_K[/tex]

which is the multiplication of ##K## integrals

[tex]\phi(jvY)=\left(\int_{x_1=0}^{\infty}e^{jvx_1}f_X(x_1)\,dx_1\right)\left(\int_{x_2>x_1}e^{jvx_2}f_X(x_2)\,dx_2\right)\cdots \left(\int_{x_K>x_{K-1}}e^{jvx_K}f_X(x_K)\,dx_K\right)[/tex]

Is this right?
 

Related to The characteristic function of order statistics

1. What is the characteristic function of order statistics?

The characteristic function of order statistics is a mathematical function that describes the probability distribution of the order statistics of a set of random variables. It provides information about the probabilities of different orders in which the variables can occur.

2. How is the characteristic function of order statistics different from the characteristic function of a single random variable?

The characteristic function of a single random variable describes the probability distribution of that variable alone. In contrast, the characteristic function of order statistics describes the probability distribution of the order that a set of random variables will occur in.

3. What are the applications of the characteristic function of order statistics?

The characteristic function of order statistics is commonly used in statistics and probability theory for analyzing the behavior of random variables in a set. It can also be used in fields such as economics, finance, and engineering for modeling and predicting data.

4. Can the characteristic function of order statistics be used for non-parametric distributions?

Yes, the characteristic function of order statistics can be used for both parametric and non-parametric distributions. It is a versatile tool for analyzing the behavior of random variables, regardless of their underlying distribution.

5. How is the characteristic function of order statistics calculated?

The characteristic function of order statistics can be calculated using mathematical formulas that involve the characteristic function of the underlying distribution and the order of the statistics. There are also statistical software programs available that can calculate the characteristic function of order statistics for a given set of random variables.

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