# Maximum Order Statistic Question

• daneault23
In summary, Ray provides a solution to the homework equation that includes the following: -The distribution of the solution U-The value of any associated parameters-The variance of the solution
daneault23

## Homework Statement

Let Yi∼iid,uniform[0,θ]. Let U=max{Yi}. Derive the distribution of U and give the value of any associated parameters. Also calculate E(U) and Var(U).

## Homework Equations

f(y)=1/Θ and F(y)=y/Θ

## The Attempt at a Solution

Since we have a product of iid random variables, we can multiply the cdf's a total of n times, giving us F(yn)=[F(y)]^n=(y/Θ)^n, so f(u)=n(y/Θ)^n-1, meaning U~Be(n,1) with α=n and β=1.

I'm stuck on the E(u) part. This is what I have, ∫(from 0 to Θ)of u*n(y/Θ)^n-1 du. Please help.

daneault23 said:

## Homework Statement

Let Yi∼iid,uniform[0,θ]. Let U=max{Yi}. Derive the distribution of U and give the value of any associated parameters. Also calculate E(U) and Var(U).

## Homework Equations

f(y)=1/Θ and F(y)=y/Θ

## The Attempt at a Solution

Since we have a product of iid random variables, we can multiply the cdf's a total of n times, giving us F(yn)=[F(y)]^n=(y/Θ)^n, so f(u)=n(y/Θ)^n-1, meaning U~Be(n,1) with α=n and β=1.

I'm stuck on the E(u) part. This is what I have, ∫(from 0 to Θ)of u*n(y/Θ)^n-1 du. Please help.

You cannot hope to get a correct answer if you are careless. From ##F_n(y) = (y/\theta)^n## we have the density ##f_n(y) = dF_n(y)/dy = (n/\theta) (y/\theta)^{n-1},## which is not what you wrote. I don't know why you write f(u) instead of f(y).

Anyway, the answer is given by a simple, calculus 101 integral. Just write it out and think about it.

Ray Vickson said:
You cannot hope to get a correct answer if you are careless. From ##F_n(y) = (y/\theta)^n## we have the density ##f_n(y) = dF_n(y)/dy = (n/\theta) (y/\theta)^{n-1},## which is not what you wrote. I don't know why you write f(u) instead of f(y).

Anyway, the answer is given by a simple, calculus 101 integral. Just write it out and think about it.

Ray, so we would have ∫from 0 to Θ of (y*n/Θ)(y/Θ)^(n-1) then correct?

If I'm doing things correctly here, I get E(U) = (nΘ)/n+1 and with the usual calculations, Var(U)=(nΘ^2)/(n+2)-((nΘ)/(n+1))^2

## 1. What is the maximum order statistic question?

The maximum order statistic question is a statistical problem that involves finding the largest value in a set of data, also known as the maximum value. This value is used as a measure of central tendency and can provide insights into the distribution of the data.

## 2. How is the maximum order statistic calculated?

To calculate the maximum order statistic, the data set is first arranged in ascending or descending order. The maximum value, also known as the largest value, is then identified from this ordered list. This can be done manually or using statistical software.

## 3. What is the significance of the maximum order statistic?

The maximum order statistic is significant because it can provide insights into the distribution of the data. It can help identify outliers and skewness in the data set, which can affect the overall analysis and conclusions drawn from the data.

## 4. How is the maximum order statistic used in hypothesis testing?

In hypothesis testing, the maximum order statistic can be used to determine the critical value for a given significance level. This critical value helps determine whether to reject or fail to reject the null hypothesis based on the test statistic calculated from the data.

## 5. Can the maximum order statistic be affected by sample size?

Yes, the maximum order statistic can be affected by sample size. As the sample size increases, the maximum value may change, especially if there are outliers in the data set. However, if the data set is normally distributed, the maximum order statistic will not change significantly with increasing sample size.

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