Method of Moments Estimation for Rectangular Distribution

[grimster]

ok, X_1,X_2,...,X_n are independant(and unbiased) rectangular distributed random variables over the interval [0,a]

It is known that T(X)=max(X_1,X_2,...,X_n) is sufficient. i am supposed to find the moment estimator for a using the method of moments.

i know I'm supposed to equate the first k sample moments to the corresponding k population moments and solve the resulting system of equations, but i have a few questions:

1.is the rectangular distribution the same as the uniform distribution?

2.to get me started so i understand what to do, what is the first sample and population moment?

 

[grimster]

something like this:
sample moment = the population moment:
x_bar=a/2 -> a=2*x_bar

 

[tacman]

The rectangular distribution is similar to the uniform distribution in that it has a constant probability density function over a given interval. However, the main difference is that the uniform distribution has equal probabilities for all values within the interval, while the rectangular distribution has a constant probability for a specific range within the interval.

To find the moment estimator for a using the method of moments, we can follow the steps outlined in the question. First, we need to equate the first k sample moments to the corresponding k population moments. The first sample moment is the sample mean, which can be calculated by taking the sum of all the observations and dividing by the sample size (n). The first population moment is the expected value of the distribution, which can be calculated by taking the integral of the probability density function over the interval [0,a]. So, we can set the first sample moment equal to the first population moment and solve for a to get our moment estimator for a.

To make this more concrete, let's consider the first sample and population moments for a rectangular distribution. The first sample moment is the sample mean, denoted as M_1, and can be calculated as:

M_1 = (X_1 + X_2 + ... + X_n)/n

The first population moment is the expected value of the distribution, denoted as μ_1, and can be calculated as:

μ_1 = ∫_0^a x*f(x) dx

where f(x) is the probability density function for the rectangular distribution. Since the rectangular distribution has a constant probability density of 1/a over the interval [0,a], we can simplify this to:

μ_1 = ∫_0^a x*(1/a) dx = a/2

Now, we can set M_1 equal to μ_1 and solve for a to get our moment estimator:

M_1 = μ_1
(X_1 + X_2 + ... + X_n)/n = a/2
a = 2*M_1

So, our moment estimator for a is simply twice the sample mean. We can continue this process for higher moments to get more moment estimators, but the first moment estimator is a good starting point.