Properties Of Estimators Question

[dipset1011]

Homework Statement

suppose that 14, 10, 18, 21 constitute a random sample of size 4 drawn from a uniform pdf defined over the interval [0, theta], where theta is unknown. Find an unbiased estimator for theta and y'3, the third order statistic. What numerical value does the estimator have for these particular observations? Is it possible that we would know that an estimate for theta based on y'3 was incorrect, even if we have no idea what the true value of theta might be? explain.

Homework Equations

I am not entirely sure how to go about solving this problem and help or a kick in the right direction would be great.

The Attempt at a Solution

Is the third order statistic 18? isn't the estimator just the summation of all the observations divided by n, the number of samples or observations. Please, any help would be greatly appreciated.

 

[mighty2000]

Hello there! Thank you for posting this question in the forum. Let me help you with finding an unbiased estimator for theta and y'3, the third order statistic.

Firstly, let's define some terms and equations that will help us solve this problem.

- A random sample is a subset of a population that is selected in a way that each member of the population has an equal probability of being chosen.
- A uniform probability density function (pdf) is a type of probability distribution in which all outcomes are equally likely.
- The third order statistic, y'3, is the third smallest value in a set of observations.

Now, to find an unbiased estimator for theta, we can use the method of moments. This involves equating the theoretical moments of a distribution to the sample moments calculated from the data.

In this case, since we have a uniform distribution over the interval [0, theta], the theoretical mean and variance are given by:

μ = (theta + 0)/2 = theta/2
σ^2 = (theta^2 - 0^2)/12 = theta^2/12

The sample mean and variance can be calculated from the given data as:

x̄ = (14 + 10 + 18 + 21)/4 = 15.75
s^2 = [(14-15.75)^2 + (10-15.75)^2 + (18-15.75)^2 + (21-15.75)^2]/3 = 12.25

Now, equating the theoretical and sample moments, we get:

theta/2 = 15.75
theta^2/12 = 12.25

Solving these equations simultaneously, we get theta = 18.

Therefore, an unbiased estimator for theta is 18.

Moving on to finding the third order statistic, y'3, we can simply arrange the given data in ascending order: 10, 14, 18, 21. The third smallest value is 18, so y'3 = 18.

To answer your question about whether it is possible to know if an estimate for theta based on y'3 is incorrect, the answer is yes. This is because we can compare our estimate of theta with other unbiased estimators or with the true value of theta (if known). If our estimate is significantly different from these values, then we can conclude that our estimate