What is the Maximum Likelihood Estimator for Uniform Distribution Endpoints?

In summary, a maximum likelihood estimator is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function. It differs from other types of estimation methods by considering the entire shape of the distribution, and it has several assumptions including independence and a differentiable likelihood function. The estimator is calculated by finding the maximum of the likelihood function, and it has advantages such as versatility, simplicity, and unbiased and efficient estimates. It is commonly used in fields like machine learning and econometrics.
  • #1
das1
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I need help on this problem, anyone know how to do it?

Suppose you have n independent observations from a uniform distribution over the interval [𝜃1, 𝜃2].

a. Find the maximum likelihood estimator for each of the endpoints θ1 and θ2.
b. Based on your result in part (a), what would you expect the maximum likelihood estimator to be for the mean? Explain or prove your result.
 
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  • #2
The answer will depend on what you know. For example, do you know an expression for the likelihood?
 

1. What is a maximum likelihood estimator (MLE)?

A maximum likelihood estimator is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function. It is often used to find the most likely values for the unknown parameters of a model based on observed data.

2. How is maximum likelihood estimation different from other types of estimation?

Maximum likelihood estimation differs from other types of estimation methods, such as least squares estimation, because it takes into account the entire shape of the probability distribution instead of just a single point. This allows for a more accurate estimation of the parameters.

3. What are the assumptions of maximum likelihood estimation?

The main assumption of maximum likelihood estimation is that the data follows a specific probability distribution. Additionally, it assumes that the observations are independent and identically distributed, and that the likelihood function is differentiable and has a unique maximum.

4. How is the maximum likelihood estimator calculated?

The maximum likelihood estimator is calculated by finding the values of the parameters that maximize the likelihood function. This is typically done using numerical optimization methods such as gradient descent or the Newton-Raphson method.

5. What are the advantages of using maximum likelihood estimation?

Maximum likelihood estimation has several advantages, including its ability to handle a wide range of data types and distributions, its simplicity in implementation, and its ability to provide unbiased and efficient estimates. It is also a popular method in fields such as machine learning and econometrics.

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