How Do You Calculate Maximum Likelihood Estimates for Different Distributions?

In summary, the conversation discusses two questions related to the negative binomial experiment and finding the maximum likelihood estimate for the parameter mu in a normal distribution with known variance, sigma^2 = sigma_0^2. The maximum likelihood estimate is the value of the parameter that gives the largest probability for the actual sample, and it is not the same as the minimum variance unbiased estimator.
  • #1
probabilityst
3
0
I actually have two questions, both of which are on the same topic

Homework Statement


Consider X = number of independent trials until an event A occurs. Show that X has the probability mass function f(x) = (1-p)^x p, x = 1,2,3,..., where p is the probability that A occurs in a single trial. Also show that the maximum likelihood estimate of p(^p) is 1/(sample mean), where (sample mean) = (sum from 1 to n of x)/n. This experiment is referred to as the negative binomial experiment

2. Find the maximum likelihood estimate for the paramter mu of the normal distribution with known variance sigma^2 = sigma_0_^2
_0_ is subscript

Homework Equations


The Attempt at a Solution


Um, I'm not actually sure how to go about solving this problem. Is maximum likelihood estimate similar to minimum variance unbiased estimator?

Any help is greatly appreciated, thank you
 
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  • #2
No. The "maximum likelihood estimator" for a parameter is the value of the parameter that gives the largest possible probability for the actual sample.
 

FAQ: How Do You Calculate Maximum Likelihood Estimates for Different Distributions?

What is Maximum Likelihood Estimate (MLE)?

Maximum Likelihood Estimate (MLE) is a statistical method used to estimate the parameters of a probability distribution by finding the set of values that maximizes the probability of observing the data. It is commonly used in regression analysis, machine learning, and other statistical modeling techniques.

How is MLE different from other parameter estimation techniques?

MLE is different from other parameter estimation techniques because it is based on the principle of maximum likelihood - finding the values that maximize the likelihood or probability of observing the data. Other techniques, such as least squares estimation, minimize the sum of squared errors instead.

What are the assumptions of MLE?

MLE assumes that the data follows a known probability distribution and that the observations are independent and identically distributed. It also assumes that the data is generated from a single population or process and that the parameters being estimated are constant.

How is MLE used in practice?

In practice, MLE is used to estimate the parameters of a probability distribution that best fits the observed data. This can be done by using mathematical formulas or numerical optimization algorithms. The estimated parameters can then be used to make predictions or draw conclusions about the underlying population.

What are the advantages of using MLE?

MLE has several advantages, including being a consistent and efficient estimator. It also provides a measure of uncertainty through confidence intervals and can be easily applied to a wide range of statistical models. Additionally, MLE is based on a well-defined mathematical framework, making it a reliable and widely accepted method in statistical analysis.

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