# Estimate Standard Deviation of Mean Average w/ Maximum Likelihood

• Thinkmarble
In summary, to estimate the standard deviation for the mean average of a Poisson distribution, the maximum-likelihood method is used by graphing the likelihood in dependence of the mean average. The standard deviation is determined by finding the value at which the likelihood becomes maximal. The minimum of the log-likelihood is obtained at a mean average of a, and the log-likelihood has the value of 101 at a mean average of b, so the standard deviation is a-b. However, for traditional approaches to Maximum Likelihood Estimators for the Poisson distribution, the variance is given by dividing the sample mean by the sample size. The standard deviation is then the square root of the variance.
Thinkmarble
How do I estimate the standart deviation for the mean average of an poisson-distribution ?
The mean average was estimated with the maximum-likelihood method by graphing the likelihood in dependence of the mean average, then just reading off the value for which the likelihood became maximal.
Up to this point I had not problem.
But I also have to determine the standard deviation for my estimation of the mean average.
And that is where I run into problems.
I'm told that "by an deviation from (...) the mean average of the standart deviation the -2*ln(L) function increases by one unit compared with the minimum".
If I understand that correctly:
Minimum of the log-likelihood is 100 at an mean average of a.
At an mean average of b my log-likelihood has the value 101(99).
So my standart deviation is a-b.

Is this interpretation correct ?

Thinkmarble said:
How do I estimate the standart deviation for the mean average of an poisson-distribution ?
The mean average was estimated with the maximum-likelihood method by graphing the likelihood in dependence of the mean average, then just reading off the value for which the likelihood became maximal.
Up to this point I had not problem.
But I also have to determine the standard deviation for my estimation of the mean average.
And that is where I run into problems.
I'm told that "by an deviation from (...) the mean average of the standart deviation the -2*ln(L) function increases by one unit compared with the minimum".
If I understand that correctly:
Minimum of the log-likelihood is 100 at an mean average of a.
At an mean average of b my log-likelihood has the value 101(99).
So my standart deviation is a-b.

Is this interpretation correct ?
Not exactly clear what you're trying to say. For traditional approaches to Maximum Likelihood Estimators, the Likelihood Function "L(θ)" is obtained, and the Variance of the Maximum Likelihood Estimator $$\mathbf{\hat{\theta}}$$ is given by:

$$1: \ \ \ \ \ Var(\mathbf{\hat{\theta}}) \ = \ \left ( - \, \frac{d^{2}L(\theta) } {d \theta^{2}} \right )^{\mathbf{-1}}_{\mathbf{ \theta = \hat{\theta}} }$$

Thus, for the Poisson Distribution with Log Likelihood Function "logL(λ)" (and using the fact that the Maximum Likelihood Estimator of "$\lambda$" is {$$\hat{\lambda} = \overline{x}$$}:

$$2: \ \ \ \ \ \ Var(\hat{\lambda}) \ = \ Var(\overline{x}}) \ = \ \frac {(\hat{\lambda})^{2}} {\sum_{i=1}^{n} \, x_{i} } \ = \ \frac {(\bar{x})^{2}} {n \bar{x} } \ = \ \frac { \overline x} {n}$$

Thus, for the Poisson Distribution, the Variance of "($\overline{x}$)" is estimated by dividing the value of "($\overline{x}$)" by the sample size "n". Remember that the Standard Deviation will be the Sqrt of the Variance.

(Note: FYI, recall for the Poisson Distribution, we have {λ = μ = σ2}).

~~

Last edited:

Yes, your interpretation is correct. The standard deviation for the mean average of a Poisson distribution can be estimated using the maximum likelihood method. This is done by graphing the likelihood function in terms of the mean average and finding the value that maximizes the likelihood. This value will be the estimated mean average. To determine the standard deviation, you can use the fact that a deviation of one unit from the mean average will result in a change of -2*ln(L) in the likelihood function. This means that the difference between the estimated mean average and the true mean average is equal to the standard deviation. In your example, if the minimum of the log-likelihood is at a mean average of a=100 and at a mean average of b=101, the standard deviation would be 1. This approach is a valid way to estimate the standard deviation for the mean average of a Poisson distribution.

## 1. What is the maximum likelihood method for estimating standard deviation of mean average?

The maximum likelihood method is a statistical approach used to estimate the parameters of a probability distribution by maximizing the likelihood function. In the context of estimating standard deviation of mean average, this method involves finding the value of the standard deviation that makes the observed data most likely to occur.

## 2. How does the maximum likelihood method differ from other methods of estimating standard deviation of mean average?

The maximum likelihood method differs from other methods in that it takes into account the entire sample of data and aims to find the value of the standard deviation that best fits the observed data. Other methods, such as the method of moments, involve using specific equations or assumptions to calculate the standard deviation.

## 3. What are the advantages of using the maximum likelihood method to estimate standard deviation of mean average?

One advantage of using the maximum likelihood method is that it is a data-driven approach and does not require any specific assumptions about the underlying distribution of the data. It also takes into account the entire sample, rather than just a summary statistic like the mean or median.

## 4. Are there any limitations to using the maximum likelihood method to estimate standard deviation of mean average?

One limitation of the maximum likelihood method is that it can be computationally intensive, especially for large data sets. Additionally, if the underlying distribution of the data is not well-defined or if the data is skewed, the results of the maximum likelihood method may not be accurate.

## 5. Can the maximum likelihood method be used to estimate standard deviation of mean average for any type of data?

Yes, the maximum likelihood method can be used for any type of data as long as the data is independent and identically distributed. This means that each observation is not influenced by previous observations and that the data follows the same distribution. However, as mentioned before, the accuracy of the results may be affected by the underlying distribution and skewness of the data.

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