# Method of moments

Method of moments |
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**Method of moments** is a statistical technique used in management to estimate the parameters of a population from a sample of data. It involves calculating the sample moments of the data, which are then used to estimate the population moments of the distribution. This method is commonly used in finance, business and economics to estimate the mean, variance and other characteristics of a population from a sample of data. It is a versatile method that can be adapted to different types of data and can be used to estimate parameters for a variety of distributions.

## Example of method of moments

- A common example of the method of moments is estimating the mean and variance of a population from a sample of data. For example, a company may have a sample of 100 sales transactions and wish to estimate the mean and variance of the total sales for the entire population of transactions. The company can use the method of moments to calculate the sample mean and variance from the sample of 100 transactions and then use them to estimate the population mean and variance.
- Another example is estimating the parameters of a normal distribution from a sample of data. Suppose a researcher has collected a sample of data and wishes to estimate the mean and variance of the population from which the sample was drawn. The researcher can use the method of moments to calculate the sample mean and variance, and then use them to estimate the population mean and variance of the normal distribution.
- Finally, the method of moments can be used to estimate the parameters of a Poisson distribution from a sample of data. Suppose a researcher has collected a sample of data and wishes to estimate the mean and variance of the population from which the sample was drawn. The researcher can use the method of moments to calculate the sample mean and variance, and then use them to estimate the population mean and variance of the Poisson distribution.

## Formula of method of moments

The method of moments is based on the idea that the sample moments of a population should match the population moments. The sample moments can be estimated from a sample of data, and these estimates can then be used to estimate the population moments.

The formula for the sample mean is

$$\mu_{\text{sample}} = \frac{1}{n}\sum_{i=1}^n x_i$$

where n is the sample size, and $$x_i$$ is the ith observation in the sample. The sample mean is the average value of all the observations in the sample.

The formula for the sample variance is

$$\sigma_{\text{sample}}^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i-\mu_{\text{sample}})^2$$

where n is the sample size, and $$x_i$$ is the ith observation in the sample. The sample variance is a measure of the spread of the observations around the sample mean.

The method of moments estimates the population mean and variance by equating the sample moments to the population moments. The formula for the population mean is

$$\mu_{\text{population}} = \frac{1}{N}\sum_{i=1}^N \mu_i$$

where N is the population size, and \mu_i is the ith population mean. The population mean is the average value of all the population means.

The formula for the population variance is

$$\sigma_{\text{population}}^2 = \frac{1}{N-1}\sum_{i=1}^N (\mu_i-\mu_{\text{population}})^2$$

where N is the population size, and $$\mu_i$$ is the ith population mean. The population variance is a measure of the spread of the population means around the population mean.

By equating the sample moments to the population moments, the population mean and variance can be estimated from a sample of data. This method is commonly used in finance, business and economics to estimate the mean and variance of a population from a sample of data.

## When to use method of moments

Method of moments is a versatile technique that can be used to estimate the parameters of a variety of distributions. It can be applied in a range of areas, including finance, business and economics. The method is particularly useful when data is limited and information about the population is scarce. Here are some applications of the method of moments:

**Estimating the mean from a sample of data**: the method of moments can be used to estimate the mean of a population from a sample of data. This is done by calculating the sample moments, and then using them to estimate the population moments of the distribution.**Estimating the variance from a sample of data**: the method of moments can also be used to estimate the variance of a population from a sample of data. This is done by calculating the sample moments, and then using them to estimate the population moments of the distribution.**Estimating the higher order moments from a sample of data**: the method of moments can also be used to estimate the higher order moments of a population from a sample of data. This is done by calculating the sample moments, and then using them to estimate the population moments of the distribution.**Estimating the parameters of a distribution**: the method of moments can also be used to estimate the parameters of a distribution from a sample of data. This is done by calculating the sample moments, and then using them to estimate the population moments of the distribution.**Estimating the parameters of a time series**: the method of moments can also be used to estimate the parameters of a time series from a sample of data. This is done by calculating the sample moments, and then using them to estimate the population moments of the distribution.

## Types of method of moments

The method of moments is a versatile statistical technique that can be used to estimate the parameters of a population from a sample of data. There are several types of methods of moments that are commonly used:

- The method of raw moments estimates the parameters of a population by calculating the raw moments of the sample data. This is the most basic method of moments and is applicable to any type of data.
- The method of central moments estimates the parameters of a population by calculating the central moments of the sample data. This method is more accurate than the method of raw moments and is applicable to data with a non-normal distribution.
- The method of standardized moments estimates the parameters of a population by calculating the standardized moments of the sample data. This method is more accurate than the method of central moments and is applicable to data with a non-normal distribution.
- The method of cumulative moments estimates the parameters of a population by calculating the cumulative moments of the sample data. This method is the most accurate method of moments and is applicable to data with a non-normal distribution.

## Steps of method of moments

**Step 1**: Choose the population distribution for which parameters are to be estimated.**Step 2**: Calculate the sample moments from the data. This involves calculating the mean, variance, skewness, and kurtosis, as well as any other moments that are relevant to the population distribution.**Step 3**: Derive the population moments from the population distribution. This involves deriving the expected mean, variance, skewness, and kurtosis, as well as any other moments that are relevant to the population distribution.**Step 4**: Compare the sample moments with the population moments. If the sample moments and population moments match, then the parameters of the population distribution can be estimated.**Step 5**: Estimate the parameters of the population distribution using the sample moments. This can be done using a variety of methods, including least squares estimation, maximum likelihood estimation, and Bayesian inference.**Step 6**: Validate the estimated parameters. This involves checking if the estimated parameters are consistent with the population distribution. If the estimated parameters are not consistent with the population distribution, then the estimation process should be repeated.

## Advantages of method of moments

The method of moments is a powerful and versatile tool for estimating population parameters from a sample of data. There are several advantages to using this method, including:

**Flexibility**: The method of moments is flexible enough to be used to estimate parameters for a variety of distributions, making it useful for a wide range of applications.**Efficiency**: Calculating the sample moments is often faster and easier than maximum likelihood estimation, which is another commonly used method for estimating population parameters.**Goodness of fit**: The method of moments can be used to assess the quality of fit of a given distribution to a set of data. This can be useful for selecting the most appropriate distribution to model the data.**Robustness**: The method of moments is relatively robust to outliers and other forms of data contamination. This makes it well suited to estimating parameters from noisy or incomplete data.

## Limitations of method of moments

Method of moments is a powerful statistical tool for estimating population parameters from sample data. However, it has several limitations that should be taken into account. These include:

- The estimates of parameters obtained from the method of moments are subject to sampling variability, especially when the sample size is small.
- The method is limited to estimating the mean, variance and other first and second order moments of a distribution. It cannot be used to estimate higher order moments or other population characteristics.
- The method of moments relies on the assumption that the sample data is normally distributed. If the data is not normally distributed, the estimates of the population parameters may not be accurate.
- The method of moments is not suitable for estimating parameters from complex distributions, such as the multivariate normal distribution.
- The method of moments is sensitive to outliers in the data and may produce biased estimates of the population parameters if the data contains outliers.

Method of Moments is a powerful tool used to estimate the parameters of a population from a sample of data. Other related approaches include:

**Maximum Likelihood Estimation (MLE)**: MLE is an estimation technique used to find the parameters of a probability distribution that maximizes the likelihood of the observed data. It is often used to estimate the parameters of a distribution when the data follows a specific distribution.**Bayesian Estimation**: Bayesian estimation is a statistical technique used to estimate a probability distribution based on prior knowledge and data. The prior knowledge is used to formulate a prior distribution and then Bayes theorem is used to update the prior distribution with the data.**Bootstrapping**: Bootstrapping is a statistical method used to estimate the population parameters from a sample of data. It involves resampling the data with replacement and calculating the estimated parameters for each sample. The parameters are then combined to estimate the population parameters.**Monte Carlo Simulation**: Monte Carlo simulation is a computer-based simulation technique used to estimate population parameters. It involves randomly generating data with a specific distribution and then calculating the estimated parameters.

In summary, Method of Moments is a powerful technique used to estimate the parameters of a population from a sample of data. Other related approaches include Maximum Likelihood Estimation, Bayesian Estimation, Bootstrapping and Monte Carlo Simulation.

## Suggested literature

- Wooldridge, J. M. (2001).
*Applications of generalized method of moments estimation*. Journal of Economic perspectives, 15(4), 87-100. - Hall, A. R. (2004).
*Generalized method of moments*. OUP Oxford.