Method of moments

The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).

$$\begin{array}{ccl} \overbrace{\mu}^{\substack{\text{parameters of}\\\text{population distribution}\\\text{in terms of moments}}} &=& \overbrace{\frac{1}{N}\sum_{i=1}^N{x_i}}^{\text{sample moments}}\\ \sigma^2 &=& \frac{1}{N}\sum_{i=1}^N{(x_i-\mu)^2} \end{array}$$

Whenever you are setting these two equal then you are employing the method of moments.

You can use this method also when you are not dealing with a normal distribution.

Example: betabinomial distribution

Say we have a population that follows a betabinomial distribution with a fixed size parameter $n$ and unknown parameters $\alpha$ and $\beta$. For this case we can also parameterize the distribution in terms of the mean and variance

$$\begin{array}{rcl} \frac{n \alpha}{\alpha + \beta} &=& \mu\\ \frac{n\alpha\beta(n+\alpha+\beta)}{(\alpha +\beta)^2(\alpha+\beta+1)} &=& \sigma^2 \end{array}$$

and set it equal to the sample moments

$$\begin{array}{rcccccl} \frac{n \hat\alpha}{\hat\alpha + \hat\beta}&=& \hat{\mu} &=& \bar{x} &=&\frac{1}{N}\sum_{i=1}^N{x_i}\\ \frac{n\hat\alpha\hat\beta(n+\hat\alpha+\hat\beta)}{(\hat\alpha +\hat\beta)^2(\hat\alpha+\hat\beta+1)}&=& \hat{\sigma}^2 &=& s^2 &=&\frac{1}{N}\sum_{i=1}^N{(x_i-\bar{x})^2} \end{array}$$

From which estimates for the distribution follow

$$\begin{array}{rcl} \hat\alpha &=& \frac{ n\hat{x}-s^2-\hat{x}^2 }{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \\ \hat\beta &=&\frac{( n-\hat{x} ) ( n-{\frac {s^2+\hat{x}^2}{\hat{x}}} )}{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \end{array}$$

With the above estimates $\hat{alpha}$ and $\hat{beta}$ the estimated population has the same mean and variance as the sample.

Note

In the case of estimating the parameters of a normal distribution, then the method of moments coincides with the maximum likelihood method.

Method of moments

The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).

$$\begin{array}{ccl} \overbrace{\mu}^{\substack{\text{parameters of}\\\text{population distribution}\\\text{in terms of moments}}} &=& \overbrace{\frac{1}{N}\sum_{i=1}^N{x_i}}^{\text{sample moments}}\\ \sigma^2 &=& \frac{1}{N}\sum_{i=1}^N{(x_i-\mu)^2} \end{array}$$

Whenever you are setting these two equal then you are employing the method of moments.

You can use this method also when you are not dealing with a normal distribution.

Example: betabinomial distribution

Say we have a population that follows a betabinomial distribution with a fixed size parameter $n$ and unknown parameters $\alpha$ and $\beta$. For this case we can also parameterize the distribution in terms of the mean and variance

$$\begin{array}{rcl} \frac{n \alpha}{\alpha + \beta} &=& \mu\\ \frac{n\alpha\beta(n+\alpha+\beta)}{(\alpha +\beta)^2(\alpha+\beta+1)} &=& \sigma^2 \end{array}$$

and set it equal to the sample moments

$$\begin{array}{rcccccl} \frac{n \hat\alpha}{\hat\alpha + \hat\beta}&=& \hat{\mu} &=& \bar{x} &=&\frac{1}{N}\sum_{i=1}^N{x_i}\\ \frac{n\hat\alpha\hat\beta(n+\hat\alpha+\hat\beta)}{(\hat\alpha +\hat\beta)^2(\hat\alpha+\hat\beta+1)}&=& \hat{\sigma}^2 &=& s^2 &=&\frac{1}{N}\sum_{i=1}^N{(x_i-\bar{x})^2} \end{array}$$

From which estimates for the distribution follow

$$\begin{array}{rcl} \hat\alpha &=& \frac{ n\hat{x}-s^2-\hat{x}^2 }{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \\ \hat\beta &=&\frac{( n-\hat{x} ) ( n-{\frac {s^2+\hat{x}^2}{\hat{x}}} )}{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \end{array}$$

With the above estimates $\hat{\alpha}$ and $\hat{\beta}$ the estimated population has the same mean and variance as the sample.

Note

In the case of estimating the parameters of a normal distribution, then the method of moments coincides with the maximum likelihood method.

Method of moments

The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).

$$\begin{array}{ccl} \overbrace{\mu}^{\substack{\text{parameters of}\\\text{population distribution}\\\text{in terms of moments}}} &=& \overbrace{\frac{1}{N}\sum_{i=1}^N{x_i}}^{\text{sample moments}}\\ \sigma^2 &=& \frac{1}{N}\sum_{i=1}^N{(x_i-\mu)^2} \end{array}$$

Whenever you are setting these two equal then you are employing the method of moments.

You can use this method also when you are not dealing with a normal distribution.

Example: betabinomial distribution

Say we have a population that follows a betabinomial distribution with a fixed size parameter $n$ and unknown parameters $\alpha$ and $\beta$. For this case we can also parameterize the distribution in terms of the mean and variance and set it equal to the sample moments

$$\begin{array}{rcl} \mu &=& \frac{n \alpha}{\alpha + \beta}&=& \frac{1}{N}\sum_{i=1}^N{x_i} &=& \bar{x}\\ \sigma^2 &=& \frac{n\alpha\beta(n+\alpha+\beta)}{(\alpha +\beta)^2(\alpha+\beta+1)}&=& \frac{1}{N}\sum_{i=1}^N{(x_i-\bar{x})^2} &=& s^2\\ \end{array}$$

From which estimates for the distribution follow

$$\begin{array}{rcl} \hat\alpha &=& \frac{ n\hat{x}-s^2-\hat{x}^2 }{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \\ \hat\beta &=&\frac{( n-\hat{x} ) ( n-{\frac {s^2+\hat{x}^2}{\hat{x}}} )}{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \end{array}$$

With the above estimates $\hat{alpha}$ and $\hat{beta}$ the estimated population has the same mean and variance as the sample.

Note

In the case of estimating the parameters of a normal distribution, then the method of moments coincides with the maximum likelihood method.

Method of moments

The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).

$$\begin{array}{ccl} \overbrace{\mu}^{\substack{\text{parameters of}\\\text{population distribution}\\\text{in terms of moments}}} &=& \overbrace{\frac{1}{N}\sum_{i=1}^N{x_i}}^{\text{sample moments}}\\ \sigma^2 &=& \frac{1}{N}\sum_{i=1}^N{(x_i-\mu)^2} \end{array}$$

Whenever you are setting these two equal then you are employing the method of moments.

You can use this method also when you are not dealing with a normal distribution.

Example: betabinomial distribution

Say we have a population that follows a betabinomial distribution with a fixed size parameter $n$ and unknown parameters $\alpha$ and $\beta$. For this case we can also parameterize the distribution in terms of the mean and variance

$$\begin{array}{rcl} \frac{n \alpha}{\alpha + \beta} &=& \mu\\ \frac{n\alpha\beta(n+\alpha+\beta)}{(\alpha +\beta)^2(\alpha+\beta+1)} &=& \sigma^2 \end{array}$$

and set it equal to the sample moments

$$\begin{array}{rcccccl} \frac{n \hat\alpha}{\hat\alpha + \hat\beta}&=& \hat{\mu} &=& \bar{x} &=&\frac{1}{N}\sum_{i=1}^N{x_i}\\ \frac{n\hat\alpha\hat\beta(n+\hat\alpha+\hat\beta)}{(\hat\alpha +\hat\beta)^2(\hat\alpha+\hat\beta+1)}&=& \hat{\sigma}^2 &=& s^2 &=&\frac{1}{N}\sum_{i=1}^N{(x_i-\bar{x})^2} \end{array}$$

From which estimates for the distribution follow

$$\begin{array}{rcl} \hat\alpha &=& \frac{ n\hat{x}-s^2-\hat{x}^2 }{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \\ \hat\beta &=&\frac{( n-\hat{x} ) ( n-{\frac {s^2+\hat{x}^2}{\hat{x}}} )}{n ( \frac {s^2}{\hat{x}}-1 ) +\hat{x}} \end{array}$$

With the above estimates $\hat{alpha}$ and $\hat{beta}$ the estimated population has the same mean and variance as the sample.

Note

In the case of estimating the parameters of a normal distribution, then the method of moments coincides with the maximum likelihood method.