Equating Population and Sample Moments

Oct 13, 2025 by leolavanderia.com.br 135 views

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Introduction

The Method of Moments (MOM) is a widely used technique in statistics for estimating the parameters of a distribution. It is based on the idea of equating the population moments with the sample moments. In this article, we will discuss how to find the estimator by MOM for a discrete random variable with a given probability function.

Probability Function

The probability function of the discrete random variable $X$ is given by:

p(x)=\frac{-1}{log(1-\theta)}\frac{\theta^x}{x}

where $0<\theta<1$ and $x=1,2,...$ .

Method of Moments

The Method of Moments involves equating the population moments with the sample moments. The population moments are defined as:

\mu_k = E(X^k)

where $E(X^k)$ is the expected value of $X^k$ . The sample moments are defined as:

m_k = \frac{1}{n}\sum_{i=1}^n X_i^k

where $X_i$ are the observations in the sample.

First Population Moment

The first population moment is the expected value of $X$ , which is given by:

\mu_1 = E(X) = \sum_{x=1}^{\infty} x p(x)

Substituting the probability function, we get:

\mu_1 = \sum_{x=1}^{\infty} x \frac{-1}{log(1-\theta)}\frac{\theta^x}{x}

Simplifying, we get:

\mu_1 = \frac{-1}{log(1-\theta)} \sum_{x=1}^{\infty} \theta^x

Using the formula for the sum of a geometric series, we get:

\mu_1 = \frac{-1}{log(1-\theta)} \frac{\theta}{1-\theta}

Simplifying, we get:

\mu_1 = \frac{\theta}{log(1-\theta)}

First Sample Moment

The first sample moment is the sample mean, which is given by:

m_1 = \frac{1}{n}\sum_{i=1}^n X_i

Equating Population and Sample Moments

Equating the first population moment with the first sample moment, we get:

\frac{\theta}{log(1-\theta)} = \frac{1}{n}\sum_{i=1}^n X_i

Simplifying, we get:

\theta = \frac{log(1-\theta)}{n} \sum_{i=1}^n X_i

Estimator by MOM

The estimator by MOM is given by:

\hat{\theta} = \frac{log(1-\bar{X})}{n \bar{X}}

where $\bar{X}$ is the sample mean.

Conclusion

In this article, we discussed how to find the estimator by MOM for a discrete random variable with a given probability function. We derived the first population moment and the first sample moment, and equated them to obtain the estimator by MOM. The estimator by MOM is given by:

\hat{\theta} = \frac{log(1-\bar{X})}{n \bar{X}}

where $\bar{X}$ is the sample mean.

References

[1] Hogg, R. V., & Craig, A. T. (1978). Introduction to mathematical statistics. Macmillan.
[2] Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the theory of statistics. McGraw-Hill.
[3] Casella, G., & Berger, R. L. (2002). Statistical inference. Duxbury.

Future Work

In future work, we can extend this result to find the estimator by MOM for other distributions. We can also investigate the properties of the estimator by MOM, such as its bias and variance.

Code

The following R code can be used to implement the estimator by MOM:

# Define the probability function
p <- function(x, theta) {
  -1 / log(1 - theta) * theta^x / x
}
mu1 <- function(theta) {
-1 / log(1 - theta) * theta / (1 - theta)
}
m1 <- function(x) {
mean(x)
}
theta_hat <- function(x) {
log(1 - m1(x)) / (length(x) * m1(x))
}
x <- rpois(100, 1)
theta_hat(x)

Note that this code is for illustrative purposes only and may not be suitable for actual use.

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Introduction

In our previous article, we discussed how to find the estimator by Method of Moments (MOM) for a discrete random variable with a given probability function. In this article, we will answer some frequently asked questions about the estimator by MOM.

Q: What is the Method of Moments?

A: The Method of Moments is a widely used technique in statistics for estimating the parameters of a distribution. It is based on the idea of equating the population moments with the sample moments.

Q: What are the population moments?

A: The population moments are defined as:

\mu_k = E(X^k)

where $E(X^k)$ is the expected value of $X^k$ .

Q: What are the sample moments?

A: The sample moments are defined as:

m_k = \frac{1}{n}\sum_{i=1}^n X_i^k

where $X_i$ are the observations in the sample.

Q: How do I find the estimator by MOM?

A: To find the estimator by MOM, you need to equate the population moments with the sample moments. In our previous article, we derived the estimator by MOM for a discrete random variable with a given probability function.

Q: What is the estimator by MOM for a discrete random variable with a given probability function?

A: The estimator by MOM for a discrete random variable with a given probability function is given by:

\hat{\theta} = \frac{log(1-\bar{X})}{n \bar{X}}

where $\bar{X}$ is the sample mean.

Q: What are the advantages of the Method of Moments?

A: The Method of Moments has several advantages, including:

It is a simple and intuitive technique.
It is widely used in statistics and econometrics.
It can be used to estimate the parameters of a distribution even when the distribution is not known.

Q: What are the disadvantages of the Method of Moments?

A: The Method of Moments has several disadvantages, including:

It assumes that the population moments are known, which may not be the case in practice.
It may not be robust to outliers or non-normality.
It may not be efficient in terms of estimating the parameters of a distribution.

Q: Can the Method of Moments be used for other distributions?

A: Yes, the Method of Moments can be used for other distributions. However, the estimator by MOM may not be as simple or intuitive as it is for a discrete random variable with a given probability function.

Q: How do I implement the Method of Moments in R?

A: The following R code can be used to implement the Method of Moments:

# Define the probability function
p <- function(x, theta) {
  -1 / log(1 - theta) * theta^x / x
}
mu1 <- function(theta) {
-1 / log(1 - theta) * theta / (1 - theta)
}
m1 <- function(x) {
mean(x)
}
theta_hat <- function(x) {
log(1 - m1(x)) / (length(x) * m1(x))
}
x <- rpois(100, 1)
theta_hat(x)

Note that this code is for illustrative purposes only and may not be suitable for actual use.

Conclusion

In this article, we answered some frequently asked questions about the estimator by Method of Moments (MOM) for a discrete random variable with a given probability function. We also provided some R code to implement the Method of Moments.

References

[1] Hogg, R. V., & Craig, A. T. (1978). Introduction to mathematical statistics. Macmillan.
[2] Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the theory of statistics. McGraw-Hill.
[3] Casella, G., & Berger, R. L. (2002). Statistical inference. Duxbury.

Future Work

In future work, we can extend this result to find the estimator by MOM for other distributions. We can also investigate the properties of the estimator by MOM, such as its bias and variance.