Estimating the Scale Parameters of a Normal Distribution with gmm Package

As a data analyst or researcher, working with normal distributions is common in various fields such as statistics, engineering, and economics. One of the key aspects of working with normal distributions is estimating its scale parameters, which can be challenging when dealing with high-dimensional data.

In this article, we’ll delve into the world of maximum likelihood estimation (MLE) using the Generalized Method of Moments (GMM) package in R to estimate the scale parameters of a normal distribution. We’ll explore how to specify moment conditions for the normal distribution, how to identify and over-identify moment conditions, and how to use GMM to estimate the scale parameters.

Introduction to the Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, with the majority of data points clustered around the mean. The normal distribution has two primary scale parameters: the mean (μ) and the standard deviation (σ).

In this article, we’ll focus on estimating the standard deviation (σ) using the GMM package in R.

Moment Conditions for the Normal Distribution

Moment conditions are mathematical equations that describe how a sample of data should be related to the true parameters of a distribution. In the context of normal distributions, moment conditions can be derived from the first two moments of the distribution.

The first moment condition is based on the expected value of the variable:

E(x) = μ

The second moment condition is based on the variance of the variable:

Var(x) = σ^2

These two moment conditions are often referred to as the “standard” or “basic” moment conditions for a normal distribution.

Identifying Moment Conditions

Identifying moment conditions involves specifying functions that map the sample data to the true parameters. In this case, we have two moment conditions:

g0(x) = (x - μ)

g1(x) = (x - μ)^2 - σ^2

These functions are often referred to as the “identifying” or “first-order” moment conditions.

Over-Identifying Moment Conditions

Over-identifying moment conditions involve specifying additional moment conditions beyond the minimum required by the model. In this case, we have an additional moment condition:

g2(x) = (x - μ / σ)^3

This function is often referred to as the “second-order” or “over-identified” moment condition.

Using GMM to Estimate Scale Parameters

The Generalized Method of Moments (GMM) package in R provides a flexible framework for estimating model parameters using moment conditions. In this section, we’ll show how to use GMM to estimate the standard deviation (σ) of a normal distribution.

First, we need to specify the moment conditions:

g0(x) = (x - μ)

g1(x) = (x - μ)^2 - σ^2

We can define these functions in R as follows:

g0 <- function(tet, x) {
  +     m1 <- (tet[1] - x)
  +     m2 <- (tet[2]^2 - (x - tet[1])^2)
  +     f <- cbind(m1, m2)
  +     return(f)

print(res0 <- gmm(g0, x, c(mu = 0, sig = 0)))

In this code snippet, we define the g0 function, which takes two inputs: tet and x. The tet vector contains the true parameters (μ and σ), while the x vector contains the sample data.

We then call the gmm function from the GMM package, passing in the g0 function, the sample data x, and the initial values for the true parameters (c(mu = 0, sig = 0)).

The output of this code snippet is a summary object that contains the estimated parameters.

Interpreting the Results

Let’s summarize the estimated parameters:

summary(res0)

This code snippet prints out a summary of the estimated parameters, including the estimated standard deviation (σ).

Note that the gmm function returns an estimate of the variance-covariance matrix of the moment conditions. We can extract the estimated standard deviation from this matrix as follows:

sigma_hat <- sqrt(diag(res0$var))

This code snippet extracts the diagonal elements of the variance-covariance matrix and takes their square root to obtain the estimated standard deviation.

Conclusion

In this article, we explored how to estimate the scale parameters of a normal distribution using the Generalized Method of Moments (GMM) package in R. We specified moment conditions for the normal distribution, identified and over-identified moment conditions, and used GMM to estimate the standard deviation.

We hope that this article has provided you with a solid understanding of how to use GMM to estimate model parameters in R.

Last modified on 2023-10-27