An Introduction to Point Estimation in Statistics

In statistics, point estimation refers to the process of estimating an unknown parameter of a population based on a sample from that population. The parameter of interest could be a mean, a variance, a proportion, or any other characteristic that describes the population.

A point estimator is a statistic that is used to estimate the parameter of interest. For example, the sample mean is a point estimator for the population mean, the sample variance is a point estimator for the population variance, and the sample proportion is a point estimator for the population proportion.

The quality of a point estimator is measured by its bias and its precision. Bias refers to the extent to which the estimator consistently overestimates or underestimates the true value of the parameter. Precision refers to the extent to which the estimator varies from sample to sample. A good point estimator should have low bias and high precision.

The most commonly used point estimator for a population parameter is the maximum likelihood estimator (MLE), which is the value of the parameter that maximizes the likelihood function based on the observed sample data.

Other commonly used point estimators include the method of moments estimator (MME), which equates the sample moments with the corresponding population moments, and the minimum variance unbiased estimator (MVUE), which has the smallest possible variance among all unbiased estimators.

It’s important to note that point estimates are subject to sampling error, which means that the point estimate can be different from the true value of the parameter due to chance variation in the sample. Therefore, it’s often useful to also report the precision of the point estimate by calculating a confidence interval or a margin of error.

What is the Definition of Point Estimation?

Point estimators are defined as functions that can be used to find the approximate value of a particular point from a given population parameter. The sample data of a population is used to find a point estimate or a statistic that can act as the best estimate of an unknown parameter that is given for a population.

The key thing in statistical inference is, based on sample information draw conclusion about the population from where the sample was drawn.

 

There are two types of statistical inference methods. We can estimate population parameters and we test hypothesis about these parameters.

There are two ways to estimate the value of a population parameter.

The first one is so called point estimate. It is a single number that is the best guess for the population parameters. And the second one is the interval estimate. It is a range of values within which we expect the parameters to fall around.

• The statistic calculated from the sample is a point estimate of the corresponding population parameter.

For example:

– The sample average is a point estimate of the true population mean.

– The sample proportion is a point estimate of the population proportion.

 

The Standard Error (SE) of the statistic provides a measure of the precision of the estimate

– A larger SE indicates a less precise point estimate

– A smaller SE indicates a more precise point estimate

 

Let’s take an example. Imagine we would like to estimate one things.

(1) What is the average height of south Indian man?

We’re going to consider the south India as population and collected a simple random sample of 20,000 people from this population.

Point estimates

We want to estimate the population mean based on the sample. The most intuitive way to do this is to simply take the sample mean. That is, to estimate the average height of all south Indian people, take the average height for the sample. Let’s think all 20,000 samples that we collected the sample mean ¯x = 172.72 cm. Then the height 172.72 cm is called a point estimate of the population mean.  If we can only choose one value to estimate the population mean, this is our best guess.

Suppose we take a new sample of another 30,000 people and recompute the mean; we will probably not get the exact same answer that we got first time. Point estimates generally vary from one sample to another and this sampling variation suggests our estimate may be close, but it may not be exactly equal to the parameter. So, the moral of the story is point estimates are not exact and we should not expect our estimate to be very good.

What are the Properties of Point Estimators?

It is desirable for a point estimate to be the following :

• Consistent – We can say that the larger is the sample size, the more accurate is the estimate.

• Unbiased – The expectation of the observed values of various samples equals the corresponding population parameter. Let’s take, for example, We can say that sample mean is an unbiased estimator for the population mean.

• Most Efficient That is also Known as Best Unbiased – of all the various consistent, unbiased estimates, the one possessing the smallest variance (a measure of the amount of dispersion away from the estimate). In simple words, we can say that the estimator varies least from sample to sample and this generally depends on the particular distribution of the population. For example, the mean is more efficient than the median (that is the middle value) for the normal distribution but not for more “skewed” ( also known as asymmetrical) distributions

What are the Methods Used to Calculate Point Estimators?

There are various methods that can be used to calculate point estimators, including:

1. Method of moments: This method involves setting the sample moments (such as the sample mean or sample variance) equal to the corresponding population moments and solving for the parameter of interest.
2. Maximum likelihood estimation (MLE): This method involves finding the parameter value that maximizes the likelihood function, which is a measure of how well the parameter value fits the observed data.
3. Bayesian estimation: This method involves specifying a prior probability distribution for the parameter of interest and then updating this distribution based on the observed data to obtain a posterior probability distribution. The point estimator can then be calculated using the posterior distribution.
4. Least squares: This method is commonly used in regression analysis and involves finding the parameter values that minimize the sum of the squared differences between the observed data and the predicted values.
5. Quantile regression: This method involves finding the parameter values that minimize the sum of the absolute differences between the observed data and the predicted quantiles.

These are some of the most commonly used methods for calculating point estimators, but there are many other methods as well, each with its own strengths and weaknesses.

What is the Formula  to Measure Point Estimators?

The formula used to measure point estimators is the mean squared error (MSE) formula.

MSE measures the average squared difference between the estimated values and the true values of a population parameter. The MSE is calculated by subtracting the true value of the parameter from the estimator, squaring the difference, and taking the average of these squared differences.

The formula for MSE is:

MSE = E[(θ_hat – θ)^2]

where:
θ_hat is the point estimator
θ is the true value of the parameter
E is the expected value operator

By calculating the MSE, you can assess the accuracy of a point estimator. A smaller MSE indicates that the estimator is closer to the true value of the parameter, while a larger MSE indicates that the estimator is further away from the true value of the parameter.

Bias-Variance Tradeoff: Modifying an estimator to reduce its bias increases its variance, and vice versa.

Balancing bias and variance is a central issue in data science.

Unknown Parameters, Statistics, and Point Estimators