Point Estimation in Statistics

What is Point Estimation?

In statistics, point estimation basically involves the use of sample data to calculate a single value (which is also known as a point estimate since it identifies a point in some parameter space) and it is to serve as a "best estimate" or "best guess" of any given unknown population parameter. Point estimation is known to be the application of a point estimator to the data to acquire a certain point estimate.

Definition of Point Estimation

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.

Various Methods Used to Calculate the Estimator

Several methods are used to calculate the estimator.

The maximum likelihood method is a frequently used method that uses differential calculus to determine the maximum of the probability function of a number of sample parameters. 

The Bayesian method is a method which is named after the mathematician Thomas Bayes, differs from the traditional methods by introducing a frequency function for the parameter which is being estimated. 

Let’s discuss one of the known drawbacks to the Bayesian method is that sufficient information on the distribution of the parameter is usually not obtainable and one advantage of the Bayesian method is that the estimation can be easily adjusted as additional information becomes obtainable. 

What are Point Estimators?

  • Basically point estimators are known to be functions that are used to find an approximate or an estimated value of a population parameter from various random samples of the population. 

  • To calculate a point estimate point estimators generally use the sample data of a population or they use a statistic that serves as the best estimate of an unknown parameter of a given population.

Properties of Point Estimators

The following are the important characteristics of point estimators:

  • Bias

We can define the bias of a point estimator as the difference between the expected value of the estimator as well as the value of the parameter being estimated. When the estimated value of the parameter as well as the value of the parameter being estimated is equal, then we can say that the estimator is unbiased.

Also, the closer is the expected value of a parameter to the value of the parameter that is to be measured, the lesser is the bias.

  • Consistency

Generally, consistency tells us how close the point estimator stays to the value of the parameter as it grows in size. The point estimator generally requires a huge sample size for it to be more consistent and for it to be more accurate. 

We can also check whether a point estimator is consistent or not by looking at its corresponding expected value as well as its variance. A point estimator is said to be consistent, the value that is expected should move towards the true or actual value of the parameter.

  • Most Efficient or Unbiased

The most efficient point estimator is the one with the smallest variance of all the unbiased as well as consistent estimators. Generally, the variance measures the level of dispersion from the estimate, as well as the smallest variance, which should vary the least from one sample to the other.

Generally, the efficiency of the estimator is said to be dependent on the distribution of the population. 

Point Estimate Formulas

Four different point estimate formulas can be used:

  1.  Maximum Likelihood Estimation (MLE)

  2.  Wilson Estimation, Laplace Estimation

  3.  Jeffrey Estimation

To Calculate the Point Estimate, You Will Need the Following Values That are Listed below:

  1. The number of successes, denoted by S: for example, the number of heads you got while tossing the coin.

  2. The number of trials denoted by T: in the coin example, it's the total number of tosses.

  • Confidence interval: that is the probability that your best point estimate is correct (within the margin of error).

  1. Z-score, denoted by z: it will be calculated automatically from the confidence interval.

Once You Know All the Values Listed Above, You Can Start Calculating the Point Estimate According to the Following Given Equations:

  • Maximum Likelihood Estimation: MLE = S / T

  • Laplace Estimation: Laplace equals (S + 1) / (T + 2)

  • Jeffrey Estimation: Jeffrey equals (S + 0.5) / (T + 1)

  • Wilson Estimation: Wilson equals (S + z²/2) / (T + z²)

Once All Four Values have been Calculated, You Need to Choose the Most Accurate One. This Should be done According to the Following Rules Listed below:

  • If the value of  MLE ≤ 0.5, the Wilson Estimation is the most accurate.

  • If the value of MLE - 0.5 < MLE < 0.9, then the Maximum Likelihood Estimation is the most accurate.

  • If 0.9 < MLE, then the smaller of Jeffrey and Laplace Estimations is said to be the most accurate.

FAQs (Frequently Asked Questions)

Question 1) What is Point Estimation and Interval Estimation?

Answer) A point estimate can be defined as a single value estimate of a parameter. For instance, a sample mean is also known to be a point estimate of a population mean. An interval estimate generally gives you a range of values where the parameter is expected to lie and a confidence interval is the most common type of interval estimate.

Question 2) What is the Purpose of Point Estimation? What is a Point Estimator?

Answer) Basically point estimators are known to be functions that are used to find an approximate or an estimated value of a population parameter from various random samples of the population.