An Introduction to Point Estimation in Statistics

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In Statistics, Estimation Theory and Hypothesis Testing play a major role in determining solutions to certain problems. Point estimation is one of the areas that help people involved in Statistical analysis come to conclusions regarding many different kinds of questions. Point estimation means using data to calculate the value or the point as it serves as a best guess of any given parameter that may be unknown. 

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. 

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?

The maximum likelihood method is a popularly used way to calculate point estimators. This method uses differential calculus to understand the probability function from a given number of sample parameters. 

Named after Thomas Bayes, the Bayesian method is another way using which the frequency function of a parameter can be understood. This is a more non-traditional approach. However, in this case, enough information on the distribution of the parameter is not always given but in case it is, then the estimation can be done fairly easily. 

What are the Formulae that Can be Used to Measure Point Estimators? 

Some common formulae include: 

What are the Values Needed to Calculate Point Estimators?

• The number of successes is shown by S.

• The number of trials is shown by T.

• The Z–score is shown by z. 

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.

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