Estimation & its characteristics || Estimation & Hypothesis || Bcis Notes

Estimation

The statistical method of estimating population parameters from the sample observation drawn from the population is called estimation. The estimation of the population parameter can be done in either of the following two ways:

1. Point estimation

If a single value of sample statistics is used to estimate the true value of the population parameter then the estimation is known as point estimation.

e.g. (a) the average height of Nepalese is 5.2 ft. (b) the average time that children spend watching television 22 hours per week.

2. Interval estimation

If the population parameter is estimated by giving a certain range within which the true value of population parameters lies with a certain degree of confidence then such estimation is called interval estimation.

e.g. the average height of Nepalese is 4.9 ft to 5.3 ft.

Characteristic/Requirement/Criteria of good estimation:

A good estimator is one that is close to the true value of the population parameter as much as possible. A good estimator has the following four parameters.

1. Unbiasedness

Let t_n be the sample statistics based on ‘n’ sample observations x₁, x₂,… ….x_n. Then t_n is said to be an unbiased estimator of population parameter θ if E(t_n) = θ.

If E(t_n) > θ then t_n is said to be positively biased.

If E(t_n) > θ then t_n is said to be negatively biased.

Then, the amount of biasedness is given by b(t_n, θ) = | E(t_n) – θ|

2. Consistency

Let t_n be the sample statistics based on ‘n’ sample observations x₁, x₂,… ….x_n. Then t_n is said to be consistent estimator of population parameter θ if.

OR,

Where € is a very small positive quantity.

3. Efficiency

Let t₁ and t₂ are two consistent estimators of a population parameter θ such that var (t₁) < var (t₂) for all n. Then, t₁ is said to be more efficient than t₂. In other words, an estimator with lesser variance is the more efficient estimator.

If t is the most efficient estimator of a parameter θ with variance v and t₁ is any other estimator with variance v₁ then the efficiency E of t₁ is defined as E =

4. Sufficiency

A sample statistic is said to be sufficient estimator of population parameter if all the information about the population parameter can be obtained from that sample statistic. For example, the statistic means is use all the sample value in its computation while mean and median don’t. Hence mean is a better estimator in this case.