The process of drawing a sample from a population and to gather information about the parameter through a reasonably close function is known as estimation. This is done in order to find out the unknown parameters which we encounter in a population and which hinders our estimation and conclusion about a population. Thus this process is used to find out the parameters. The value obtained from such a process is known as estimated value and the function is called estimator.
An estimator is supposed to possess the qualities stated below to perform well. They are:
 Un biasedness
A statistic t is an unbiased estimator of a parameter , if E[t] =
If not, then the estimator is biased.
There are quite a few theorems stated below which proves this.
Theorem 1:
Prove that the sample mean x is an unbiased estimator of the population mean µ
Proof:
Let x₁, x₂, …, , be a simple random sample with replacement from a finite population of size N, say, X₁, X₂,… ,
µ=
Prove E(x) =µ
While drawing xi, it can be one of the population members i.e., the probability distribution of Xi can be taken as follows:
X₁ 
X₂ 
… 

probability 
1/N 
1/N 
1/N 
Therefore,
E ( = X₁* + X₂* +….+ *
=
= µ, i= 1, 2,…., n
E ( = E [( ]
= [E (
If the population is finite or the sampling is done without replacement, the same result will be obtained.
Theorem 2:
The sample variance S²= is a biased estimator of variance σ².
Proof:
Let x₁, x₂, …, be a random sample from an infinite population with mean σ and variance σ².
Then, E (x) = µ, Variance (x_{i}) = E (xi – µ)² = σ², where i= 1, 2, … ,n.
S²=
=
= where xi – µ and standard deviation is unaffected by change in origin.
= – µ)²
E (s²) = – µ)²
= )= σ²
Thus, s² is a biased estimator of σ².
Also, Let S²=
E (s²) =
= σ²
= σ²
Therefore, s² is a biased estimator of σ².
Example: A population consists of 4 values 3, 7, 11, 15. Draw all possible sample of size two with replacement. Verify that the sample mean is an unbiased estimator of the population mean.
Solution:
No. of samples = 42 = 16, which are as below:
(3, 3), (7, 3), (11, 3), (15, 3)
(11 , 7), (15, 7), (11 , 11), (15, 11)
(11, 15), (15, 15), (3, 7), (7, 7),
(3, 11), (7, 11), (3, 15), (7, 15)
Population mean µ= = = 9
Sampling distribution of mean
Sample mean 
Frequency f 
.f 
3  1  3 
5  2  10 
7  3  21 
9  4  36 
11  3  33 
13  2  26 
15  1  15 
Total  16  144 
Mean of sample = = 9
Since E ()= µ
Therefore, sample mean is an unbiased estimator of population mean.
 Consistency
A statistic obtained from a random sample of size n is said to be a consistent estimator of a parameter if it converges in probability to θ as n tends to infinity.
Alternatively, If E [ ] θ and Var [ ] 0 as n ∞, then the statistic is said to be consistent estimator of θ.
Example:
When sampling from a population N,
E )= µ and )= → 0 as n→
Therefore, sample mean is a consistent estimator of population mean.
 Efficiency
A parameter might comprise of more than one consistent estimator. Let T_{1} and T_{2} be two consistent estimators of a parameter θ. If Var (T₁) < Var (T₂) for all n, then T₁ is said to be more efficient than T₂ for all size.
 Sufficiency
Let x₁, x ₂, … , be a random sample from a population whose pmfor pdfisf (x, 8). Then T is said to be a sufficient estimator of e if f (x₁, ). f(x₂, )…..f( , ) = g_{1}(T , ). g₂( x_{1},x_{2},…,
Where g_{1}(T , ) is the sampling distribution and g₂( x_{1},x_{2},…, is independent of .
Even though sufficient estimators exists only in certain cases, but when random sampling for a normal population, the sampling mean x is a sufficient estimator of µ.
Links of Previous Main Topic:
 Introduction to statistics
 Knowledge of central tendency or location
 Definition of dispersion
 Moments
 Bivariate distribution
 Theorem of total probability addition theorem
 Random variable
 Binomial distribution
 What is sampling
Links of Next Statistics Topics:
 Statistical hypothesis and related terms
 Analysis of variance introduction
 Definition of stochastic process
 Introduction operations research
 Introduction and mathematical formulation in transportation problems
 Introduction and mathematical formulation
 Queuing theory introduction
 Inventory control introduction
 Simulation introduction
 Time calculations in network
 Introduction of game theory