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![ Consistency
An estimator is said to be consistent if the statistic to be used as estimator
becomes closer and closer to the population parameter being estimator as the
sample size n increases.
lim
𝑛→∞
𝑃[|Ô − θ| ≤ 𝑒] = 1
A consistent estimator may or may not be unbiased.
The sample mean X bar=
1
𝑛 𝑖=1
𝑛
𝑥 which is an unbiased estimator of µ, is a
consistent estimator of the mean µ.](https://image.slidesharecdn.com/pointestimation-150517082733-lva1-app6891/85/Point-estimation-11-320.jpg)
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Point estimation
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- 4. Contents Statistical Inference Estimation A. Point Estimation B. Estimator C. Estimate Criteria of a good Estimator A. Unbiasedness B. Consistency C. Efficiency D. Sufficiency
- 5. Statistical Inference Usually thepopulation is not known completely we can obtain information about population parameters by use of samples drawn from it. Statistical inference deals with such problems.it is defined as “The art of drawing conclusions or inferences about the unknown parameters of the populations from the limited information contained in the sample” Two important parts of statistical inference are: 1. Estimation of parameter 2. Testing of Hypotheses
- 6. Estimation It is aprocedure of making judgment about the true but unknown values of the population parameters from the sample observations It is further divided into two parts Point Estimation Interval Estimation
- 7. Point Estimation If weexpress an estimate by a single value, it is called point Estimation. For example the value of X bar (the sample mean) computed from a sample of size n is a point estimate of the population parameter µ Estimator A rule used to estimate a numerical value is called estimator. The estimator of mean is given below X bar= 𝑖=1 𝑛 𝑋 𝑛
- 8. Estimate An estimate isa numerical value of the unknown parameter obtained by applying a formula (estimator) to a particular sample. If θ is a parameter Ô denotes it’s estimate Example: Let a sample of size 5 be 2,4,5,9,10.then an estimate of the population mean µ,obtained by applying an estimator X bar= 𝑖=1 𝑛 𝑋 𝑛 Estimator X bar= 2+4+5+9+10 5 X bar= 30 5 =6 Estimate
- 9. Criteria for goodpoint Estimators A point estimator is considered a good estimator if it is satisfies various criteria. Four of criteria are A. Unbiasedness B. Consistency C. Efficiency D. Sufficiency
- 10. Unbiasedness The biasof an estimator Ô is defines as the difference between it’s expected value and the true parameter θ Bias=E(Ô) – θ An estimator is defined to be unbiased if the statistic used an estimator has it’s expected value equal to the true value of the population parameter being estimated. E(Ô)=θ The estimator is defined to be positively biased when E(Ô)>θ The estimator is defined to be negatively biased when E(Ô)<θ
- 11. Consistency An estimatoris said to be consistent if the statistic to be used as estimator becomes closer and closer to the population parameter being estimator as the sample size n increases. lim 𝑛→∞ 𝑃[|Ô − θ| ≤ 𝑒] = 1 A consistent estimator may or may not be unbiased. The sample mean X bar= 1 𝑛 𝑖=1 𝑛 𝑥 which is an unbiased estimator of µ, is a consistent estimator of the mean µ.
- 12. To prove thatan estimator is consistent, we may state a criterion that is sometimes quite useful, as follows “Let Ô be an estimator of θ based on a sample of size n. Then Ô is a consistent estimator of θ,if Var(Ô) approaches 0 as n approaches infinity.”
- 13. Efficiency An unbiasedestimator is defined to be efficient if the variance of its sampling distribution is smaller than that of sampling distribution of any other any other unbiased estimator of the same parameter .IN other words suppose there are two unbiased estimator T1 and T2 of the same parameter θ , then T1 will be said to be more efficient estimator than T2.If Var(T1)<Var(T2). i. An estimator is more efficient if it is unbiased as well as has the minimum variance as compared with any other unbiased estimator ii. The relative efficiency of T1 compare to T2 is given by the ratio Ef=Var(T2)/Var(T1) which is greater than 1
- 14. Mean Var(x bar)= σ2 n Median Var(x bar)= 𝝅 𝟐 . σ2 n =(1.57) σ2 n
- 15. Sufficiency An estimatoris defined to be sufficient if the statistic used as estimator uses all the information that contained in the sample. Any statistic that is not computed from all the values in the sample is not a sufficient estimator. The sample mean X bar is a sufficient estimator of µ.This implies that x bar contains all the information in the sample relative to the estimation of population parameter µ and no other estimator such as the sample median etc.
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