EconHypothesisTesting

# EconHypothesisTesting - Introductory Statistics Stats 210...

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1 Introductory Statistics Stats 210 Estimation & Hypothesis Testing

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2 Estimation Remember We have described a population We have taken a sample from the population We have calculated the sample mean Q: Why have we calculated the sample mean? A: To estimate the population mean
3 Population vs. Sample Population (random) Samples

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5 Estimation Estimator = function of the sample data to be drawn randomly from the population Note: An estimator is a random variable Estimate = numerical value of the estimator when it is actually computed using data from a specific sample Note: An estimate is a non-random number

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6 Linear Estimators We are only going to focus on linear estimators Linear estimators are of the following form The estimator is a linear combination of the sample observations Neither the weights nor the sample observations are non-linear functions (i.e. x 2 , x 3 , 1/x, log(x)) N N x a x a x a x a X + + + + = ......... ˆ 3 3 2 2 1 1
7 Estimators We used the sample mean to estimate the population mean However, there are many other linear estimators we could have used Assume we draw a sample of size N=4 Possible linear estimators: 2 4 1 x x X + = ± 2 4 3 2 1 x x x x X + + + = ² 4 3 2 1 3 4 3 3 3 2 3 1 ˆ x x x x X + + + = = = N i i x N X 1 1

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8 Estimators In general we can summarize all linear estimators as: For the sample mean we have for all i = = N i i i x a X 1 ~ N a i 1 =
9 Estimators How do we choose between alternative estimators? We will look at the following properties of estimators Bias/Unbiasedness Efficiency Mean Square Error Consistency

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10 Unbiasedness Definition = An estimator, , is unbiased if its expected value equals the true population parameter Example: We have shown that the sample mean is an unbiased estimator of the true population mean ( ) Y Y E μ = ( ) θθ = ˆ E ˆ θ
11 Bias Let’s look at a biased estimator As always we are taking a i.i.d random sample (N=4) 4 3 2 1 8 1 4 1 4 1 8 1 ˆ x x x x X + + + = () 4 3 2 1 8 1 4 1 4 1 8 1 ˆ x E x E x E x E X E + + + = X X μ = + + + = 75 . 0 8 1 4 1 4 1 8 1

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12 Bias Let’s look at the expected value of the general form of the linear estimator ( ) X X E Bias μ = ˆ X X X = = 25 . 0 75 . 0
13 Mean of General Linear Estimator As before assume i.i.d. sample Let’s check for bias Result: Unbiased as long as = = N i i i x a X 1 ~ () = = = = = = N i i Y N i i i N i i i a x E a x a E X E 1 1 1 ) ( ~ μ 1 1 = = N i i a

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14 Efficiency Definition = an estimator is more efficient than an alternative estimator if its variance is smaller than the variance of the alternative estimator Let’s check out how the sample mean compares to the general linear estimator in terms of Variance
15 Efficiency: Sample Mean What is the variance of the general estimator Compare with variance of sample mean To have a smaller variance than the sample mean we need () = = =

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EconHypothesisTesting - Introductory Statistics Stats 210...

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