# Topic 9: Sampling Distributions of Estimators

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1 Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0

2 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be derived from the joit distributio of X 1... X. It is called the samplig distributio because it is based o the joit distributio of the radom sample. Give a samplig distributio, we ca make appropriate trade-offs betwee sample size ad precisio of our estimator sice samplig distributios o sample size. obtai iterval estimates rather tha poit estimates after we have a sample- a iterval estimate is a radom iterval such that the true parameter lies withi this iterval with a give probability (say 95%). choose betwee to estimators- we ca, for istace, calculate the mea-squared error of the estimator, E θ [( ˆθ θ) 2 ] usig the distributio of ˆθ. Page 1

3 Examples: Applicatio: sample size ad precisio 1. What if X i N(θ, 4), ad we wat E( X θ) 2.1? This is simply the variace of X, ad we kow X N(θ, 4/). 4.1 if Cosider a radom sample of size from a Uiform distributio o [0, θ], ad the statistic U = max{x 1,..., X }. The CDF of U is give by: 0 if u 0 ( ) F(X) = uθ if 0 < u < θ 1 if u θ We ca ow use this to see how large our sample must be if we wat a certai level of precisio i our estimate for θ. Suppose we wat the probability that our estimate lies withi.1θ for ay level of θ to be bigger tha 0.95: Pr( U θ.1θ) = Pr(θ U.1θ) = Pr(U.9θ) = 1 F(.9θ) = We wat this to be bigger tha 0.95, or With the LHS decreasig i, we choose log(.05) log(.9) = Our miimum sample size is therefore 29. Page 2

4 Joit distributio of sample mea ad sample variace For a radom sample from a ormal distributio, we kow that the M.L.E.s are the sample mea ad the sample variace 1 (X i X ) 2. We kow that X N(µ, σ2 ) ad ( X i µ σ )2 χ 2 ( sum of squares of stadard ormals) If we replace the populatio mea µ with the sample mea X, the resultig sum of squares, has a χ 2 1 distributio, which is idepedet of the distributio of X. This is stated formally below: Theorem: If X 1,... X form a radom sample from a ormal distributio with mea µ ad variace σ 2, the the sample mea X ad the sample variace 1 (X i X ) 2 are idepedet radom variables ad X N(µ, σ2 ) (X i X ) 2 σ 2 χ 2 1 Note: This is oly for ormal samples. Page 3