Illustration of CLT: Poisson Sampling

Transcription

1 Illustration of CLT: Poisson Sampling Justin L. Tobias 1 1 Iowa State University Department of Economics September 23, 2007 Tobias (ISU) Econ 671 September 23, / 15

2 As argued in class, a central limit theorem is a powerful tool for approximating sampling distributions in finite samples. Tobias (ISU) Econ 671 September 23, / 15

3 As argued in class, a central limit theorem is a powerful tool for approximating sampling distributions in finite samples. In some cases, the finite sample behavior of an estimator is difficult to ascertain (or the model is not rich enough to even allow for the possiblility), and thus a large-sample approximation can be used for testing and inference purposes. Tobias (ISU) Econ 671 September 23, / 15

4 As argued in class, a central limit theorem is a powerful tool for approximating sampling distributions in finite samples. In some cases, the finite sample behavior of an estimator is difficult to ascertain (or the model is not rich enough to even allow for the possiblility), and thus a large-sample approximation can be used for testing and inference purposes. Here we provide a simple example of the CLT approximation when the (finite sample) sampling distribution can be analytically obtained. Tobias (ISU) Econ 671 September 23, / 15

5 As argued in class, a central limit theorem is a powerful tool for approximating sampling distributions in finite samples. In some cases, the finite sample behavior of an estimator is difficult to ascertain (or the model is not rich enough to even allow for the possiblility), and thus a large-sample approximation can be used for testing and inference purposes. Here we provide a simple example of the CLT approximation when the (finite sample) sampling distribution can be analytically obtained. We then compare the exact results to those based on the CLT. Tobias (ISU) Econ 671 September 23, / 15

6 Consider the random variable Tobias (ISU) Econ 671 September 23, / 15

7 Consider the random variable Y T = X 1 + X X T, Tobias (ISU) Econ 671 September 23, / 15

8 Consider the random variable Y T = X 1 + X X T, where X t iid Poisson(λ) t = 1, 2,..., T. Tobias (ISU) Econ 671 September 23, / 15

9 Consider the random variable Y T = X 1 + X X T, where That is, X t iid Poisson(λ) t = 1, 2,..., T. p(x t ) = λxt exp( λ), x t = 0, 1, 2,..., t. x t! Tobias (ISU) Econ 671 September 23, / 15

10 Consider the random variable where That is, Y T = X 1 + X X T, X t iid Poisson(λ) t = 1, 2,..., T. p(x t ) = λxt exp( λ), x t = 0, 1, 2,..., t. x t! Using the moment generating function approach (as shown in class), we know that Y T Poisson(T λ). Tobias (ISU) Econ 671 September 23, / 15

11 Thus, Pr(Y T = c) = [T λ]c exp( T λ), c = 0, 1, 2,.... c! Tobias (ISU) Econ 671 September 23, / 15

12 Thus, Pr(Y T = c) = [T λ]c exp( T λ), c = 0, 1, 2,.... c! Now, consider the distribution of the sample average, which we define as W T : W T 1 T Y T. Tobias (ISU) Econ 671 September 23, / 15

13 Thus, Pr(Y T = c) = [T λ]c exp( T λ), c = 0, 1, 2,.... c! Now, consider the distribution of the sample average, which we define as W T : W T 1 T Y T. Note that W T can take values in the set {0, 1 T, 2 T,..., 1, T + 1 T,...} so that the support of f WT (w T ) gets finer and finer as T increases. Tobias (ISU) Econ 671 September 23, / 15

14 In addition, note Tobias (ISU) Econ 671 September 23, / 15

15 In addition, note Pr(Y T = c) = Pr(TW T = c) Tobias (ISU) Econ 671 September 23, / 15

16 In addition, note Pr(Y T = c) = Pr(TW T = c) = Pr(W T = c/t ) Tobias (ISU) Econ 671 September 23, / 15

17 In addition, note Pr(Y T = c) = Pr(TW T = c) = Pr(W T = c/t ) = [T λ]c exp( T λ), c = 0, 1, 2,... c! Tobias (ISU) Econ 671 September 23, / 15

18 In addition, note Pr(Y T = c) = Pr(TW T = c) = Pr(W T = c/t ) = [T λ]c exp( T λ), c = 0, 1, 2,... c! We can compute these probabilities c to give the (exact) sampling distribution of the sample average for various values of T. Tobias (ISU) Econ 671 September 23, / 15

19 Note, also that E(X i ) = λ, Var(X i ) = λ (known moments of the Poisson) so that the Lindberg-Levy CLT gives [ ] WT λ d T N (0, 1). λ Tobias (ISU) Econ 671 September 23, / 15

20 Note, also that E(X i ) = λ, Var(X i ) = λ (known moments of the Poisson) so that the Lindberg-Levy CLT gives [ ] WT λ d T N (0, 1). λ To compare the normal approximation to the exact finite sample result, let ψ T [ ] WT λ T. λ Tobias (ISU) Econ 671 September 23, / 15

21 We note Tobias (ISU) Econ 671 September 23, / 15

22 We note Pr(Y T = c) = Pr(W T = c/t ) Tobias (ISU) Econ 671 September 23, / 15

23 We note Pr(Y T = c) = Pr(W T = c/t ) ( T [ c ] ) = Pr ψ T = λ T λ, which we plot over c = 0, 1, 2,.... Tobias (ISU) Econ 671 September 23, / 15

24 We note Pr(Y T = c) = Pr(W T = c/t ) ( T [ c ] ) = Pr ψ T = λ T λ, which we plot over c = 0, 1, 2,.... We then compare this (exact) finite sample result with the large sample standard normal approximation. Tobias (ISU) Econ 671 September 23, / 15

25 We note Pr(Y T = c) = Pr(W T = c/t ) ( T [ c ] ) = Pr ψ T = λ T λ, which we plot over c = 0, 1, 2,.... We then compare this (exact) finite sample result with the large sample standard normal approximation. We set λ = 1 to fix ideas. Tobias (ISU) Econ 671 September 23, / 15

26 T=1 Sampling Dist. Std. Normal Tobias (ISU) Econ 671 September 23, / 15

27 Note that, when T = 1 and λ = 1, Tobias (ISU) Econ 671 September 23, / 15

28 Note that, when T = 1 and λ = 1, T [ c ] ψ T = λ T λ Tobias (ISU) Econ 671 September 23, / 15

29 Note that, when T = 1 and λ = 1, ψ T = T [ c ] λ T λ = c λ Tobias (ISU) Econ 671 September 23, / 15

30 Note that, when T = 1 and λ = 1, ψ T = T [ c ] λ T λ = c λ so that the range of possible values for ψ 1 is { 1, 0,...}, as shown on the previous figure. Tobias (ISU) Econ 671 September 23, / 15

31 T=2 Sampling Dist. Std. Normal Tobias (ISU) Econ 671 September 23, / 15

32 Note that, when T = 2 and λ = 1, Tobias (ISU) Econ 671 September 23, / 15

33 Note that, when T = 2 and λ = 1, T [ c ] ψ T = λ T λ Tobias (ISU) Econ 671 September 23, / 15

34 Note that, when T = 2 and λ = 1, T [ c ] ψ T = λ T λ = 2[(c/2) 1] so that the range of possible values for ψ 2 is { 2, 1/ 2, 0, 1/ 2,...}. Tobias (ISU) Econ 671 September 23, / 15

35 Note that, when T = 2 and λ = 1, T [ c ] ψ T = λ T λ = 2[(c/2) 1] so that the range of possible values for ψ 2 is { 2, 1/ 2, 0, 1/ 2,...}. The range for ψ T continutes to spread out as T grows. For T = 4, for example, ψ 4 { 2, 3/2, 1, 1/2, 0, 1/2,...}. Tobias (ISU) Econ 671 September 23, / 15

36 T=5 Sampling Dist. Std. Normal Tobias (ISU) Econ 671 September 23, / 15

37 T=25 Sampling Dist. Std. Normal Tobias (ISU) Econ 671 September 23, / 15

38 Note that the shape of the bar graph is approaching the shape of the normal density. Tobias (ISU) Econ 671 September 23, / 15

39 Note that the shape of the bar graph is approaching the shape of the normal density. The fact that the bar graph is smaller is an artifact of discrete probabilities versus a continuous density. In the bar graph, the probabilities sum to one, while the area under the normal curve integrates to one. Tobias (ISU) Econ 671 September 23, / 15

40 Note that the shape of the bar graph is approaching the shape of the normal density. The fact that the bar graph is smaller is an artifact of discrete probabilities versus a continuous density. In the bar graph, the probabilities sum to one, while the area under the normal curve integrates to one. When rescaling the bar graph so that it integrates to one, we obtain the following figure: Tobias (ISU) Econ 671 September 23, / 15

41 T=25, Scaled up by 5 Sampling Dist. Std. Normal Tobias (ISU) Econ 671 September 23, / 15