Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Definition 5(9.5)[Power of Test] If C is the rejection region of a test of the null hypothesis lo:0∈Θo,then the functionπ(9)=Pa(Xn∈C)is called the power of the test with the rejection region C,where Pe(.) is the probability measure when the random sample follows the distribution fxn (x",0). Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 21/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 21/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Definition 5 (9.5) [Power of Test]
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks: The power function r(0)is the probability of rejecting HIo· In Example 9.4,the power of the test statistic T(X")= vn(Xn -uo)/o is given by =P(">a） Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 22/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 22/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks:
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Definition 6 (9.6)[Type I and Type ll Errors] If Ho 0E o holds and the observed data x"falls into the critical region C,then a Type I error is made.The probability of making Type I error is ax(0)≡P(Xn∈CHo). If HA:0e6 holds and the observed data x"is in the accep- tance region,then a Type II error is made.The probability of making Type II error is B(O)≡P(Xm∈AIA) 1-P9(X”∈CIA)、 Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 23/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 23/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Definition 6 (9.6) [Type I and Type II Errors]
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks: Under Ho:0∈©o,the power functionπ(O)is the proba bility of making Type I error,namely incorrectly reject- ing a correct null hypothesis. Type I errors are unavoidable because under Ho,because a test statistic T(Xm)may still take large values with nontrivial (though small)probabilities. Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 24/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 24/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks:
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Under HA:0E e6,the power function (0)is the prob- ability of rejecting a false null hypothesis,and it is equal to 1-B(0),where B(0)is the probability of making Type II error,namely accepting an false null hypothesis. .A test is called unbiased if P[T(Xm)>cH]>PT(X")> cHo.That is,the probability to reject Ho when Ho is false is strictly larger than when it is true. Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 25/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 25/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing