Confidence Interval For The Weighted Sum Of Two Binomial Proportions Wojciech Zieli´ nski Department of Econometrics and Statistics SGGW XLII Wi´ slana Konferencja „Statystyka Matematyczna” B˛ edlewo, 2 grudnia 2016 Wojciech Zieli´ nski Confidence Interval For...
Probability θ of success is estimated. Wojciech Zieli´ nski Confidence Interval For...
Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Wojciech Zieli´ nski Confidence Interval For...
Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Let ξ be the number of successes. Wojciech Zieli´ nski Confidence Interval For...
Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Let ξ be the number of successes. ξ is a random variable binomially distributed. Wojciech Zieli´ nski Confidence Interval For...
Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Let ξ be the number of successes. ξ is a random variable binomially distributed. The statistical model ( { 0 , 1 , . . . , n } , { Bin ( n, θ ) , θ ∈ (0 , 1) } ) Wojciech Zieli´ nski Confidence Interval For...
UMV ( θ ) is ˆ θ c = ξ n Variance of that estimator equals θ c = θ (1 − θ ) θ ˆ D 2 for all θ. n Wojciech Zieli´ nski Confidence Interval For...
The Clopper-Pearson symmetric confidence interval at confidence level γ for θ has the form ( θ c L ( u ) , θ c U ( u )) , where � 0 for u = 0 , θ c L ( u ) = B − 1 � � n (1 − u ) + 1 , nu ; 1+ γ for u > 0 , 2 � 1 for u = 1 , θ c U ( u ) = B − 1 � � n (1 − u ) , nu + 1; 1 − γ for u < 0 . 2 Here B − 1 ( · , · ; · ) is the quantile of the beta distribution. Wojciech Zieli´ nski Confidence Interval For...
Suppose θ = w 1 θ 1 + w 2 θ 2 w 1 , w 2 ∈ (0 , 1) w 1 + w 2 = 1 Wojciech Zieli´ nski Confidence Interval For...
Motivation Wojciech Zieli´ nski Confidence Interval For...
Motivation Suppliers Wojciech Zieli´ nski Confidence Interval For...
Motivation Suppliers DECROUEZ, G. & ROBINSON, A. P . (2012) Aust. N. Z. J. Stat. Wojciech Zieli´ nski Confidence Interval For...
Motivation Suppliers DECROUEZ, G. & ROBINSON, A. P . (2012) Aust. N. Z. J. Stat. Stratification Wojciech Zieli´ nski Confidence Interval For...
Je˙ zeli ζ n jest zmienn ˛ a losow ˛ a o rozkładzie Bin ( n, π ) , to ( ∀ π ) ( ∀ ε > 0) ( ∃ N ( π )) ( ∀ n > N ( π )) � � � � ζ n − nπ � � sup � P ≤ x − Φ( x ) � < ε � � � � � nπ (1 − π ) x ∈ R Wojciech Zieli´ nski Confidence Interval For...
Je˙ zeli ζ n jest zmienn ˛ a losow ˛ a o rozkładzie Bin ( n, π ) , to ( ∀ π ) ( ∀ ε > 0) ( ∃ N ( π )) ( ∀ n > N ( π )) � � � � ζ n − nπ � � sup � P ≤ x − Φ( x ) � < ε � � � � � nπ (1 − π ) x ∈ R ale nie zachodzi ( ∀ ε > 0) ( ∃ N ) ( ∀ n > N ) ( ∀ π ) � � � � ζ n − nπ � � sup � P ≤ x − Φ( x ) � < ε � � � � � nπ (1 − π ) x ∈ R Wojciech Zieli´ nski Confidence Interval For...
Je˙ zeli ζ n jest zmienn ˛ a losow ˛ a o rozkładzie Bin ( n, π ) , to ( ∀ π ) ( ∀ ε > 0) ( ∃ N ( π )) ( ∀ n > N ( π )) � � � � ζ n − nπ � � sup � P ≤ x − Φ( x ) � < ε � � � � � nπ (1 − π ) x ∈ R ale nie zachodzi ( ∀ ε > 0) ( ∃ N ) ( ∀ n > N ) ( ∀ π ) � � � � ζ n − nπ � � sup � P ≤ x − Φ( x ) � < ε � � � � � nπ (1 − π ) x ∈ R R. Zieli´ nski, Applicationes Mathematicae (2004) Wojciech Zieli´ nski Confidence Interval For...
θ = w 1 θ 1 + w 2 θ 2 , w 1 , w 2 ∈ (0 , 1) , w 1 + w 2 = 1 Wojciech Zieli´ nski Confidence Interval For...
θ = w 1 θ 1 + w 2 θ 2 , w 1 , w 2 ∈ (0 , 1) , w 1 + w 2 = 1 Let n = n 1 + n 2 . Wojciech Zieli´ nski Confidence Interval For...
θ = w 1 θ 1 + w 2 θ 2 , w 1 , w 2 ∈ (0 , 1) , w 1 + w 2 = 1 Let n = n 1 + n 2 . There are two random variables ξ 1 ∼ Bin ( n 1 , θ 1 ) , ξ 2 ∼ Bin ( n 2 , θ 2 ) , Wojciech Zieli´ nski Confidence Interval For...
θ = w 1 θ 1 + w 2 θ 2 , w 1 , w 2 ∈ (0 , 1) , w 1 + w 2 = 1 Let n = n 1 + n 2 . There are two random variables ξ 1 ∼ Bin ( n 1 , θ 1 ) , ξ 2 ∼ Bin ( n 2 , θ 2 ) , Let ξ 1 ξ 2 ˆ θ w = w 1 + w 2 n 1 n 2 Wojciech Zieli´ nski Confidence Interval For...
The estimator ˆ θ w is an unbiased estimator of θ . Wojciech Zieli´ nski Confidence Interval For...
The estimator ˆ θ w is an unbiased estimator of θ . For a given θ there are infinitely many θ 1 and θ 2 giving θ . Hence we average with respect to θ 1 assuming the uniform distribution of θ 1 on the interval ( a θ , b θ ) , where � 0 , θ − w 2 � � 1 , θ � a θ = max ; b θ = min . w 1 w 1 Wojciech Zieli´ nski Confidence Interval For...
� ξ 1 ξ 2 � E θ ˆ θ w = E θ w 1 + w 2 n 1 n 2 � b θ � w 1 � 1 E θ 1 ξ 1 + w 2 = E θ − w 1 θ 1 ξ 2 dθ 1 b θ − a θ n 1 n 2 a θ w 2 � b θ 1 � w 1 n 1 θ 1 + w 2 � θ − w 1 θ 1 �� = n 2 dθ 1 b θ − a θ n 1 n 2 w 2 a θ = θ. Wojciech Zieli´ nski Confidence Interval For...
Let ˆ θ w = u be observed. The (symmetric) confidence interval for θ at confidence level γ is ( θ w L ( u ) , θ w U ( u )) , where � 0 for u = 0 , θ w L ( u ) = max { θ : P θ { ˆ θ w < u }} = 1+ γ for u > 0 , 2 � 1 for u = 1 , θ w U ( u ) = for u < 1 . . min { θ : P θ { ˆ θ w ≤ u }} = 1 − γ 2 Wojciech Zieli´ nski Confidence Interval For...
P θ { ˆ θ w ≤ u } � ξ 1 ξ 2 � = P θ w 1 + w 2 ≤ u n 1 n 2 � b ( θ ) n 2 1 � ξ 1 ≤ n 1 � i 2 �� � = P θ 1 u − w 2 P θ 2 { ξ 2 = i 2 } dθ 1 L ( θ ) w 1 n 2 a ( θ ) i 2 =0 θ 2 = ( θ − w 1 θ 1 ) /w 2 Wojciech Zieli´ nski Confidence Interval For...
P θ { ˆ θ w < u } � ξ 1 ξ 2 � = P θ w 1 + w 2 < u n 1 n 2 � b ( θ ) n 2 1 � ξ 1 ≤ n 1 � i 2 � � � = P θ 1 u − w 2 − 1 P θ 2 { ξ 2 = i 2 } dθ 1 L ( θ ) w 1 n 2 a ( θ ) i 2 =0 θ 2 = ( θ − w 1 θ 1 ) /w 2 Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
For given θ ∈ (0 , 1) , the expected length of the confidence interval equals � L ( u )) P θ { ˆ l w ( θ ) = ( θ w U ( u ) − θ w θ w = u } 1 ( θ w U ( u )) ( θ ) , L ( u ) ,θ w u ∈U where � k 1 k 2 � U = u = w 1 + w 2 : k 1 = 0 , 1 , . . . , n 1 , k 2 = 0 , 1 , . . . , n 2 n 1 n 2 and � ξ 1 ξ 2 � P θ { ˆ θ w = u } = P θ w 1 + w 2 = u = n 1 n 2 � b ( θ ) n 2 1 � ξ 1 = n 1 � i 2 �� � P θ 1 u − w 2 P θ − w 1 θ 1 { ξ 2 = i 2 } dθ 1 . L ( θ ) w 1 n 2 a ( θ ) w 2 i 2 =0 Wojciech Zieli´ nski Confidence Interval For...
For given θ ∈ (0 , 1) , the expected length of the Clopper-Pearson confidence interval equals n � n � � l c ( θ ) = ( θ c U ( u ) − θ c θ u (1 − θ ) n − u 1 ( θ c L ( u )) U ( u )) ( θ ) , L ( u ) ,θ c u u =1 Wojciech Zieli´ nski Confidence Interval For...
For comparison of l c ( θ ) and l w ( θ ) it is enough to compare F − 1 ( γ ) − F − 1 (1 − γ ) and F − 1 w ( γ ) − F − 1 w (1 − γ ) , where F c and c c F w are the cdf’s of ˆ θ c and ˆ θ w , respectively. Wojciech Zieli´ nski Confidence Interval For...
For comparison of l c ( θ ) and l w ( θ ) it is enough to compare F − 1 ( γ ) − F − 1 (1 − γ ) and F − 1 w ( γ ) − F − 1 w (1 − γ ) , where F c and c c F w are the cdf’s of ˆ θ c and ˆ θ w , respectively. Since distributions of ˆ θ c and ˆ θ w have the same support (0 , 1) are unimodal have the same expected value hence, for the length comparison it is sufficient to compare the variances of ˆ θ c and ˆ θ w . Wojciech Zieli´ nski Confidence Interval For...
θ c = θ (1 − θ ) θ ˆ D 2 n θ (3 n 1 +3 nw 1 − 6 n 1 w 1 − 2 nθ ) , 0 < θ < w 1 , 6 n 1 ( n − n 1 ) nw 2 1 − 3 n 1 w 1 +6 n 1 θ (1 − θ ) D 2 θ ˆ θ w = , w 1 ≤ θ ≤ 1 − w 1 , 6 n 1 ( n − n 1 ) (1 − θ )(3 n 1 +3 nw 1 − 6 n 1 w 1 − 2 n (1 − θ )) , 1 − w 1 < θ < 1 . 6 n 1 ( n − n 1 ) (for w 1 ≤ 0 . 5 ) Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
Wojciech Zieli´ nski Confidence Interval For...
f (1 − 2 w 1 ) + 2 w 2 θ c = 1 θ w = 1 1 / 3 0 . 5 ˆ 0 . 5 ˆ D 2 4 n and D 2 4 n f (1 − f ) (where f = n 1 /n ) Wojciech Zieli´ nski Confidence Interval For...
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