Drafting unit:Working Group of the Subcommittee on Data Processing and Interpretation of the National Technical Committee for Standardization of Statistical Methods
Focal point unit:National Technical Committee for Application of Statistical Methods and Standardization
Proposing unit:National Technical Committee for Application of Statistical Methods and Standardization
Publishing department:National Bureau of Standards
The statistical terms used in this standard can be found in the national standard GB 3358-82 "Statistical Terms and Symbols". GB 4089-1983 Statistical processing and interpretation of data Estimation of Poisson distribution parameters GB4089-1983 standard download decompression password: www.bzxz.net
The statistical terms used in this standard can be found in the national standard GB 3358-82 "Statistical Terms and Symbols".
Some standard content:
1 Introduction National Standard of the People's Republic of China Statistical processing and interpretation of data Estimation of parameter in Poisson distribution Statistical fnterpretation of data Estimation of parameter in Poisson distribution 1.1 The statistical terms used in this standard can be found in the national standard GB3358--82 "Statistical Terms and Symbols". 1.2 The population discussed in this standard is the Poisson distribution p(x=xlai= x=0,12, UDC 519.25 GB4089--83 Where 0 is the distribution parameter. This standard specifies the point estimation and interval estimation methods for the parameter λ based on independent random samples x, x2,, x. This standard can be used when there is sufficient reason to believe that the population follows a Poisson distribution. 1.3 Sample x, ×2, 2 point estimate λ is denoted by interval estimate , and the sum of x is denoted by is T, that is, =x. There are three common forms of confidence intervals for elements: a. a two-sided confidence interval (,), where 0<<<. b. a one-sided confidence interval with only a lower confidence limit (, +), where >0. c. a one-sided confidence interval with only an upper confidence limit (0,), where >0. Which type of confidence interval to choose depends on the nature of the problem being studied. The confidence interval obtained contains the true input value with a given probability, and specifically should satisfy the two-sided confidence interval: -a One-sided confidence interval: ()1-a Probability 1-α is called the confidence level. According to different requirements, the value of 1α is usually selected from 0.90, 0.95, and 0.99. Due to the discreteness of the Poisson distribution, it is not always possible to find the value that makes the above equation valid. In this case, we take, which satisfies: ()1a National Bureau of Standards 1983-12-21 Issued (Two-sided case) (One-sided lower limit) (One-sided upper limit) 1984-10-01 Implementation Method for finding confidence limits 4.1 One-sided lower confidence limit GB 4089 -83 (2) where ×(2T) represents the a quantile of the distribution with 2 degrees of freedom. Their values can be found in the distribution table (see national standard GB4084.2-83 "Statistical Distribution Numerical Table X Distribution"). 4.2 One-sided upper confidence limit -a (2T+ 2) where ×-(2T+2) represents the 1-α quantile of the ×2 distribution with 2+2 degrees of freedom. 4.3 Confidence limits of two-sided confidence intervals Two-sided lower confidence limit: Two-sided upper confidence limit: xan(2T) X1 -α/2(2 T + 2) (3) ·(5) Where ×(2) represents the α/2 quantile of the distribution with 2T degrees of freedom. ×α/2(2+2) represents the 1α/2 quantile of the x2 distribution with 2T+2 degrees of freedom. 4.4 Example When n=10,T = xi = 14, 1 ~α = 0.95, find,. One-sided upper confidence limit From the ×2 distribution table, we can directly find: x -a (2 T + 2) = x3.9s (30) = 43.7730Au So the one-sided confidence interval is: (0, 2.189). b. The one-sided lower confidence limit is directly bent from the x2 distribution table to: X# (2T) = x3.0s (28) = 16.9279± =16-9279 = 0.846 So the one-sided confidence interval is: (0.846, +α). The two-sided confidence interval is directly bent from ×? Directly look up in the distribution table: X2a/2 (2 T) =X3.025 (28) = 15.307915.3079 X -a/2( 2 T + 2) =X3.07s (30) = 46.9792au (0.765, 2.349) So the two-sided confidence interval is: Approximate method of confidence interval GB 4089 --83 When 2T≥30, the approximate method given below can be used. 5.1 Approximate method for calculating the one-sided lower confidence limit The approximate calculation formula for the one-sided lower confidence limit is: (c + (VT+2c -0.5 - uie+ 2 Where: c= ui- is the 1-α quantile of the standard normal distribution. 5.2 Approximate method for calculating the one-sided upper confidence limit The approximate formula for the one-sided upper confidence limit is: Where: c, u.-Same as 5.1. Ic+ (VT + 2c+0.5+- 5.3 Approximate method for calculating the confidence limits of the two-sided confidence interval The approximate calculation formula for the two-sided upper and lower confidence limits is: 3 Where: (c+(VT+2c+0.5+\1-==)2) tc + (VT +2c- 0.5 u1-n/2 is the 1.-α/2 quantile of the standard normal distribution. 5.4 Example When n=15, = x = 18, 1 -α = 0.95, find: one-sided upper confidence limit Check the normal distribution table (see national standard GB4086.1-83 "Statistical Distribution Numerical Table Love Normal Distribution"): ui-a = uo.95 = 1.64485 Calculate: ur + 2 (1.64485)2 +2 =0,13071bZxz.net VT + 2 c+ 0.5 = 4.33145 (c + (VT + 2c+0.5+\-° (0.13071 +(4.33145 +0.82243)2= 1.780 The one-sided confidence interval is (0, 1.780). b. One-sided lower confidence limit GB 4089 —83 ur-a = uo.95 = 1.64485 uz + 2 = 0.13071 T + 2 c - 0.5 = 4.21443 (c+ (T+2c-0.5-\1g)2) {0.13071 +(4.21443 - 0.82243)*)15 The one-sided confidence interval is (0.776, +α) The two-sided confidence interval ul -a /2 = uo.975 = 1.95996 ui -α/2 + 2 (1.95996)2+2 1-/2 = 0.87998 VT + 2c-0.5= 4.22191 VT + 2 c + 0.5 = 4.33872 = 0.16226 ul -a/2- ) 2 1 +c + (T +2c= 0.5 - (0.16226 + (4.22191-0.87998) 2 )=0.755 (c + (VT +2c+0.5+ (0.16226 + (4.33872 +0.87998)2)=1.826 The two-sided confidence interval is (0.755, 1.826). 131 GB4089—83 Appendix A Bayesian estimation method (reference) A.1 The Bayesian estimation method may be adopted with the agreement of the relevant parties and the consent of the competent authorities. A.2 Conditions of use Prior knowledge: 1. It obeys the distribution of ", and the distribution density is: ba Xa-te-bx When ×>0 When x≤0 Formula ia and b are unknown parameters, and there are empirical mean u and variance o. For example, there are a large number of reliable numerical records in the past, and the empirical mean u and variance o can be calculated based on these historical data. A.3 Sampling method The sample size n is predetermined. The sample is randomly and independently drawn from the population. A.4 Estimator The size of a sample is given by u and the calculated values a, b: T is the sum of the sample x, ×2, x#, that is, T = x1 Additional remarks: This standard is proposed by the National Technical Committee for the Application of Statistical Methods for Standardization. (AI) (A2) (A3) This standard was drafted by the Working Group of the Data Processing and Interpretation Subcommittee of the National Technical Committee for the Application of Statistical Methods for Standardization. The main drafters of this standard are Sun Shanze, Yu Xiulin and Zheng Zhongguo. 132 Tip: This standard content only shows part of the intercepted content of the complete standard. 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