Statistical interpretation of data;Estimation of parameter in Poisson distribution

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".

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
=0,13071
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≤0bzxz.net
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)
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
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