For details, please refer to this standard GB 4087.1-1983 Statistical processing and interpretation of data Point estimation of binomial distribution parameters GB4087.1-1983 Standard download decompression password: www.bzxz.net
For details, please refer to this standard

1 Introduction
National Standard of the People's Republic of China
Statistical interpretation of dataPoint estimation of parameter in binomial distribution1.1 The statistical terms used in this standard can be found in GB3358-82 "Statistical Terms and Symbols". UDC 519.25
GB 4087.1-83
1.2 Suppose that some individuals in the population have a certain characteristic, and p is the proportion of individuals with this characteristic in the population. For example, p can be the proportion of defective products in a batch of products. A number of individuals are randomly and independently selected from the population as samples. This standard specifies the method for making point estimates of the population parameter p based on such samples. 1.3 For a finite population, let its size be N and the sample size be n. When the sampling is with replacement, or when the sampling is without replacement, but N
0.1, the n samplings can be considered independent. 2 Classical estimation method
2.1 Sample extraction method
The sample size n is predetermined. The sample is randomly and independently extracted from the population. At this time, the number x of individuals with a certain characteristic in the sample is an observation of the random variable X that follows the binomial distribution. The probability that X takes the value x is P(X=xin, p) =(
2.2 Estimator
The estimator of p is denoted as β
n)px(1bZxz.net
, x=0,1,2,\, n
（1）
Where: n—sample size. Its determination method can be found in Appendix A (reference) formula (A1). x-——the number of individuals with specified characteristics in the sample. 2.3 Example
To estimate the defective rate of a batch of products (about 1000 pieces), 40 pieces are randomly selected as samples, of which 5 are defective. Then
3 Sequential sample estimation method
3.1 Sample selection method
Published by National Bureau of Standards on December 21, 1983
1984 - 10- 01 Implementation
GB 4087.1—83
The sample size is not specified in advance, but samples are drawn sequentially, randomly and independently from the population. Each time a sample is drawn, it is immediately checked whether the individual has the specified characteristics, and the number of individuals with the specified characteristics is continuously accumulated. When the number of individuals with the specified characteristics reaches the predetermined number c (an integer greater than or equal to 2), sampling is stopped. At this time, the total number of individuals drawn cumulatively n is a random variable that obeys the negative binomial distribution, and the probability that n takes the value k is p(n=klc, p)= (
3.2Estimator
)pc(1p)-, k=c,c+l,
Where: c-
（2）
The number of individuals with specified characteristics that are specified in advance. The determination method can be found in Appendix A (reference) formula (A2).
The total number of individuals drawn when c individuals with specified characteristics are reached. 3.3 Example
To estimate the defective rate of a batch of products (about 1000 pieces), sampling is carried out one by one in sequence. It is stipulated that sampling will stop when 5 defective pieces are found. When the 35th piece is drawn, the 5th defective piece is found, then c=5
p=C-1-=4
GB 4087.1- 83
Appendix A
Method for determining and c based on the requirements for accuracy (reference)
When it is hoped that the absolute difference between the point estimate β and the estimated value does not exceed pl-a
with a probability of approximately 1-α, a rough estimate po of p can be determined based on past records or experience. At this time, n in the classical estimation method can be determined as follows: na(
c in the sequential sample estimation method can be determined as follows: Ca
u 1-a/ 2
)po(1
u+-al ?
)pa(1po)
α——determined by the required guaranteed probability 1-a; where:
u-a1 2 1-α/2 quantile of the standard normal distribution. 32
..(AI)
+++++++*++*+*++++*++o+e (A2)GB 4087.1— 83
Appendix B
Other estimation methods
(reference)
In addition to the estimation methods in the standard text, two more estimation methods are given here. These estimation methods can be used when the users reach a consensus and the competent authorities agree. B.1 Bayesian Estimation
B.1.1 Conditions of Use
Prior knowledge of p: It obeys the β distribution, and its probability density function is f(x)
r(a+b)
F(a)r(b)
1(1 - x)b-1
When x<0 or x≥1
where a, b are unknown parameters, and the empirical mean u and square of p are known. For example, when someone has a reliable numerical record of the past, the empirical mean and square of p can be calculated based on these historical data. B.1.2 Sample Extraction Method
The sample size is specified in advance. The sample is randomly and independently extracted from the population. B.1.3 Estimator
The values ​​of a and b are calculated from u and u as follows
a=u[u(1
b=(-u)
Formula: n——sample size,
The number of individuals with specified characteristics in the sample. Minimax estimation
Usage conditions
(B3)
When the sample size n is relatively small and the value is 1/2, the classical low estimate will cause a large mean square error. At this time, the minimax estimate can make the maximum mean square error reach the minimum value. B.2.2 Sample selection method
The sample size n is predetermined. The sample is randomly and independently selected from the population. B.2.3 Estimator
Formula t: n—sample size,
GB 4087.1—83
-The number of individuals with the specified characteristics in the sample. Additional remarks:
This standard was proposed by the National Technical Committee for the Application of Statistical Methods. (BA)
This standard was drafted by the Working Group 1 of the Data Processing and Interpretation Subcommittee of the National Technical Committee for the Application of Statistical Methods. The main drafters of this standard are Sun Shanze, Yu Xiulin and Zheng Zhongguo. 34
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