Addenda B
Some Concepts and Properties of Integral and Statistical Inference
B.1. Some Properties of Integral
B.1.1. Functions of Bounded Variation
Definition B.1.1
is a finite function on 
. Points of division on are 
. Define
The supremum of V is called total variation of on denoted by 
.
on denoted by .When 
, is called a function of bounded variation on , or has a bounded variation on .
, is called a function of bounded variation on , or has a bounded variation on .Proposition B.1.1
A monotonic function is a function of bounded variation.
Proposition B.1.2
A function of bounded variation is bounded.
Proposition B.1.3
The sum, difference and product of two functions of bounded variation are still functions of bounded variation.
Proposition B.1.4
If both and 
have a bounded variation and 
, then 
is still a function of bounded variation.
and have a bounded variation and , then is still a function of bounded variation.
Proposition B.1.6
The necessary and sufficient condition that function has a bounded variation is that can be represented by the difference of two increasing functions.
has a bounded variation is that can be represented by the difference of two increasing functions.Proposition B.1.7
If has a bounded variation on , then 
is finite almost everywhere on , and is integrable on , where is the differential of .
has a bounded variation on , then is finite almost everywhere on , and is integrable on , where is the differential of .Proposition B.1.8
Any function of bounded variation can be represented by the sum of its jump function and a continuous function of bounded variation.
Proposition B.1.9 (Herlly’ Principle of Selection)
Define an infinite number 
of functions of bounded variation on , and denoted by 
. If there is a constant 
such that 
and 
, then a sequence 
of everywhere convergent functions on can be selected from F and its limit function 
still has a bounded variation.
of functions of bounded variation on , and denoted by . If there is a constant such that and , then a sequence of everywhere convergent functions on can be selected from F and its limit function still has a bounded variation.B.1.2. LS Integral
Definition B.2.1
and are two finite functions on . Points of division on are . Choose any point 
from each interval 
and construct a sum as follows
When 
, if the sum converges to the same limit 
independent of the selection of , then limit is called 
integral of with respect to denoted by
, if the sum converges to the same limit independent of the selection of , then limit is called integral of with respect to denoted by
Proposition B.2.2
If is continuous on , has differential 
everywhere and is 
integrable (Riemann integrable), then
is continuous on , has differential everywhere and is integrable (Riemann integrable), then
Proposition B.2.3
is continuous on and has a bounded variation, then
.Proposition B.2.4
If is a function of bounded variation on , 
is a sequence of continuous functions on and uniformly converges to a continuous function , then 
.
is a function of bounded variation on , is a sequence of continuous functions on and uniformly converges to a continuous function , then .Proposition B.2.5
is a continuous function on . 
on converges to a finite function . If 
, 
, then 
.Definition B.2.2
If 
, 
is measurable set and has its corresponding value 
, then 
is called a set function on . Given , when 
have 
, then is called an absolutely continuous function, where 
is the measure of .
, is measurable set and has its corresponding value , then is called a set function on . Given , when have , then is called an absolutely continuous function, where is the measure of .For countable mutually disjoint measurable sets 
, have 
, then is called a completely additive set function.
, have , then is called a completely additive set function.Definition B.2.3
is a bounded measurable function on . is a completely additive set function on . Assume that 
. Interval 
is partitioned as follows
Define 
on 
and construct a sum as
on and construct a sum as
If 
and 
have the same limit independent of the selection of 
, then is called 
integral (Lebesgue-Stieltjes integral) of with respect to 
.
and have the same limit independent of the selection of , then is called integral (Lebesgue-Stieltjes integral) of with respect to .B.1.3. Limit Under Integral Symbol
Proposition B.3.1
is a sequence of measurable functions on 
and converge in measure to 
. If there exists integrable function 
such that , 
, then
Definition B.3.1
Assume that 
is a family of integrable functions on . If for 
, there exists 
, when 
and 
, for all 
, 
uniformly holds, 
is called absolutely equicontinous integral on .
is a family of integrable functions on . If for , there exists , when and , for all , uniformly holds, is called absolutely equicontinous integral on .Proposition B.3.2 (Vitali Theorem)
A sequence of functions converges in measure to on , is integrable on and has absolutely equicontinuous integral on , then is integrable on and
of functions converges in measure to on , is integrable on and has absolutely equicontinuous integral on , then is integrable on and
Proposition B.3.3
is a sequence of integrable functions on a measurable set . For 
, is measurable, if 
holds, then has absolutely equicontinuous integral.Corollary B.3.1
Assume that is a sequence of integrable functions and is an integrable function on a measurable set . For any measurable set , if has absolutely equicontinuous integral on .
is a sequence of integrable functions and is an integrable function on a measurable set . For any measurable set , if
has absolutely equicontinuous integral on .Proposition B.3.4
is a sequence of integrable functions on and converges in measure to an integrable function , then 
(e is measurable), we have
The necessary and sufficient condition of the above result is that has absolutely equicontinuous integral on .
has absolutely equicontinuous integral on .Proposition B.3.5 (Vallee-Poussin Theorem)
is a family of measurable functions on a measurable set . If there is a positive increasing function 
, 
, such that 
and for 
, have 
, where 
is a constant independent of , then each is integrable on and has absolutely equicontinuous function.Proposition B.3.6
If 
, is a family of functions on having absolutely equicontinuous integral, then there exists a monotonically increasing function 
such that and 
, where is a constant independent of .
, is a family of functions on having absolutely equicontinuous integral, then there exists a monotonically increasing function such that and , where is a constant independent of .Proposition B.3.7
is an integrable function on . If 
, have 
, then 
( is zero almost everywhere).B.2. Central Limit Theorem
are independently random variables. Let
Proposition B.2.1
Corollary B.2.1
is 
and has a non-zero variance, then when 
, for 
the following formula uniformly holds
is its mean and 
, 
is its variance.Proposition B.2.2
is an independently random variable. If there exists a positive constant such that when ,
, for , the following formula uniformly holds
Definition B.2.1
is a discrete random variable. If there exist constants 
such that all possible values of can be represented by form 
, where 
, then is called having sieve distribution, or is a sieve variable.Proposition B.2.3
Proposition B.2.4
is and has finite mean and variance. When 
( 
is a fixed integer) let the distribution density function of 
be 
. Then, the necessary and sufficient condition that , for 
the formula 
uniformly holds, is that there exists an integer 
such that 
function is bounded.B.3. Statistical Inference
B.3.1. SPRT Method
Definition 3.1.1
is and its distribution depends on parameter 
, denoted by 
.A hypothesis testing problem: the simple null hypothesis 
and the simple alternative hypothesis 
. Let
and the simple alternative hypothesis . Let
The testing procedure is the following
Given constants and 
. Assume that 
is the first observation of the subsample. Calculate 
.
and . Assume that is the first observation of the subsample. Calculate .If 
, then stop the observation and reject the null hypothesis 
.
, then stop the observation and reject the null hypothesis .If 
, then stop the observation and accept the null hypothesis .
, then stop the observation and accept the null hypothesis .If 
, then continue to get the second observation 
.
, then continue to get the second observation .Generally, if from the 
-th observation the ‘stopping decision’ cannot be made, then continue to get the n-th observation 
and calculate 
.
-th observation the ‘stopping decision’ cannot be made, then continue to get the n-th observation and calculate .If 
, then stop sampling and reject .
, then stop sampling and reject .If 
, then stop sampling and accept .
, then stop sampling and accept .If 
, then continue sampling.
, then continue sampling.The above testing procedure is called Sequential Probability Ratio Test denoted by SPRT. Constants and are called the stopping boundaries of SPRT.
and are called the stopping boundaries of SPRT.Proposition 3.1.1
If 
stops with probability 1, its stopping boundaries are constants and , and significance level is 
, then
stops with probability 1, its stopping boundaries are constants and , and significance level is , then
Proposition 3.1.2
If stops with probability 1, stopping boundary 
and significance level 
, then 
.
stops with probability 1, stopping boundary and significance level , then .Let
The stopping rule of is the following.
is the following.If 
, then reject .
, then reject .If 
, then accept .
, then accept .If 
, then continue sampling.
, then continue sampling.Proposition 3.1.3
If for a given parameter , have 
, where 
, then there exist 
, and 
such that.
, have , where , then there exist , and such that
.Proposition 3.1.4
Assume that . If for , have 
, then 
.
. If for , have , then .Proposition 3.1.5
is i.i.d., is a measurable function and 
. Let N be a stopping variable and 
. If 
, then
Especially, if 
, then
, then
Proposition 3.1.6
Assume that 
. For a with stopping probability one and significance level , the following formula holds
. For a with stopping probability one and significance level , the following formula holds
Proposition 3.1.7
For simple null hypothesis and simple alternative hypothesis testing, among the testing methods, including sequential and non-sequential, that have 
(reject )
, 
(accept )
and 
, the with significance level has the minimums of 
and 
.
and simple alternative hypothesis testing, among the testing methods, including sequential and non-sequential, that have (reject ), (accept ) and , the with significance level has the minimums of and .B.3.2. ASM Method
3.2.1. Normal Distribution
is i.i.d and its distribution function is 
, 
, 
. Given credibility probability 
. When 
is known, there exists a fixed size of samples 
, where 
is the minimal integer satisfying the following formula
, 
is a normal function, i.e., 
. Then 
, have
(II.1)
When is unknown, define a sampling process and assume that 
is its stopping variable (when sampling stops Formula (II.1) holds). If satisfies the following formula
is unknown, define a sampling process and assume that is its stopping variable (when sampling stops Formula (II.1) holds). If satisfies the following formula
Then the process is called asymptotically efficient. The corresponding method is called asymptotically efficient testing method with fixed width of the mean confident interval, denoted by ASM. The distribution of xi is assumed to be 
.
.Definition 3.2.1
Let n1, n2
. For each , calculate 
. Define stopping variable as the minimal integer satisfying the following formula
is a series of positive constants and converges to 
, .
. For each , calculate . Define stopping variable as the minimal integer satisfying the following formula
(II.2)
is a series of positive constants and converges to , .Proposition 3.2.1
Assume that is a stopping variable defined by Formula (II.2). Then, we have the following properties.
is a stopping variable defined by Formula (II.2). Then, we have the following properties.(1) 
(2) If 
, then 
and 
, then and where, symbol a.s means almost everywhere.
(3) 
, 
holds
, holds(4) 
(5) 
(6) 
Proposition 3.2.2
Proposition 3.2.3
B.3.2. General Cases
Definition 3.2.2
Define as the minimal integer satisfying the following formula a series of positive constants and converges to , 
.
as the minimal integer satisfying the following formula
(II.3)
a series of positive constants and converges to , .Proposition 3.2.4
Assume that 
is a series of positive random variables 
and 
. Let 
be a series of constants satisfying the following condition
is a series of positive random variables and . Let be a series of constants satisfying the following condition
For 
, define 
as the minimal integer satisfying the following formula
, define as the minimal integer satisfying the following formula
Then, is a non-decreasing stopping variable of t and
is a non-decreasing stopping variable of t and(1) 
(2) 
(3) 
(4) 
.
.If 
again, then 
.
again, then .Proposition 3.2.5 (Chow-Robbins Theorem)
Assume that is a stopping variable defined by Formula (II.3). Then,
is a stopping variable defined by Formula (II.3). Then,
Proposition 3.2.6
is i.i.d., 
and 
. Let 
. is a positive random integer, 
, and satisfies
Then, we have
is a convergent in measure limit and 
.
is a convergent in measure limit and .The above proposition is the extension of common central limit theorem. In the common theorem N is a constant variable but is a random variable.
is a random variable.Proposition 3.2.7
Let be a sequence of random variables satisfying the following properties
be a sequence of random variables satisfying the following properties(1) There exist real number , distribution function and a series 
of real such that for all continuous points of F the following formula holds
, distribution function and a series of real such that for all continuous points of F the following formula holds
(2) 
, there exists a sufficiently large and sufficient small positive number c such that when have
, there exists a sufficiently large and sufficient small positive number c such that when have
Let 
be a sequence of ascending integers and 
. Let be a stopping variable, and 
. Then, for all continuous points of , we have
be a sequence of ascending integers and . Let be a stopping variable, and . Then, for all continuous points of , we have
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