Sequential Statistics
Designed for students with familiarity in advanced calculus, probability and statistical inference, this text provides only elementary proofs, and maintains mathematics and statistics at a basic level.
TABLE OF CONTENTS Sequential Statistics
4.9:
Confidence Intervals for P(X
4.9 Confidence Intervals for P(X In reliability problems the parameter p which is the probability that one random variable is stochastically larger than the other is of much interest.
Sequential confidence interval for p=P(X has been considered by Govindarajulu (1974). Let (X, Y) have a bivariate normal distribution with an unknown mean vector and an unknown covariance-matrix. Assume that we observe pairs of observations (X i , Y i ), i=1, 2,...
Let D i =Y i ?X i , D n = Y n ? X n , 
where Y n and X n denote means of samples of size n. Then, it is well known that p= ? ( ? D / ? D ) where ? D =E(Y ?X), 
denotes the variance of (Y ?X). Whatever be the covariance structure of X and Y, a reasonable estimate of p is 
. Then, we have the following result. For the sake of simplicity, we shall suppress all the subscripts in ?, ?, s and 
.
Theorem 4.9.1
Let a 2=(1+ ? 2/2 ? 2) and 
. Then, we have
where the subscript n in D n is suppressed for the sake of implicity.
Proof
Let 
. First we will show that
Toward this, write 
where t lies between D and ?. However, since D converges to ?
Copyright World Scientific Publishing Co. Pte. Ltd. 2004 under license agreement with Books24x7
4.9 Confidence Intervals for P(X In reliability problems the parameter p which is the probability that one random variable is stochastically larger than the other is of much interest.
Sequential confidence interval for p=P(X has been considered by Govindarajulu (1974). Let (X, Y) have a bivariate normal distribution with an unknown mean vector and an unknown covariance-matrix. Assume that we observe pairs of observations (X i , Y i ), i=1, 2,...
Let D i =Y i ?X i , D n = Y n ? X n , where Y n and X n denote means of samples of size n. Then, it is well known that p= ? ( ? D / ? D ) where ? D =E(Y ?X), denotes the variance of (Y ?X). Whatever be the covariance structure of X and Y, a reasonable estimate of p is . Then, we have the following result. For the sake of simplicity, we shall suppress all the subscripts in ?, ?, s and .
Theorem 4.9.1
Let a 2=(1+ ? 2/2 ? 2) and . Then, we have
where the subscript n in D n is suppressed for the sake of implicity.
Proof
Let . First we will show that
Toward this, write where t lies between D and ?. However, since D converges to ?
In reliability problems the parameter p which is the probability that one random variable is stochastically larger than the other is of much interest.
Sequential confidence interval for p=P(X
Let D i =Y i ?X i , D n = Y n ? X n , where Y n and X n denote means of samples of size n. Then, it is well known that p= ? ( ? D / ? D ) where ? D =E(Y ?X), denotes the variance of (Y ?X). Whatever be the covariance structure of X and Y, a reasonable estimate of p is . Then, we have the following result. For the sake of simplicity, we shall suppress all the subscripts in ?, ?, s and .
Theorem 4.9.1
Let a 2=(1+ ? 2/2 ? 2) and . Then, we have
where the subscript n in D n is suppressed for the sake of implicity.
Proof
Let . First we will show that
Toward this, write where t lies between D and ?. However, since D converges to ?
Copyright World Scientific Publishing Co. Pte. Ltd. 2004 under license agreement with Books24x7
