Point Estimation
Point Estimation
Example (a variant of Problem 62, Ch5)
Manufacture of a certain component requires three different maching operations. The total time for manufacturing one such component is known to have a normal distribution. However, the mean µ and variance
σ
2 for the normal distribution are unknown. If we did an experiment in which we manufactured 10 components and record the operation time, and the sample time is given as following:
1 2 3 4 5 time 63.8
60.5
65.3
65.7
61.9
6 7 8 9 10 time 68.2
68.1
64.8
65.8
65.4
What can we say about the population mean µ and population variance
σ
2
?
Liang Zhang (UofU)
July 14, 2008 1 / 9
Point Estimation
Example (a variant of Problem 64, Ch5)
Suppose the waiting time for a certain bus in the morning is uniformly distributed on [0 , θ ] , where θ is unknown. If we record 10 waiting times as follwos:
1 2 3 4 5 time 7.6
1.8
4.8
3.9
7.1
6 7 8 9 10 time 6.1
3.6
0.1
6.5
3.5
What can we say about the parameter θ ?
Liang Zhang (UofU)
July 14, 2008 2 / 9
Point Estimation
Definition
A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ . A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of θ .
e.g. X = P
10 i =1
X i
/ 10 is a point estimator for µ for the normal distribution example.
The largest sample data X
10 , 10 distribution example.
is a point estimator for θ for the uniform
Liang Zhang (UofU)
July 14, 2008 3 / 9
Point Estimation
Problem: when there are more then one point estimator for parameter θ , which one of them should we use?
There are a few criteria for us to select the best point estimator : unbiasedness , minimum variance , and mean square error .
Liang Zhang (UofU)
July 14, 2008 4 / 9
Point Estimation
Definition
A point estimator ˆ is said to be an unbiased estimator of θ if E θ ) = θ for every possible value of θ . If ˆ is not unbiased, the difference E θ ) − θ is called the bias of ˆ .
Principle of Unbiased Estimation
When choosing among several different estimators of θ , select one that is unbiased.
Liang Zhang (UofU)
July 14, 2008 5 / 9
Point Estimation
Proposition
Let X
1
, X
2
, . . . , X n be a random sample from a distribution with mean µ and variance σ
2
. Then the estimators
ˆ = X =
P n i =1 n
X i and ˆ
2
= S
2
=
P n i =1
( X i
− X )
2 n − 1 are unbiased estimator of µ and σ
2
, respectively.
If in addition the distribution is continuous and symmetric, then e any trimmed mean are also unbiased estimators of µ .
Liang Zhang (UofU)
July 14, 2008 6 / 9
Point Estimation
Principle of Minimum Variance Unbiased Estimation
Among all estimators of θ that are unbiased, choose the one that has minimum variance. The resulting
ˆ is called the minimum variance unbiased estimator ( MVUE ) of θ .
Theorem
Let X
1
, X
2
, . . . , X n be a random sample from a normal distribution with mean µ and variance σ
2
. Then the estimator ˆ = X is the MVUE for µ .
Liang Zhang (UofU)
July 14, 2008 7 / 9
Point Estimation
Definition
Let ˆ be a point estimator of parameter θ . Then the quantity E [(ˆ − θ )
2
] is called the mean square error ( MSE ) of ˆ .
Proposition
MSE = E [(ˆ − θ )
2
] = V (ˆ ) + [ E (ˆ ) − θ ]
2
Liang Zhang (UofU)
July 14, 2008 8 / 9
Point Estimation
Definition
The standard error of an estimator ˆ is its standard deviation q
σ = V (ˆ ). If the standard error itself involves unknown parameters whose values can be estimated, substitution of these estimates into σ yields the estimated standard error (estimated standard deviation) of or by s .
Liang Zhang (UofU)
July 14, 2008 9 / 9
