Point Estimation : August 19, 2011 1
Definition. Apoint estimate of a parameter θ(unknown) is a single number, computed from the sample data, that
can be regarded as an educated guess for the value of the parameter θ.
Example.
a. Suppose we want to estimate the proportion pof all crashes that results in no damages in a population of
automobiles from a certain manufacturer. In a sample of 25 cars, 15 cars result in such crashes, what is the
estimate of the proportion p?
b. 20 observations on the dielectric breakdown voltage for pieces of epoxy resin yield the following statistcis: sample
mean ¯
X= 27.793 and the median M= 27.960.
What is the estimate of the mean, µ, of the population of interest?
Remark. A point estimate is obtained by selecting a suitable statistic and computing its value from the sample data.
The selected statistic is called the point estimator.
Definition (MSE: accuracy and precision of the estimator).The mean square error of an estimator, ˆ
θ, of θis the
quantity
MSE(ˆ
θ) = E[(ˆ
θ−θ)2].
It measures the accuracy and the precision of the estimator.
Definition. A point estimator ˆ
θis said to be unbiased estimator of θif E[ˆ
θ] = θfor every possible value of θ;
otherwise ˆ
θis said to be biased and the difference E[ˆ
θ]−θis called the bias of ˆ
θ.
Example. Suppose that X, the reaction time to a certain stimulus, has a uniform distribution on the interval (0,θ )
( where θis unknown)What is the estimator of θ? Is it unbiased ? Can you find an unbiased estimator based on the
first one ? Given a random sample of n = 5 reaction times x1= 4.2x2= 1.7x3= 2.4x4= 3.9x5= 1.3, Find the
estimate of θ.
Proposition 1. When Xis a binomial random variable with parameters nand p, the sample proportion ˆp=X/n
is an unbiased estimator of p.
Proposition 2. If X1, X2,· · · , Xnis a random sample from a distribution with mean µand variance σ2, then ¯
Xis
an unbiased estimator of µand S2is an unbiased estimator of σ2. If in addition the distribution is continuous and
symmetric, then the median Mand any trimmed mean are also unbiased estimators of µ.
Principle of Minimum variance Unbiased Estimation
Among all estimators of θthat are unbiased, choose the one that has minimum variance. The resulting ˆ
θis called
the the minimum variance unbiased estimator (MVUE) of θ.
Theorem. Let X1, X2,· · · , Xnbe a random sample from the normal distribution with parameters µand σ. Then,
the estimator ˆµ=¯
Xis the MVUE for µ.
So in example 1-b, ¯
Xshould be used to estimate µ, because the underlining population has a normal distribution.
Remark. In general, the best estimator for µdepends crucially on which distribution is being sampled.
Definition. The standard error of an estimator ˆ
θis its standard deviation qV ar[ˆ
θ]. If the standard error itself
involves unknown parameters whose values can be estimated, substitution of these estimate into qV ar[ˆ
θ]yields the
estimated standard error of the estimator,and is denoted by
\
qV ar[ˆ
θ].
Example.
Assuming that the breakdown voltage is normally distributed, ˆµ=¯
Xis the best unbiased estimator of µ. Find the
standard error of ¯
Xin the following 2 cases (a) the value of σis known to be 1.5, (b) the value of σis unknown and
the sample standard deviation is s= 1.462
Homework. 6.1[pages 240-243] 3, 4, 13.