This presentation lists out the properties that should hold for an estimator to be Best Unbiased Linear Estimator (BLUE)

1. PROPERTIES OF
ESTIMATORS (BLUE)
KSHITIZ GUPTA

2. WHAT IS AN ESTIMATOR?
• In statistics, an estimator is a rule for calculating an
estimate of a given quantity based on observed data
• Example-
i. X follows a normal distribution, but we do not know the
parameters of our distribution, namely mean (μ) and
variance (σ2 )
ii. To estimate the unknowns, the usual procedure is to
draw a random sample of size ‘n’ and use the sample
data to estimate parameters.

3. TWO TYPES OF ESTIMATORS
• Point Estimators
A point estimate of a population parameter is a single value of a
statistic.
For example, the sample mean x is a point estimate of the
population mean μ. Similarly, the sample proportion p is a point
estimate of the population proportion P.
• Interval Estimators
An interval estimate is defined by two numbers, between which
a population parameter is said to lie. For example, a < x < b is
an interval estimate of the population mean μ. It indicates that
the population mean is greater than a but less than b.

4. PROPERTIES OF BLUE
• B-BEST
• L-LINEAR
• U-UNBIASED
• E-ESTIMATOR
An estimator is BLUE if the following hold:
1. It is linear (Regression model)
2. It is unbiased
3. It is an efficient estimator(unbiased estimator with least
variance)

5. LINEARITY
• An estimator is said to be a linear estimator of (β) if it is a
linear function of the sample observations
• Sample mean is a linear estimator because it is a linear
function of the X values.

6. UNBIASEDNESS
• A desirable property of a distribution of estimates iS that
its mean equals the true mean of the variables being
estimated
• Formally, an estimator is an unbiased estimator if its
sampling distribution has as its expected value equal to
the true value of population.
• We also write this as follows:
Similarly, if this is not the case, we say that the estimator is
biased

7. • Similarly, if this is not the case, we say that the estimator
is biased
• Bias=E( ) - β

9. MINIMUM VARIANCE
• Just as we wanted the mean of the sampling distribution to be
centered around the true population , so too it is desirable for
the sampling distribution to be as narrow (or precise) as
possible.
– Centering around “the truth” but with high variability might be of
very little use
• One way of narrowing the sampling distribution is to increase
the sampling size

11. BLUE