Section 6.1Some General Concepts of Point Estimation: Key Definitions of this section: Statistical Inference: Drawing conclusions about a population from a random sample drawn from it Population: a well-defined collection of objects of interest Sample: A subset of the population Population Parameters: Sample Statistics:
Point Estimates: We are familiar with certain distribution parameters already, for example, for the normal distribution, we call µ the population mean and σ the population standard deviation. For other distributions like the Poisson, the population parameter of interest would be λ . Because different distributions use different letters to describe that population’s parameter, when we generalize and discuss any population’s parameter, we will use θ . In this section we will discuss point estimation. The purpose of point estimation is to obtain a single number based on sample data that represents a sensible value for θ .
Formal definition of a Point Estimate:
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So the numerical value you calculate from the sample is the The statistic you choose (a random variable/its accompanying general formula) is called the
Example 1: (#2 in exercises) A sample of 20 students who had recently taken elementary statistics yielded the following information on brand of calculator owned (T =Texas Instruments, H = Hewlett Packard, C = Casio, S = Sharp). T
T
H
T
C
T
T
S
C
H
S
S
T
H
C
T
T
T
H
T
a. Estimate the true proportion of all such students who own a TI calculator.
b. Of the 10 students who owned a TI calculator, 4 had graphing calculators. Estimate the proportion of students who do not own a TI graphing calculator.
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Example 2: ( exercise #7) a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each house. The resulting observations are 103, 156, 118, 89, 125, 147, 122, 109, 138, 99. Let μ denote the average gas usage during January by all houses in this area. Compute a point estimate for μ.
b. Suppose there are 10,000 houses in this area that use natural gas for heating. Let τ denote the total amount of gas used by all of these houses during January. Estimate τ using the data in part a.
c. Use the data in part a to estimate p, the true proportion of all homes that used at least 100 therms.
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d. Give a point estimate of the population median usage (the middle value in the population of all houses) based on the sample data in part 89, 99,103, 109, 118, 122, 125, 147, 138,156,
Unbiased Estimators ^
It would be nice if θ always was = to θ(or was predictably always very close to it). But because estimators are random variables, the estimate they produce are based on the sample data observed, it is not possible to minimize the error of the estimator. Instead, we concentrate on a different property of estimators, called unbiasedness. Definition of Unbiased Estimators: ^
A point estimator θ is said to be and unbiased estimator of θ if
^
This means if θ is unbiased, its distribution is centered at the true value of the parameter. It is not necessary to know the actual value of the parameter to make this claim. For example we know that X (provided n is > 30) is approximately normally distributed with mean μ, hence X is unbiased. Note: .
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Another brief example: E p = ^
When X is a binomial RV with parameters n and p, the sample proportion, ^
p=
X is an unbiased estimator of p. n
More unbiased Estimators: X is an unbiased estimator for μ.
Proof:
^
σ 2 = S2
Proof:
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is an unbiased estimator of
Example 3(exercise #12) Suppose a certain type of fertilizer has an expected yield per accrue of μ1 with variance σ2., whereas the expected yield for a second type of fertilizer is μ1 with the same variance, σ2. Let S12 and S22 denote the sample variances of yields based on sample sizes n1 and n2, respectively of the two fertilizers. Show that the pooled estimator
Is an unbiased estimator or σ2
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Estimators with minimum variance: Suppose are two estimators of that are both unbiased. Then although the distribution of each estimator is centered at the true value of the spreads of the distributions about the true value may be different. Since the variance of a distribution measures the spread of the distribution, distributions with larger variance would imply that there is a more probability associated with the tails (away from the mean) as opposed to those with smaller variances, whose probability is more clustered around the mean. So it would imply that the unbiased estimator with the smallest variance would be the optimal choice.
Principle of Minimum Variance Unbiased Estimators: Among all estimators of θ that are unbiased, choose the one that has minimum variance. The resulting 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 parameters μ and σ. Then the estimator
X is the MVUE for μ
This theorem only works if your sample is normally distributed. We cannot say universally that c is a good estimator for μ for every population.
Here are a few other estimators for μ and which populations they serve well: Page 7 of 8
middle value of the data, if n odd 1. The median: X = (n / 2)th + (n / 2+ 1)th value in data set , if n is even 2 ~
2.
X e : The average of the two extreme observations:
3. The trimmed mean: X tr (10) you discard the smallest 10% and the largest 10% of the sample, and then find the average with what remains.
The Standard Error: Standard Error gives some idea as to how precise your estimate is. Standard Error of an estimator is defined as
is its standard deviation
. If the standard error itself involves unknown parameters whose values can be estimated, substitution of these estimates into yields the estimated standard error (estimated standard deviation) of the estimator. The estimated standard error can be donated either by
When the point estimator is approximately normally distributed, which is often the case when n is large, then we can be reasonably confident that the true value of lies within two standard errors (deviations) of If is unbiased (but not normal) the estimate will deviate from standard errors at most 6% of the time. Page 8 of 8
by 4
