Chapter 7. Statistical Estimation and Sampling Distributions

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Chapter 7. Statistical Estimation and Sampling Distributions. 7.1 Point Estimates 7.2 Properties of Point Estimates 7.3 Sampling Distributions 7.4 Constructing Parameter Estimates 7.5 Supplementary Problems. 7.1 Point Estimates 7.1.1 Parameters. Parameters

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### Chapter 7. Statistical Estimation and Sampling Distributions

7.1 Point Estimates

7.2 Properties of Point Estimates

7.3 Sampling Distributions

7.4 Constructing Parameter Estimates

7.5 Supplementary Problems

7.1 Point Estimates7.1.1 Parameters
• Parameters
• In statistical inference, the term parameter is used to denote a quantity , say, that is a property of an unknown probability distribution.
• For example, the mean, variance, or a particular quantile of the probability distribution
• Parameters are unknown, and one of the goals of statistical inference is to estimate them.
7.1.2 Statistics
• Statistics
• In statistical inference, the term statistics is used to denote a quantity that is a property of a sample.
• Statistics are functions of a random sample. For example, the sample mean, sample variance, or a particular sample quantile.
• Statistics are random variables whose observed values can be calculated from a set of observed data.
7.1.3 Estimation
• Estimation
• A procedure of “guessing” properties of the population from which data are collected.
• A point estimate of an unknown parameter is

a statistic that represents a “guess” at

the value of .

• Example 1 (Machine breakdowns)
• How to estimate

P(machine breakdown due to operator misuse) ?

• Example 2 (Rolling mill scrap)
• How to estimate the mean and variance of the probability distribution of % scrap ( ) ?
• Definitions

- A point estimate for a parameter is said to be

unbiased if

- If a point estimate is not unbiased, then its bias is defined

to be

7.2.1. Unbiased Estimates (2/5)
• Point estimate of a success probability

-

7.2.1. Unbiased Estimates(3/5)
• Point estimate of a population mean

-

7.2.1. Unbiased Estimates(4/5)
• Point estimate of a population variance
7.2.2. Minimum Variance Estimates (1/4)
• Which is the better of two unbiased point estimates?
7.2.2. Minimum Variance Estimates (3/4)
• An unbiased point estimate whose variance is smaller than any other unbiased point estimate: minimum variance unbised estimate (MVUE)
• Relative efficiency
• Mean squared error (MSE)
• How is it decomposed ?
• Why is it useful ?
Example: two independent measurements

Point estimates of the unknown C

They are both unbiased estimates since

The relative efficiency of to is

Let us consider a new estimate

Then, this estimate is unbiased since

What is the optimal value of p that results in having

the smallest possible mean square error (MSE)?

Let the variance of be given by

Differentiating with respect to p yields that

The value of p that minimizes

Therefore, in this example,

The variance of

The relative efficiency of to is

In general, assuming that we have n independent and unbiased

estimates having variance

respectively for a parameter , we can set the unbiased

estimator as

The variance of this estimator is

Mean square error (MSE):

Let us consider a point estimate

Then, the mean square error is defined by

Moreover, notice that

7.3.1 Sample Proportion (2/2)
• Standard error of the sample mean
7.3.2 Sample Mean (1/3)
• Distribution of Sample Mean
7.3.2 Sample Mean (2/3)
• Standard error of the sample mean
7.3.3 Sample Variance (1/2)
• Distribution of Sample Variance
Theorem: if is a sample from a normal population having mean and variance , then are independent random variables, with being normal with mean and variance and being chi-square with n-1 degrees of freedom.

(proof)

Let Then,

or equivalently,

Dividing this equation by we get

Cf. Let X and Y be independent chi-square random variables with m and n degrees of freedom respectively. Then, Z=X+Y is a chi-square random variable with m+n degrees of freedom.

• Method of moments point estimate for One Parameter
7.4.1The Method of Moments (2/3)
• Method of moments point estimates for Two Parameters
7.4.1The Method of Moments (3/3)
• Examples

-

-

• What if the distribution is exponential with the parameter ?
7.4.2Maximum Likelihood Estimates (1/4)
• Maximum Likelihood Estimate for One Parameter
7.4.2Maximum Likelihood Estimates (3/4)
• Maximum Likelihood Estimate for Two Parameters
7.4.3 Examples (1/6)
• Glass Sheet Flaws

- The method of moment

7.4.3 Examples (2/6)
• The maximum likelihood estimate:
7.4.3 Examples (3/6)
• Example 26: Fish Tagging and Recapture
7.4.3 Examples (5/6)
• Example 36: Bee Colonies
MLE for
• For some distribution, the MLE may not be found by differentiation. You have to look at the curve of the likelihood function itself.