Chapter 7 Parameter Estimation Using the Sample Mean

Yates and Goodman
Chapter 7 Parameter Estimation Using the Sample Mean

statistical inference

http://en.wikipedia.org/wiki/Statistical_inference
Statistical inference or statistical induction comprises the use of statistics and random sampling to make inferences concerning some unknown aspect of a population


7.1  Sample Mean: Expected Value and Variance

The sample mean converges to a constant as the number of repetitions of an experiment increases.

Althouth the result of a single experiment is unpredictable, predictable patterns emerge as we collect more and more data.


sample mean
= numerical average of the observations
: the sum of the sample values divided by the number of trials


7.2 Deviation of a Random Variable from the Expected Value

Markov Inequality
: an upper bound on thte probability that a sample value of a nonnegative random variable exceeds the expected value by any arbitrary factor

http://en.wikipedia.org/wiki/Markov_inequality


Chebyshev Inequality 
: The probability of a large deviation from the mean is inversely proportional to the square of the deviation

http://en.wikipedia.org/wiki/Chebyshev_inequality


7.3 Point Estimates of Model Parameters

http://en.wikipedia.org/wiki/Estimation_theory
estimating the values of parameters based on measured/empirical data. The parameters describe an underlying physical setting in such a way that the value of the parameters affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

http://en.wikipedia.org/wiki/Point_estimation
the use of sample data to calculate a single value (known as a statistic) which is to serve as a "best guess" for an unknown (fixed or random) population parameter


relative frequency (of an event)

point estimates :
bias
consistency
accuracy

consistent estimator
: sequence of estimates which converges in probability to a parameter of the probability model.

The sample mean is an unbiased, consistent estimator of the expected value of a random variable.

The sample variance is a biased estimate of the variance of a random variable.

mean square error
: expected squared difference between an estimate and the estimated parameter

 The standard error of the estimate of the expected value converges to zero as n grows without bound.

http://en.wikipedia.org/wiki/Law_of_large_numbers


7.4 Confidence Intervals

accuracy of estimate

confidence interval
: difference between a random variable and its expected value

confidence coefficient
: probability that a sample value of the random variable will be within the confidence interval