Topic 7 - Estimation Flashcards
What is an estimator
- An estimator of a population parameter is a random variable that depends on sample information
What does an estimator provide
- An approximation to an unkown parameter
What is an estimate
- A specific value of the estimator random variable
What are point and interval estimates
- A point estimate is a single number
- A confidence interval provides additional information about variablility
What is the first property of an estimator
- Unbiasedness
- A point estimator θ hat is said to be an unbiased estimator of the parameter θ if the expected value, or mean of the sampling distribution of θ hat is θ
- E(θ hat) = θ
What does it mean if we have a biased estimator
- The estimate we see comes from a distribution which is not centered around the real parameter
How is the Bias of an estimator calculated
- Bias(θ hat) = E(θ hat) - θ
- The bias of an unbiased estimator is 0
What is the second property of an estimator
- Efficiency
- If we have multiple unbiased estimators of θ, the most efficient is the one with the smallest variance
- If var(θ hat 1) < var(θ hat 2) then θ hat 1 is more efficient
How is the relative efficiency of two unbiased estimators calculated
- var(θ hat 2) / var(θ hat 1)
- Relative efficiency of theta hat 1 with respect to theta hat 2
What do both properties consistency and unbiasedness assume
- That the data consists of a fixed sample size n
What is the third property of an estimator
- Consistency
- Consistency is an asymptotic property, concerning the study of behaviour of an estimator as the sample size increases indefinetly
- Concerns how far the estimator is likely to be from the parameter it is estimating as the sample increases
When is an unbiased estimator always consistent
- If variance shrinks to 0 as the sample size grows
What is a confidence interval estimator
- A rule of determining an interval that is likely to include the parameter
- How are confidence intervals calculated
- If P(a < θ < b) = 1 - alpha then the interval from a to b is called a 100(1-alpha)% confidence interval of θ
- 100(1-alpha)% is called the confidence level of the interval (0 < alpha < 1)
- This is written as a < θ < b with 100(1-alpha)% confidence
If σ^2 is known, what is the estimate for our confidence interval
- x bar - z(alpha/2) * σ/sqrt(n) < mu < x bar + z(alpha/2) * σ/sqrt(n)
- where z(alpha/2) is the normal distribution value for a probability for alpha/2 in each tail
What is the margin of error and how can the confidence interval estimate when σ^2 is known be rewritten
- ME = z(alpha/2) * σ/sqrt(n)
- x bar - ME < mu < x bar + ME
How large is the interval width
- 2 * Margin of Error
How can the margin of error be reduced
- Population S.D reduced
- Sample size increases
- Confidence level (1 - alpha) decreases
What do we do if the population standard deviation is unkown
- We can substitute the sample standard deviation (s)
What does substiuting s for σ cause
- Greater uncertainty, as s varies from sample to sample
How is the Students t variable calculated when σ is unkown
- t = x bar - mu / (s / sqrt(n))
- With (n - 1) degrees of freedom
What happens with the t distribution as n increases
- t tends to Z
What is the confidence interval for mu if σ is unkown
- x bar - t(n-1,a/2) * S/sqrt(n) < mu < x bar + t(n-1,a/2) * S/sqrt(n)
- Where t(n-1,a/2) is the critical value of the t ditribution with n-1 degrees of freedom and an area of a/2 in each tail
