AIMS: Describe and interpret estimators of the sample mean and their properties. Describe and interpret the least squares estimator. Describe the properties of point estimators: Distinguish between unbiased and biased estimators. Define an efficient estimator and consistent estimator.
Questions:
208.1. A random sample, drawn from a population with unknown mean and variance, includes the following six outcomes: 3, 6, 6, 8, 9, 10. Please note: "random sample" implies independent and identically distributed (i.i.d.). Each of the following is TRUE except:
a. The sample variance is 6.40
b. The standard error of the sample mean is 2.61
c. The standard error of the sample mean is an estimator of the standard deviation of the sample mean
d. The sample variance employs a degrees of freedom correction (n-1). However even for this small sample, the standard error of the sample mean uses (n) or SQRT(n) in the denominator and therefore does not itself employ a degrees of freedom correction.
208.2. We assume there is a population mean for the monthly return of hedge funds that employ a certain strategy (e.g., market neutral funds in 2011). A sample of hedge fund returns is collected and the sample mean return is +1.0% per month.
a. If the returns are not a random sample (i.e., are not i.i.d.), the sample mean may be a biased estimator of the population mean
b. If the returns are a random sample (i.i.d.), the sample mean is the Best Linear Unbiased Estimator (BLUE)
c. If the returns are a random sample (i.i.d.), the sample mean is the least squares estimator
d. If the returns are a random sample (i.i.d.), the property of consistency implies that the variance of the sample mean is smaller than the variance of alternative estimators of the population mean
208.3. A backtest of a 99.0% value at risk (VaR) model over two years observes 8 exceptions in 500 trading days; i.e., the VaR loss threshold was exceeded on 1.6% of the days but the model was calibrated to expect losses in excess of the VaR for only 5 days (1.0%). Please note that we assume exceptions (exceedences) are i.i.d. with a Bernoulli distribution. What is, respectively, the standard error of the sample mean and the t-statistic? (Bonus for finding the p-value, which cannot be done with most calculators)
a. 3.9% (s.e.) and 0.88 t-statistic
b. 4.7% (s.e.) and 1.03 t-statistic
c. 5.6% (s.e.) and 1.07 t-statistic
d. 6.2% (s.e.) and 1.65 t-statistic
Answers:
Questions:
208.1. A random sample, drawn from a population with unknown mean and variance, includes the following six outcomes: 3, 6, 6, 8, 9, 10. Please note: "random sample" implies independent and identically distributed (i.i.d.). Each of the following is TRUE except:
a. The sample variance is 6.40
b. The standard error of the sample mean is 2.61
c. The standard error of the sample mean is an estimator of the standard deviation of the sample mean
d. The sample variance employs a degrees of freedom correction (n-1). However even for this small sample, the standard error of the sample mean uses (n) or SQRT(n) in the denominator and therefore does not itself employ a degrees of freedom correction.
208.2. We assume there is a population mean for the monthly return of hedge funds that employ a certain strategy (e.g., market neutral funds in 2011). A sample of hedge fund returns is collected and the sample mean return is +1.0% per month.
a. If the returns are not a random sample (i.e., are not i.i.d.), the sample mean may be a biased estimator of the population mean
b. If the returns are a random sample (i.i.d.), the sample mean is the Best Linear Unbiased Estimator (BLUE)
c. If the returns are a random sample (i.i.d.), the sample mean is the least squares estimator
d. If the returns are a random sample (i.i.d.), the property of consistency implies that the variance of the sample mean is smaller than the variance of alternative estimators of the population mean
208.3. A backtest of a 99.0% value at risk (VaR) model over two years observes 8 exceptions in 500 trading days; i.e., the VaR loss threshold was exceeded on 1.6% of the days but the model was calibrated to expect losses in excess of the VaR for only 5 days (1.0%). Please note that we assume exceptions (exceedences) are i.i.d. with a Bernoulli distribution. What is, respectively, the standard error of the sample mean and the t-statistic? (Bonus for finding the p-value, which cannot be done with most calculators)
a. 3.9% (s.e.) and 0.88 t-statistic
b. 4.7% (s.e.) and 1.03 t-statistic
c. 5.6% (s.e.) and 1.07 t-statistic
d. 6.2% (s.e.) and 1.65 t-statistic
Answers:
