Lecture 4
Sampling and Estimation
Dr Peter Wheale
Sampling
 To make inferences about the parameters of a
population, we will use a sample
 A simple random sample is one where every
population member has an equal chance of being
selected
 A sampling distribution is the distribution of
sample statistics for repeated samples of size n
 Sampling error is the difference between a
sample statistic and true population parameter
(e.g., x – 
Use of n – sample size versus N – population size
• Data set of a stock’s
returns over time:
• 12%, 25%, 34%, 15% , 19%, 44%,
54%, 33%, 22%, 28%, 17%, 24%
• µ=
12+25+34+15+19+44+54+33+22+
28+17+24 / 12 =27.25%
x = 25+34+19+54+17 / 5
= 29.8%
In the above calculations the
population size, N, is 12, and
the sample size, n, is 5.
All interval and ratio data sets
have an arithmetic mean.
• The population mean and
sample mean are both
examples of arithmetic
means – the most common
measure of central tendency.
• The arithmetic mean is
unique and the sum of the
deviations of each
observation in the data set
from the mean is zero.
• The sampling error of the
mean = 29.8% - 27.25% =
2.55.
Stratified Random Sampling
1. Create subgroups from population based on
important characteristics, e.g. identify bonds
according to: callable, ratings, maturity, coupon
2. Select samples from each subgroup in
proportion to the size of the subgroup
Used to construct bond portfolios to match a bond
index or to construct a sample that has certain
characteristics in common with the underlying
population
Time-Series vs. Cross-Sectional
Time-series data
e.g. Monthly prices for IBM stock for 5 years
Cross-sectional data
e.g. Returns on all health care stocks last month
Central Limit Theorem
 For any population with mean µ and variance
σ2, as the size of a random sample gets large,
the distribution of sample means approaches a
normal dist. with mean µ and variance σ2
 Allows us to make inferences about and
construct confidence intervals for population
means based on sample means
Semivariance and CV
• Semivariance is calculated by only including those observations that
fall below the mean ion the calculation.
• Sometimes described as “downside risk” with respect to
investments.
• Useful for skewed distributions, as it provides additional information
that the variance does not.
• Target semivariance is similar but based on observations below a
certain value, e.g values below a return of 5%.
• Coefficient of Variation (CV) = standard deviation of x
•
average value of x
• X can stand for investments for example ; CV measures the risk
(variability) per unit of expected return (mean).
CV Example
• CV calculation: = standard deviation of x
•
average value of x
• Example: Suppose you wish to calculate the CV for two
investments, the monthly return on British T-Bills and the monthly
return for the S&P 500, where: mean monthly return on T-Bills is
0.25% with SD of 0.36%, and the mean monthly return for the S&P
500 is 1.09%, with a SD of 7.30%.
• CV (T-Bills) = 0.36/0.25 = 1.44
• CV (S&P 500) = 7.30/1.09 = 6.70
• Interpretation: is the variation per unit of return, indicating that
these results indicate that there is less dispersion (risk) per unit of
monthly returns for T-Bills than there is for the S&P 500, i.e. 1.44 vs
6.70.
Standard Error of the Sample Mean
Standard error of sample mean is the standard
deviation of the distribution of sample means.
• When the population σ is known:
X =

n
 When the population σ is unknown:
S
SX =
n
Standard Error of the Sample Mean
Example: The mean P/E for a sample of 41 firms is
19.0, and the standard deviation of the population is
6.6. What is the standard error of the sample mean?
Interpretation: For samples of size n = 41, the
distribution of the sample means would have a mean
of 19.0 and a standard error of 1.03.
Point Estimate and Confidence Interval
Example: The mean P/E for a sample of 41 firms is
19.0, the standard error of the sample mean is 1.03, and
the population is normal
Point estimate of mean is 19.0
90% confidence interval is 19 +/- 1.65 (1.03)
17.3 < mean < 20.7
95% confidence interval is 19 +/- 1.96 (1.03)
17.0 < mean < 21.0
Confidence Interval: Normal
Distribution
Confidence interval: a range of values around an expected
outcome within which we expect the actual outcome to occur
some specified percent of the time.
Properties of Normal
Distribution
•
•
•
•
Completely described by mean and variance
Symmetric about the mean (skewness = 0)
Kurtosis (a measure of peakedness) = 3
Linear combination of normally distributed random
variables is also normally distributed
• Probabilities decrease further from the mean, but the
tails go on forever
Kurtosis - peakedness
• Kurtosis is a measure of the degree to which a distribution is more
or less “peaked” than a normal distribution. Leptokurtic describes a
distribution that is more peaked than a normal distribution – it will
have more returns clustered around the mean and large deviations
from the mean - and platykurtic describes a distribution that is less
peaked (flatter than a normal distribution) – having a broader spread
of deviations from the mean.
• Skewness and kurtosis are important in for risk management
because model predictions need to take account of the distribution
of returns in the tails of the distribution, which is where the risk lies.
Measures of Sample Skew and Kurtosis
• Sample skewness is equal to the sum of the cubed deviations from
the mean divided by the cubed standard deviation and by the
number of observations.
• A left skewed distribution is negative and a right skewed
distribution is positive.
• Sample kurtosis is measured as above, but using deviations raised
to the fourth power.
• Interpretation of kurtosis: calculations are compared to the value
for a normal distribution curve, which is 3.
• Excess kurtosis = sample kurtosis – 3.
Confidence Interval: Normal
Distribution
90% confidence interval = X ± 1.65s
95% confidence interval = X ± 1.96s
99% confidence interval = X ± 2.58s
• Example: The mean annual return (normally
distributed) on a portfolio over many years is 11%,
and the standard deviation of returns is 8%. A 95%
confidence interval on next year’s return is 11% +
(1.96)(8%) = –4.7% to 26.7%
Desirable Estimator Properties
1. Unbiased - expected value equal to parameter
2. Efficient - sampling distribution has smallest
variance of all unbiased estimators
3. Consistent – larger sample → better estimator
Standard error of estimate
decreases with larger sample
size
Student’s t-Distribution and
Degrees of Freedom
Properties of Student’s t-Distribution
▫ Symmetrical (bell shaped)
▫ Less peaked and fatter tails than a normal distribution
▫ Defined by single parameter, degrees of freedom (df), where df =
n–1
▫ As df increase, t-distribution approaches normal distribution
t-Distribution
The figure below shows the shape of the tdistribution with different degrees of freedom.