Non-parametric tests is useful if population is strongly non-normal, or sample size is too small to say so, or the data is only available on ordinal scale. Concern about quantities other than distribution parameters
Most of these nonparametric tests are specialized and unlikely to be tested in any detail on the CFA Level I exam.
Showing posts with label Statistics and probability. Show all posts
Showing posts with label Statistics and probability. Show all posts
Wednesday, November 12
Hypothesis Tests on the Variance
Hypothesis Tests on the Variance of a Normally Distributed Population
Hypothesis
H0: σ2 = σ02 , or σ2 ≥σ02 , or σ2 ≤ σ02
H1: σ2 ≠σ02 , or σ2 < σ02 , or σ2 > σ02
Test statistic : χ2 = (n – 1)*s2 / σ02
Where:
n = sample size
s2 = sample variance
σ02 = population variance from hypothesis
Sample variance s2 is referred to as the square sum of deviations between observed values and sample mean, degrees of freedom, or n – 1
Use Chi-squared test
Critical value comes from chi-squared table. It is an asymmetrical distribution and approaches to a normal distribution when degree of freedom increase, but bound below by zero, i.e. cannot be -ve.
{Look up df=n-1, p=significance for 1-sided test or p=significance/2 for 2-sided test}.
Hypothesis
H0: σ2 = σ02 , or σ2 ≥σ02 , or σ2 ≤ σ02
H1: σ2 ≠σ02 , or σ2 < σ02 , or σ2 > σ02
Test statistic : χ2 = (n – 1)*s2 / σ02
Where:
n = sample size
s2 = sample variance
σ02 = population variance from hypothesis
Sample variance s2 is referred to as the square sum of deviations between observed values and sample mean, degrees of freedom, or n – 1
Use Chi-squared test
Critical value comes from chi-squared table. It is an asymmetrical distribution and approaches to a normal distribution when degree of freedom increase, but bound below by zero, i.e. cannot be -ve.
{Look up df=n-1, p=significance for 1-sided test or p=significance/2 for 2-sided test}.
Equality of the Variances
Equality of the Variances of Two Normally Distributed Populations, where both Samples are Random and Independent
Hypothesis
H0: σ12 = σ22 , or σ12 ≥ σ22 , or σ12 ≤ σ22
H1: σ12 ≠σ22 , or σ12 < σ22 , or σ12 > σ22
Test statistic χ2= s12/s22
Use F-test
Critical value comes from F table.
{Look up df1=n1 - 1, df2 = n2 -1, p=significance}.,
F-distritubiton - right skewed, truncated at zero on left hand side,2 separate defree of freedom
Hypothesis
H0: σ12 = σ22 , or σ12 ≥ σ22 , or σ12 ≤ σ22
H1: σ12 ≠σ22 , or σ12 < σ22 , or σ12 > σ22
Test statistic χ2= s12/s22
Use F-test
Critical value comes from F table.
{Look up df1=n1 - 1, df2 = n2 -1, p=significance}.,
F-distritubiton - right skewed, truncated at zero on left hand side,2 separate defree of freedom
Mean difference of two normally distributed populations
Paired comparisons test
In a paired-comparisons test, the two populations have some degree of correlation or co-movement, and the calculation of test statistic takes account of this correlation.
Use t-test based on the variable being the difference between pairs of sample values. {Look up df=n-1, p=significance}.
H0=Ud-Udo
Sample mean difference, denoted by d:
d = (1/n)(d1 + d2 + d3 …. + dn)
where:
n is the number of paired observations
di is the difference between ith observation in the sample.
Sample variance = (sum of all deviations from d )2/(n – 1)
Standard deviation (sd) = (Sample variance)1/2 = [(sum of all deviations from d )2/(n – 1)]1/2
Standard error = sd/(n)1/2.
In a paired-comparisons test, the two populations have some degree of correlation or co-movement, and the calculation of test statistic takes account of this correlation.
Use t-test based on the variable being the difference between pairs of sample values. {Look up df=n-1, p=significance}.
H0=Ud-Udo
Sample mean difference, denoted by d:
d = (1/n)(d1 + d2 + d3 …. + dn)
where:
n is the number of paired observations
di is the difference between ith observation in the sample.
Sample variance = (sum of all deviations from d )2/(n – 1)
Standard deviation (sd) = (Sample variance)1/2 = [(sum of all deviations from d )2/(n – 1)]1/2
Standard error = sd/(n)1/2.
Equality of the population means
Equality of the population means of two normally distributed populations based on independent random samples with equal variance and unequal variance
With equal variance
Hypothesis
H0: u1 = u2
H1: u1 ≠ u2
Where:
u1 is the population mean of population 1
u2 is the population mean of population 2
Use t-test
Test-statistic = (X1-X2)/Standard error
Standard error = (s2/n1 + s2/n2)1/2
Estimated population variance: s2 = [(n1-1)s12 – (n2-2)s22]/(n1+n2-2)
Critical value: look up df= n1+n2-2, p=significance
Where:
n1, n2 are samples sizes,
X1, X2 are sample means
s12, s22 are sample variances.
With unequal variance
Hypothesis
H0: u1 = u2
H1: u1 ≠ u2
Where:
u1 is the population mean of population 1
u2 is the population mean of population 2
Use t-test
Test-statistic = (X1-X2)/Standard error
Standard error = (s12/n1 + s22/n2)1/2
Critical value: look up df=[ (s12/n1+s22/n2)2]/{( s12/n1)2/n1+( s22/n2)2/n2}, p=significance
Where:
n1, n2 are samples sizes
X1, X2 are sample means
s12, s22 are sample variances
With equal variance
Hypothesis
H0: u1 = u2
H1: u1 ≠ u2
Where:
u1 is the population mean of population 1
u2 is the population mean of population 2
Use t-test
Test-statistic = (X1-X2)/Standard error
Standard error = (s2/n1 + s2/n2)1/2
Estimated population variance: s2 = [(n1-1)s12 – (n2-2)s22]/(n1+n2-2)
Critical value: look up df= n1+n2-2, p=significance
Where:
n1, n2 are samples sizes,
X1, X2 are sample means
s12, s22 are sample variances.
With unequal variance
Hypothesis
H0: u1 = u2
H1: u1 ≠ u2
Where:
u1 is the population mean of population 1
u2 is the population mean of population 2
Use t-test
Test-statistic = (X1-X2)/Standard error
Standard error = (s12/n1 + s22/n2)1/2
Critical value: look up df=[ (s12/n1+s22/n2)2]/{( s12/n1)2/n1+( s22/n2)2/n2}, p=significance
Where:
n1, n2 are samples sizes
X1, X2 are sample means
s12, s22 are sample variances
Normally distributed population with unknown variance and known variance
With unkown variance
Use t-test when the population variance is unknown. But, if the sample size is large (>=30), z-test may be used.
With known variance
Use z-test when the population
For details, please see the summary of the applications of z-statisitc and t-distribution
Use t-test when the population variance is unknown. But, if the sample size is large (>=30), z-test may be used.
With known variance
Use z-test when the population
For details, please see the summary of the applications of z-statisitc and t-distribution
Relation between Confidence Intervals and Hypothesis Tests
Confidence intervals is used to estimate the population parameters as a function of "number of standard deviations away from the mean".
For example:
When we use z-statistic and with 95% confidence that our interval will include the population mean (μ), the interval is:
(-1.96)(standard error) < (sample mean) < (+1.96)(standard error).
Hypothesis tests is used to test the value of population parameters, either reject or not reject based on "number of standard deviations away from the mean".
For example:
To test the null hypothesis at the 5% significance level and use the z-statistic, not reject H0 if:
(-1.96)(standard error) < (sample mean) – (hypothesis population mean) < (+1.96)(standard error).
In Hypothesis testing, an interval within which the null will not be rejected is created, and we are 95% confident in this interval (i.e. there's a 5% chance of a type I error).
For example:
When we use z-statistic and with 95% confidence that our interval will include the population mean (μ), the interval is:
(-1.96)(standard error) < (sample mean) < (+1.96)(standard error).
Hypothesis tests is used to test the value of population parameters, either reject or not reject based on "number of standard deviations away from the mean".
For example:
To test the null hypothesis at the 5% significance level and use the z-statistic, not reject H0 if:
(-1.96)(standard error) < (sample mean) – (hypothesis population mean) < (+1.96)(standard error).
In Hypothesis testing, an interval within which the null will not be rejected is created, and we are 95% confident in this interval (i.e. there's a 5% chance of a type I error).
Step 7 - Make economic decision
The statistical result is one of many considerations for our final decision and might not be our investment decision as other factors such as risk tolerance of the client, volatility measures of the fund, transaction cost, etc. would also be taken into account.
Step 6 - Make statistical decision
If test statistic is greater than the critical value, then reject Ho and accept Ha.
If test statistic is smaller than the critical value, do not reject Ho {do NOT accept Ho either}.
Statistical decision does not automatically lead to economic decision, other factors may need to be considered.
If test statistic is smaller than the critical value, do not reject Ho {do NOT accept Ho either}.
Statistical decision does not automatically lead to economic decision, other factors may need to be considered.
Step 5 - Collect samples & Calculate test statistic
Collect the sample sand calcuate test statistic as:
Test statistic = (Sample statistic - Hypothesized value) / Standard error
Where: Standard error = Sample standard deviation / n1/2
Test statistic = (Sample statistic - Hypothesized value) / Standard error
Where: Standard error = Sample standard deviation / n1/2
Tuesday, November 11
Step 4. State decision rule
If test statistic is greater than the critical value, then reject Ho and accept Ha.
If test statistic is smaller than the critical value, do not reject Ho {do NOT accept Ho either}.
Statistical decision does not automatically lead to economic decision, other factors may need to be considered.
If test statistic is smaller than the critical value, do not reject Ho {do NOT accept Ho either}.
Statistical decision does not automatically lead to economic decision, other factors may need to be considered.
Step 3 - Specify significance level
Significance level
Denoted by α, or alpha, the significance level is the probability of making a type I error, or the probability that we will reject the null hypothesis when it is true. The common values used are 0.1, 0.05,and 0.01.
Type I error
Reject the null hypothesis when it is true.
Type II error
Fail to reject the null hypothesis when it is false.
Note:
Denoted by α, or alpha, the significance level is the probability of making a type I error, or the probability that we will reject the null hypothesis when it is true. The common values used are 0.1, 0.05,and 0.01.
Type I error
Reject the null hypothesis when it is true.
Type II error
Fail to reject the null hypothesis when it is false.
Note:
p-value :
Lowest significance level at which Ho can be rejected. Lower the p-value, stronger the evidence to reject H0. If p is smaller than α, reject H0; If p is greater than α, do not reject H0
Power of a test
It is the probability of correctly rejecting Ho when it is false. The formula is as:
Power of a test = 1 - probability of Type II error
Step 2 - Identify test statistic and its probability distribution
t-statistic:
It can be used when the population variance is unknown for large samples and small samples with normally distributed population. Comparing the t-statistic with critical value that comes from Student’s t table.
Critical value from Student’s t table
It can be used when the population variance is unknown for large samples and small samples with normally distributed population. Comparing the t-statistic with critical value that comes from Student’s t table.
Critical value from Student’s t table
- Look up df=n-1, p=significance for 1-sided test or p=significance/2 for 2-sided test
Note:
- Since it is flatter, it's more difficult to reject H0 than using z-distribution.
z- statistic:
Same test statistic as t-test but critical value comes from normal distribution. It can be used when population is normally distributed with known variance; also for large samples with unknown variance.
Test statistic
A quantity taken from a sample that is used as the basis for testing the null hypothesis (rejecting or failing to reject the null).
Test statistic = (Sample statistic - Hypothesized value) / Standard error
Where: Standard error = Sample standard deviation / n1/2
My tips:
Be remembered some commonly used values from z-table:
Singificant level Critical Value (1-sided) Critical Value (2-sided)
0.10 1.280 1.645
0.05 1.645 1.960
0.01 2.330 2.575
Step 1 - State the hypothesis
Null Hypothesis (Ho):
The hypothesis to be tested and that you want to disprove it. It is initially regarded as true until gathering enough proof to either reject the null hypothesis, or fail to reject the null hypothesis. To support or reject the hypothesis but NOT to accept it.
Alternative Hypothesis (Ha):
A statement to be accepted if the null hypothesis is rejected.
Two-tailed tests:
Ho: Mean = X
One-tailed tests:
Ho: Mean ≤X, or Ho: Mean ≥ X
The hypothesis to be tested and that you want to disprove it. It is initially regarded as true until gathering enough proof to either reject the null hypothesis, or fail to reject the null hypothesis. To support or reject the hypothesis but NOT to accept it.
Alternative Hypothesis (Ha):
A statement to be accepted if the null hypothesis is rejected.
Two-tailed tests:
Ho: Mean = X
One-tailed tests:
Ho: Mean ≤X, or Ho: Mean ≥ X
Hypothesis Testing
A hypothesis is a statement made about a population parameter. It involves the process to test whether the statement is true or not at certain confidence level.
7-Steps of hypothesis testing:
7-Steps of hypothesis testing:
Sample size and Potential mistake in sampling
Sample size
Increasing sample size benefits increase the confidence and reliability of the confidence interval, and thus the precision with which the population parameter can be estimated.
Factors that make larger sample size undesirable
Increasing sample size benefits increase the confidence and reliability of the confidence interval, and thus the precision with which the population parameter can be estimated.
Factors that make larger sample size undesirable
- Additional expenses
- Population parameters have a tendency to change over time
Potential mistakes in sampling:
Data-snooping bias
- Conclusions of one analyst are guided by conclusions of others.
Data-mining bias
- Analyst keeps searching for patterns & trading rules until he find one that matches the data.
Sample selection bias
- Refer to the tendency to exclude a certain part of a population simply because the data is not
available.
Survivorship bias
- Only funds/stocks that have survived to date are included. Big problem with stock indices.
Look-ahead bias
- Based on information that do not actually exist at the time of analysis, just based on analyst’s
assumptions.
Time-period bias
Conclusions may apply only to a specific time period and are not repeatable over longer
time periods. Either too short or too long.
t-statistic vs. z-statistic
t-statistic:
It can be used when the population variance is unknown for large samples and small samples with normally distributed population. Comparing the t-statistic with critical value that comes from Student’s t table.
Critical value from Student’s t table
Look up df=n-1, p=significance for 1-sided test or p=significance/2 for 2-sided test
t-distribution is less peaked and flater tails than normal distribution but move closer at higher degrees of freedom.
z-statistic:
It is based on the standard normal distribution and can be used when population is normally distributed with known variance; also for large samples with unknown variance.
Summary of the applications of z-statisitc and t-distribution
It can be used when the population variance is unknown for large samples and small samples with normally distributed population. Comparing the t-statistic with critical value that comes from Student’s t table.
Critical value from Student’s t table
Look up df=n-1, p=significance for 1-sided test or p=significance/2 for 2-sided test
t-distribution is less peaked and flater tails than normal distribution but move closer at higher degrees of freedom.
z-statistic:
It is based on the standard normal distribution and can be used when population is normally distributed with known variance; also for large samples with unknown variance.
Summary of the applications of z-statisitc and t-distribution
My tips:
- Memorize the following commonly used values (from z-table):
- Confidence Interval Reliability factor
90% 1.65 ; 95% 1.96 ; 99% 2.58 - Be able to look up Student's table given degrees of freedom and probability in the right tail.
Reliability Factor, Degree of freedom (df)
Reliability factor
For small samples from a population with unknown variance, reliability factor should be looked up from Student's t table {Look up df=n-1, p=significance/2 = (1-confidence)/2}.
For known variance or for large samples, you can use z-statistic.
Degree of freedom (df)
Used to determine the reliability factor of the confidence interval with the t–distribution. Degrees of freedom = n – 1, where n is the sample size.
For small samples from a population with unknown variance, reliability factor should be looked up from Student's t table {Look up df=n-1, p=significance/2 = (1-confidence)/2}.
For known variance or for large samples, you can use z-statistic.
Degree of freedom (df)
Used to determine the reliability factor of the confidence interval with the t–distribution. Degrees of freedom = n – 1, where n is the sample size.
Points estimators vs. Confidence interval (CI)
Point estimators
Use single(sample) values to estimate population paramenters
Confidence interval
Estimate range of value within which the actual parameters will lie, given the probability of 1-α. (1- α) is the degree of confidence.
Desirable properties of an estimate
Use single(sample) values to estimate population paramenters
Confidence interval
Estimate range of value within which the actual parameters will lie, given the probability of 1-α. (1- α) is the degree of confidence.
Desirable properties of an estimate
- Lack of bias, i.e. expected value of estimator = true value of population parameter)
- Efficiency, i.e. low variance
- Consistency, i.e. accuracy must increase with sample size
Construct Confidence Intervals
The general structure for a (1 – α) confidence interval is given by:
CI= Point estimate ± Reliability factor x Standard error.
Standard error of the sample mean
This refers to the standard deviation of the sample statistic.
If the population standard deviation (σ) is given, standard error is calculated as:
Standard error =σ/(n)1/2
If population standard deviation is unknown, the sample standard deviation (s) is used:
Standard error = s/(n)1/2
If the population standard deviation (σ) is given, standard error is calculated as:
Standard error =σ/(n)1/2
If population standard deviation is unknown, the sample standard deviation (s) is used:
Standard error = s/(n)1/2
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