ERM Chapter 16 Flashcards
1
Q
Outline the moments of a distribution.
A
- un is the nth moment about the mean E[(X - E[X])^n]
- coefficient of skewness = u3/o^3
- coefficient of kurtosis = k = u4/o^4
- k = 3 = mesokurtic
- k > 3 = leptokurtic
- k < 3 = platykurtic
- excess kurtosis = k - 3
2
Q
Provide examples of univariate discrete distributions.
A
- binomial and negative binomial distributions
2. poisson distribution
3
Q
Provide example of univariate continuous distributions.
A
Values from negative inifinity to infinity
- normal distribution
- normal mixture distribution
- student’s t distribution
- skewed t distribution
Values that are non-negative
- lognormal distribution
- wald distribution
- chi-squared distribution
- gamma and inverse gamma distributions
- generalised inverse gamma distribution
- exponential distribution
- frechet distribution
- pareto distribution
- generalised pareto distribution
Values are in a finite range than can be positive and/or negative
- uniform distribution
- triangular distribution
4
Q
Why might univariate continuous distributions be used even when variables can only take non-negative values?
A
Might still be used if the probability of getting a negative value is very small. This might be the case if the mean is sufficiently positive and the variance is sufficiently low.
5
Q
Outline the binomial distribution.
A
- Bin(n, p) is the sum of n independent bernoulli(p) trials
- X~Bin(n, p) is the number of successes that occur in the n trials
- the limiting distribution as n approaches infinity is the normal distribution
6
Q
Outline the negative binomial distribution.
A
- Type 1: no. of the trial on which the rth success occurs, where r is a positive integer
- Type 2: no. of failures before the rth success
- geometric distribution is a special case of the Type 1 negative binomial distribution where r = 1
- limitations include:
> CDF is laborious to calculate
> n! becomes time consuming to calculate for large values of n
7
Q
Outline the poisson distribution.
A
- models the number of events that occur in a specified time interval, where events occur one after another in a well-defined manner
- assumes that events occur singly, at a constant rate, and the no. of events that occur in separate time intervals are are independent of one another
- a sequence of binomial (n, p) distributions approaches a poisson with mean np as n approaches infinity and p approaches zero.
- poisson can be used as an approximation to the binomial if p is small enough
8
Q
Outline the normal (or Gaussian) distribution.
A
- standard normal distribution has a mean at 0 and a scaling parameter of 1
- according to CLT, it will approximate the distribution of the sum or average of a sufficiently large number of iid random variables
- it can facilitate simple analytical solutions to complex problems e.g. when it is used as an approximation to the binomial distribution
- normal distribution with zero mean and given variance can be used for error terms when modelling a random walk
- standard normal is the distribution of the test statistic Z = (X - u)/o used to determine whether the mean of an underlying population is significantly different to the assumed mean, when the value of o is known
- standard normal is the distribution of the test statistic Z = (X(hat) - u)/(o/sqrt(T)) used to determine, based on the sample mean, whether the mean of an underlying population is significantly different from u, where T is the number of observations and o is known.
9
Q
List two tests for normality.
A
- graphical approaches e.g. QQ plots
2. statistical tests e.g. Jarque-Bera test
10
Q
Outline the Normal mean-variance mixture distributions.
A
Let W be some strictly positive random variable and Z be a standard normal random variable that is independent of W. X is said to be mean normal-variance mixture distribution if X = m(W)+sqrt(W) x BZ for some function m(W) and scale parameter B.
- key benefit compared to modelling using a normal distribution is that it allows randomness into the mean and the variance.
- special cases are the generalised hyperbolic function, the generalised t-distribution, and the skewed t-distribution
11
Q
Outline the t-distribution.
A
- can be Student’s, standard and general
- if a Student’s t or standard t has y DOF then X = a + By has a generalised t-distribution with location parameter a, scaling parameter B and y DOF
- the CDF can only be determined analytically when y = 1 (Cuachy distribution)
- important for risk modelling as is has fatter tails than the normal distribution. The fact it is leptokurtic makes it an important distribution for risk modelling
12
Q
Outline the skewed t-distribution.
A
- parameters are as for the general t-distribution, but with an additional skew parameter
- the general t-distribution is a special case of the skewed distribution where the skew parameter = 0
13
Q
Outline the lognormal distribution.
A
- If Y = lnX has a normal distribution, then X is said to have a lognormal distribution
- applications include many insurance applications since it takes only positive values and is skewed
- can be used to model financial variables e.g. asset returns, with assumptions that the natural logarithm of the variable will follow a random walk
14
Q
Outline the Wald distribution.
A
- describes the time taken for a Brownian motion process to reach a given value
- is a special case of the inverse Gaussian distribution
- takes only positive values and has positive skew
15
Q
Outline the chi-squared distribution.
A
- distribution with y DOF is the distribution of the sum of y squared independent variables taken from a standard normal distribution
- is a special case of the gamma distribution
16
Q
Outline the exponential distribution.
A
- has a single scale parameter p
- provides the expected waiting times between the events of a Poisson process
- characteristics limiting its application include:
> monotonically-decreasing nature
> single parameter
> low probabilities associated with extreme values
17
Q
Outline the gamma and inverse-gamma distributions.
A
- has two positive parameters and is a versatile family
- PDF can take significantly different shapes, depending on the specific values of parameters
- exponential distribution is a special case when y = 1
- chi-squared distribution if a special case when B = 2
- can be fitted by equating sample and population moments and solving for the distribution’s parameters
18
Q
Outline the Generalised inverse Gaussian distribution.
A
- like the gamma distributions, offer significant flexibility with regard to its shape, due to having three parameters: y, B1 and B2
- when B1=0 the result is gamma(2B2, y)
- when 1/B2 approach 0, tends to InverseGamma(B1/2, -y)
- when y = -1/2 the result is a Wald distribution
19
Q
Outline the Frechet distribution.
A
- like the exponential, has a single parameter
- distribution is a special case of the generalised extreme value distribution
20
Q
Outline the Pareto distribution.
A
- has two parameters
- monotonically decreasing and, like the tails of the t-distribution, follows a power law with the shape parameter (y) determining the power
21
Q
Outline the generalised Pareto distribution.
A
- three parameter distribution is far more flexible than the two parameter distribution
- in particular, it is applied in extreme value theory
- when y=0 it is the exponential distribution
- when y>0 it is the Pareto distribution
22
Q
Outline the uniform distribution.
A
- assigns an equal probability in all outcomes in a range
23
Q
Outline the triangular distribution.
A
- used in cases where, in addition to upper and lower values, the most likely value is known
- distribution has a lower limit, mode, and upper limit
- mean is the average of the parameter values
- distribution can be negatively or positively skewed
24
Q
Outline the multivariate normal distribution.
A
- completely characterised by its mean and covariance vectors
- X~Nn(a, E) denoted an n-dimensional multivariate normal distribution with mean vector a and covariance vector E
- components of vector X are mutually independent if and only if the covariance matrix E is diagonal
- three key limitations that mean that the multivariate normal distribution is not a good description of reality in many RM applications:
> the tails of the univariate marginal distributions are too thin
> the joint tails do not assign enough weight to joint extreme outcomes
> the distribution has a strong form of symmetry, known as elliptical symmetry
25
Q
Outline the Cholesky decomposition.
A
- method of decomposing a matrix, allowing generation of a set of correlated normal variables from a set of independent standard normal variables
- positive definite matrices are always invertible and can be written in the form M = CC’
- C is a lower triangular matrix with positive diagonal entries
- CC’ is called the cholesky decompositiong of M
- Matrix C is known as the Cholesky factor and is denoted M^(1/2)
26
Q
Outline Principal Component Analysis (PCA).
A
- also known as eigenvalue decomposition
- breaks down each variable’s divergence from its mean into a weighted average of independent volatility factors
- theory of spectral decomposition states that, for any covariance matrix E there exists a decomposition E = VAV’, where V is an orthogonal that consists of the eigenvectors of E
- each pair of corresponding eigenvectors and eigenvalues is described as a principal component
27
Q
Outline multivariate mean-variance mixture distributions.
A
- special cases include the generalised hyperbolic distribution, the multivariate t-distribution, and the skewed t-distribution
28
Q
Outline the multivariate t-distribution.
A
- fatter tails the the multivariate normal distribution in two senses:
> marginal distributions have higher probabilities associated with extreme values compared to the normal distribution
> each combination of ‘jointly extreme values’ has higher probabilities than the multivariate normal distribution - the standard multivariate t-distribution is defined by only scale and shape parameters
29
Q
Outline the multivariate skewed t-distribution.
A
- defined by parameters for location (a), scale (B), shape (DOF, y) and skew (d)
30
Q
Outline spherical and elliptical distributions.
A
- distribution where each constant-probability contour of the two variables forms an elipse
- a multivariate spherical distribution is one where the marginal distributions are:
> identical
> symmetrical
> uncorrelated with each other - examples include the multivariate standard normal distribution and normal mixture distributions
- the special case of an elliptical distribution where the correlation is zero is known as a spherical distribution
- an example is the multivariate normal distribution with distinct marginal distributors
31
Q
What is the Mahalanobis distance used for?
A
- testing whether observations are from a multivariate normal distribution
- another test for multivariate normality is Mardia’s test based on Mahanalobis angle
