Let D = {xi} n i=1, where xi ∈ R, be a data set drawn independently...

Let D = {xi} n i=1, where xi ∈ R, be a data set drawn independently from a Gumbel distribution α ∈ R is the location parameter and β > 0 is the scale parameter.

a) Derive an algorithm for estimating α and β.

b) Implement the algorithm derived above and evaluate it on data sets of different sizes. First, find or develop a random number generator that creates a data set with n ∈ {100, 1000, 10000} values using some fixed α and β. Then make at least 10 data sets for each n and estimate the parameters. For each n, report the mean and standard deviation on the estimated α and β. If n = 10000 is too large for your computing resources, skip it.

c) The problem above will require you to implement an iterative estimation procedure. You will need to decide on how to initialize the parameters, how to terminate the estimation process and what the maximum number of iterations should be. Usually, some experimentation will be necessary before you run the experiments in part (b) above. Summarize what you did in a short paragraph, no more than two paragraphs.

Expert Solution

orchestra answered 2 years ago

Suppose that the random variable Y1,...,Yn satisfy Yi = ?xi + ?i i=1,...,n. where the set...

Suppose that the random variable Y1,...,Yn satisfy Yi = ?xi + ?i i=1,...,n. where the set of xi are fixed constants and ?i are iid random variables following a normal distributions of mean zero and variance ?2. ?a (with a hat on it) = ?i=1nYi xi / ?i=1nx2i is unbiased estimator for ?. The variance is ?a (with a hat on it) = ?2/ ?i=1nx2i . What is the distribation of this variance?

c) Let R be any ring and let ??(?) be the set of all n by...

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

Let Xi = lunch condition on day i (rice/noodles) P(Xi+1 = rice| Xi-1 = rice, Xi...

Let Xi = lunch condition on day i (rice/noodles) P(Xi+1 = rice| Xi-1 = rice, Xi = rice) = 0.7 P(Xi+1 = rice| Xi-1 = noodles, Xi = rice) = 0.6 P(Xi+1 = rice| Xi-1 = rice, Xi = noodles) = 0.3 P(Xi+1 = rice| Xi-1 = noodles, Xi = noodles) = 0.55 Q1. Is {Xn} a Markov Chains? Why? Q2. How to transform the process into a M.C. ?

7. Let n ∈ N with n > 1 and let P be the set of...

7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R. (a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P. (b) Let R be the set of equivalence classes of P and let F : R → P be...

Let A∈Mn(R)"> A ∈ M n ( R ) A∈Mn(R) such that I+A"> I + A I+A is invertible. Suppose that

Let A∈Mn(R)A∈Mn(R) such that I+AI+A is invertible. Suppose thatB=(I−A)(I+A)−1B=(I−A)(I+A)−1(a) Show that B=(I+A)−1(I−A)B=(I+A)−1(I−A) (b) Show that I+BI+B is invertible and express AA in terms of BB.

Let A be a set with m elements and B a set of n elements, where...

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

Let p be an element in N, and define d _p to be the set of...

Let p be an element in N, and define d _p to be the set of all pairs (l,m) in N×N such that p divides m−l. Show that d_p is an equivalence relation

Let Ti, i=1, … ,n be a set of dates, on which payments of the floating...

Let Ti, i=1, … ,n be a set of dates, on which payments of the floating leg of an interest rate swap occur. The payoff of the floating leg of the swap at time Ti is Fi + s where Fi is the reference rate of the floating leg and s is a constant spread. For simplicity, let’s assume that the floating and fixed payments happen on the same dates. Also, ri is the risk-free rate on the same tenor....

Let xi, i = 1,2,...,n be independent realizations from a population distributed like a Pareto with...

Let xi, i = 1,2,...,n be independent realizations from a population distributed like a Pareto with unknown mean. (a) Compute the Standard Error of the sample mean. [Note: By“compute”, I mean “find the formula for”]. What is the probability that the sample mean is two standard deviations larger than the popula- tion mean? (b) Suppose the sample mean is 1 and the Standard Error is 2. Con- sider a test of the hypothesis that the population mean is 0, against...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R. a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R). ii). Show...

Question