Question
In: Math
Let X ~ Geometric (p) where 0 < p <1 a) Show explicitly that this family...
Let X ~ Geometric (p) where 0 < p <1
a) Show explicitly that this family is “very regular,” that is, that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R 1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, p) is not zero and should not depend on p.
R 3. One derivative can be found with respect to p.
R 4. Two derivatives can be found with respect to p.
b) Find the maximum likelihood estimator of p, call it Yn for this problem.
c) Is Yn unbiased? Explain.
d) Show that Yn is consistent asymptotically normal and identify the asymptotic normal variance.
e) Variance-stabilize your result in (d) or show there is no need to do so.
f) Compute I (p) where I is Fisher’s Information.
g) Compute the efficiency of Yn for p (or show that you should not!).
Solutions
for query in above,
comment.
milcah answered 10 months agoRelated Solutions
2. Let X ~ Geometric (p) where 0 < p <1 a. Show explicitly that this...
2. Let X ~ Geometric (p) where 0 < p <1
a. Show explicitly that this family is “very regular,” that is,
that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R 1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, p) is not zero and should
not depend on p.
R 3. One derivative can be found with respect to p.
R 4. Two...
2. Let X ~ Pois (λ) λ > 0 a. Show explicitly that this family is...
2. Let X ~ Pois (λ) λ > 0
a. Show explicitly that this family is “very regular,” that is,
that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, λ) is not zero and should
not depend on λ .
R 3. One derivative can be found with respect to λ.
R 4. Two derivatives can...
Please answer all parts if possible. Let X ~ Geometric (p) where 0 < p <1...
Please answer all parts if possible.
Let X ~ Geometric (p) where 0 < p <1
a) Show explicitly that this family is “very regular,” that is,
that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R 1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, p) is not zero and should
not depend on p.
R 3. One derivative can be found with respect...
A geometric distribution has a pdf given by P(X=x) = p(1-p)^x, where x = 0, 1,...
A geometric distribution has a pdf given by P(X=x) = p(1-p)^x,
where x = 0, 1, 2, ..., and 0 < p < 1. This form of the
geometric starts at x = 0, not at x = 1. Given are the following
properties:
E(X) = (1-p)/p, and Var(X) = (1-p)/p^2
A random sample of size n is drawn; the data are X1, X2, ...,
Xn.
A. Derive the Fisher information function for the parameter
p.
B. Find the Cramér-Rao...
Let X~Geometric(p), with parameter p unknown, 0<p<1. a) Find I(p), the Fisher Information in X about...
Let X~Geometric(p), with parameter p unknown, 0<p<1.
a) Find I(p), the Fisher Information in X about p.
b) Suppose that pˆ is some unbiased estimator of p. Determine
the Cramér-Rao Lower Bound for Var p[ ]ˆ based on one observation
from this distribution.
c) Show that p= I {1}(X) is an unbiased estimator of
p. Does its variance achieve the Cramer-Rao Lower Bound?
Let X and Y be i.i.d. geometric random variables with parameter (probability of success) p, 0<p<1....
Let X and Y be i.i.d. geometric random variables with parameter
(probability of success) p, 0<p<1.
(a) Find P(X>Y).
(b) Find P(X+Y=n) and P(X=k|X+Y=n), for n=2,3,..., and
k=1,2,...,n−1.
I need an answer asap please. Thank you.
Let X and Y be i.i.d. geometric random variables with parameter (probability of success) p, 0...
Let X and Y be i.i.d. geometric random variables with parameter
(probability of success) p, 0 < p < 1. (a) (6pts) Find P(X
> Y ). (b) (8pts) Find P(X + Y = n) and P(X = k∣X + Y = n), for
n = 2, 3, ..., and k = 1, 2, ..., n − 1
Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,...
Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,
P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2,
P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0.
a. Determine E(X) and E(Y ).
b. Find Cov(X, Y )
c. Find Cov(2X + 3Y, Y ).
1. Let X be random variable with density p(x) = x/2 for 0 < x<...
1. Let X be random variable with density p(x) = x/2 for 0 < x < 2 and 0 otherwise. Let Y = X^2−2.
a) Compute the CDF and pdf of Y.
b) Compute P(Y >0 | X ≤ 1.8).
1. Suppose that X is a Geometric r.v. and its pmf is f(x; p) = p(1...
1. Suppose that X is a Geometric r.v. and its pmf is
f(x; p) = p(1 − p) x−1 , x = 1, 2, . . . and 0 ≤ p ≤ 1.
a. Calculate the mean of X.
b. Calculate the median of X.
c. Check and show whether the median is smaller than mean.
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