Question: An alternative negative binomial parameterization results from the gamma density formula, for which E(λ) = µ, var(λ) = µ/k. Show that this gamma mixture of

An alternative negative binomial parameterization results from the gamma density formula,

(k)** r (kμ) f (A; k, μ) - exp ( -kλ) λέμ-1, λ2 0,

for which E(λ) = µ, var(λ) = µ/k. Show that this gamma mixture of Poissons yields a negative binomial with

E(Y) = µ, var(Y) = µ(1 + k)/k.

For what limiting value of k does this reduce to the Poisson?

(k)** r (k) f (A; k, ) - exp ( -k) -1, 2 0,

Step by Step Solution

★★★★★

3.39 Rating (161 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

Using the ... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Students Have Also Explored These Related Categorical Data Analysis Questions!

Q:

Prove that the p.f. of the negative binomial distribution can be written in the following alternative form: f(xir,p) =|C)pr(-11- pD for x 0, 1. 2. otherwise.

Q:

For a negative binomial random variable whose pdf is given by Equation 4.5.1, find E(X) directly by evaluating p r (1 p) kr . 00 (1> k-1 -1. k=r

Q:

Let X1,..., Xn be iid negative binomial(r, p). We want to construct some approximate confidence intervals for the negative binomial parameters. (a) Calculate Wilks' approximation (10.4.3) and show...

Q:

Question: Scientists are interested in predicting the cancer volume (in mm3) from a blood marker measurement for patients who are known to have prostate cancer. The file prostate.csv contain the...

Q:

PHEB 609: Categorical Data Analysis Homework 1 due 09/06/2017 by end of the day 1. In the following examples, identify the response variable and the explanatory variables. a. Attitude toward gun...

Q:

Question: Assume you have two boxes An and B. Box A contains 7 dark marbles, 4 white marbles. Box B contains 7 white and 4 dark marbles. An arbitrary test is acted in two consecutive preliminaries by...

Q:

Problem 6 The power of this test against the alternative hypothesis that the chance the die lands with two spots showing is 32.45%, the chance the die lands with five spots showing is 0.88%, and the...

Q:

Exercise 15.7.11. Consider the polio vaccine example from Exercise 6.8.9, where Xy is the number of subjects in the vaccine group that contracted polio, and Xc is the number in the control group. We...

Q:

For a sequence of independent Bernoulli trials, let y = the number of successes before the kth failure. Show that y has the negative binomial distribution, f(y; ????, k) = (y + k) (k)(y + 1)????y (1 ...

Q:

The data for this exercise are from a study of the frequencies of urinary tract infections in n = 98 men infected with the HIV virus. CD4+ cell counts were also measured. They are used as an...

Q:

The polarity test is not necessary for the single-phase transformer shown in Fig. 1 so as to correctly determine of the transformer. (A) shunt branch parameters. (B) transformation ratio. (C) series...

Q:

A piece of wire is bent so that the wire forms a closed circle of radius 3 cm. The density of the wire varies. The density is given by p(x, y) = x g/cm. That means the density only depends on how far...

Q:

Mica's Manicures leases a space in the local mall for $ 4 , 5 0 0 a month. For her business, this expense is considered a ( n ) Multiple Choice implici cost of $ 4 , 5 0 0 . fixed cost of $ 4 , 5 0 0...

Q:

3. The experiment by Tolman and Honzik (1930) has traditionally been regarded as a demonstration of learning, in which learning appears to take place in the absence of any reward. The experiment has...

Q:

In Example 13 for testing H0: p = 1/3 (astrologers randomly guessing) with n = 116 when actually p = 0.50, suppose we used Ha: p 1/3. Then show that: a. A Type II error occurs if 0.248

Q:

Consider Example 14 about the power of the test used in the TT experiment for testing H0: p = 0.50 against Ha: p > 0.50, where p is the probability of a correct prediction about which hand was nearer...

Q:

Refer to the simulation in Activity 2 at the end of the section. a. Repeat the simulation, now assuming that actually p = 0.45. In a large number of simulations, what proportion of the time did you...

Q:

Question Let ,ul and p2 denote true average densities for two different types of brick. Assuming normality of the two density distributions, test: H0: p1 ,uz = 0 versus Ha: _u.1 #2 0 Use the...

Q:

Cairo Cab, Inc.. uses two dispatchers to handle reauests for service and to dispatch the cabs. The telephone calls that are made to Cairo Cab use a common telephone number. When both dispatchers are...

Q:

Geary Machine Shop is considering a four-year project to improve its production efficiency. Buying a new machine press for $931,200 is estimated to result in $310,400 in annual pretax cost savings....