Part B In randomized response study participants are asked a sensitive question. For example, how many alcoholic drinks they have per week. But instead of asking them directly, a second, less sensitive question is available. For example how many cups of coffee they drink which is known to be 3 on average. Then, a random draw is taken out of the two questions, for example, the sensitive question is drawn with probability 0.5 and the resulting question, sensitive or non-sensitive question is shown to the participant who is the only one who sees the question. Then, a response is recorded, for example the number 3. This could then mean, in the example, 3 alcoholic drinks per week or 3 cups of coffee per day. The motivation of doing it in this anonymous way is the aim to achieve a less biased response in comparison to asking directly. We assume here that the distribution of the non-sensitive question is completely known and the probability of selecting a sensitive question is 0.5. We assume further that the distribution of the sensitive question is given by a Poisson distribution with parameter 0 > 0 which is unknown and it is the goal to estimate 0. Suppose that X1,..., X are a random sample of response from the experiment described above. It is clear that X, follows the following distribution with probability mass function 1 fi(e) + exp(-0)0"/2! for x = 0, 1,2, ... and i = 1,...,n. Here fi(x) of the non-sensitive question. exp(-3)3"/a! is the known probability mass function 1. Determine the log-likelihood function of 0. 2. Show that the first derivative of the log-likelihood is exp(-0)0":/x;!+ exp(-0)0"i-1/(x; - 1)! fi(x;) + exp(-0)Oi /x;! dlogL de i=1 1 Po(xi – 1,0) – Po(xi,0) f1(x;) + Po(x;; 0) where Po(x, 0) = exp(-0)0"/x! is the Poisson probability mass function. 3. Let 0; denote a current, given value for 0. Using the result on the first derivative in 2., show that the Gauss-Newton method for constructing the maximum likelihood estimate is given by E;A;(0;) E; A;(0;)² 0j+1 = 0;+
Part B In randomized response study participants are asked a sensitive question. For example, how many alcoholic drinks they have per week. But instead of asking them directly, a second, less sensitive question is available. For example how many cups of coffee they drink which is known to be 3 on average. Then, a random draw is taken out of the two questions, for example, the sensitive question is drawn with probability 0.5 and the resulting question, sensitive or non-sensitive question is shown to the participant who is the only one who sees the question. Then, a response is recorded, for example the number 3. This could then mean, in the example, 3 alcoholic drinks per week or 3 cups of coffee per day. The motivation of doing it in this anonymous way is the aim to achieve a less biased response in comparison to asking directly. We assume here that the distribution of the non-sensitive question is completely known and the probability of selecting a sensitive question is 0.5. We assume further that the distribution of the sensitive question is given by a Poisson distribution with parameter 0 > 0 which is unknown and it is the goal to estimate 0. Suppose that X1,..., X are a random sample of response from the experiment described above. It is clear that X, follows the following distribution with probability mass function 1 fi(e) + exp(-0)0"/2! for x = 0, 1,2, ... and i = 1,...,n. Here fi(x) of the non-sensitive question. exp(-3)3"/a! is the known probability mass function 1. Determine the log-likelihood function of 0. 2. Show that the first derivative of the log-likelihood is exp(-0)0":/x;!+ exp(-0)0"i-1/(x; - 1)! fi(x;) + exp(-0)Oi /x;! dlogL de i=1 1 Po(xi – 1,0) – Po(xi,0) f1(x;) + Po(x;; 0) where Po(x, 0) = exp(-0)0"/x! is the Poisson probability mass function. 3. Let 0; denote a current, given value for 0. Using the result on the first derivative in 2., show that the Gauss-Newton method for constructing the maximum likelihood estimate is given by E;A;(0;) E; A;(0;)² 0j+1 = 0;+
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Transcribed Image Text:Part B
In randomized response study participants are asked a sensitive question. For example, how many
alcoholic drinks they have per week. But instead of asking them directly, a second, less sensitive
question is available. For example how many cups of coffee they drink which is known to be 3 on
average. Then, a random draw is taken out of the two questions, for example, the sensitive question is
drawn with probability 0.5 and the resulting question, sensitive or non-sensitive question is shown to
the participant who is the only one who sees the question. Then, a response is recorded, for example
the number 3. This could then mean, in the example, 3 alcoholic drinks per week or 3 cups of coffee
per day. The motivation of doing it in this anonymous way is the aim to achieve a less biased response
in comparison to asking directly. We assume here that the distribution of the non-sensitive question
is completely known and the probability of selecting a sensitive question is 0.5. We assume further
that the distribution of the sensitive question is given by a Poisson distribution with parameter 0 > 0
which is unknown and it is the goal to estimate 0.
Suppose that X1,..., X, are a random sample of response from the experiment described above. It is
clear that X; follows the following distribution with probability mass function
1
1
fi(2) +, exp(-0)0" /æ!
for x = 0, 1, 2, ... and i = 1,..., n. Here fi(x) = exp(-3)3"/a! is the knowm probability mass function
of the non-sensitive question.
1. Determine the log-likelihood function of .
2. Show that the first derivative of the log-likelihood is
exp(-0)0": /x;!+ exp(-0)8"i-1/(x; – 1)!
Σ
fi(xi) + exp(-0)O:/x;!
dlogL
do
i=1
1
Po(x; – 1,0) – Po(xi, 0)
fi(xi) + Po(x;; 0)
-
i=1
where Po(x, 0) = exp(-0)0"/x! is the Poisson probability mass function.
3. Let 0; denote a current, given value for 0. Using the result on the first derivative in 2., show
that the Gauss-Newton method for constructing the maximum likelihood estimate is given by
E;A:(0,)
Oj+1 = 0; +
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