3) As on previous let X_1….X_n be i.i.dwith pdf
f_θ(x)= θx^(θ-1)1(0≤θx≤1) where θ>0
a)
Assume we do not actually get to observe X_1…X_n Instead let Y_1… Y_n be our observations where Our goal is to estimate θ based on this new data.
What distribution does Y_i follow?
First, choose the type of the distribution:
Bernoulli
Poisson
Normal
Exponential
Second, enter the parameter of this distribution in terms of θ. Denote this parameter by . (If the distribution is normal, enter only 1 parameter, the mean).
m θ= ?
b)
Write down a statistical model associated to this experiment is the parameter θ identificable
Y
N
Bernoulli
a.
1/(2^theta)
b.
Y
c.
(ln(2))^2/(2^theta-1)
d.
-ln(barY_n)/ln(2)
e.
-ln(barY_n)/ln(2)
f.
(2^theta-1)/(ln(2))^2
g.
1-Phi(sqrt(n/V)*(hattheta-1))
no
a) The distribution that Y_i follows can be determined by transforming the distribution of X_i.
Since Y_i = θX_i^(θ-1)1(0≤θX_i≤1), we can see that Y_i is a transformed random variable of X_i.
To determine the distribution of Y_i, we need to find the cumulative distribution function (CDF) of Y_i.
The CDF of Y_i can be found by integrating the probability density function (PDF) of Y_i over the range of possible values.
The PDF of Y_i can be obtained by differentiating the CDF of Y_i.
To determine the type of distribution, let's examine the range of possible values for Y_i.
Since 0≤θX_i≤1, we can conclude that 0≤Y_i≤θ.
Therefore, Y_i follows a Uniform distribution on the interval [0, θ].
b) The statistical model associated with this experiment is identifiable if there is a unique value of the parameter θ that corresponds to the observed data.
In this case, the statistical model is identifiable because the parameter θ is uniquely determined by the observed data Y_1…Y_n.
a) To determine the distribution that Y_i follows, we need to derive the distribution of Y_i based on the given pdf of X_i.
Given that Y_i is an observation of X_i, we can express Y_i as a function of X_i. In this case, Y_i = θX_i^(θ-1)1(0≤θX_i≤1).
Since X_i follows a continuous uniform distribution on the interval [0,1], the probability that θX_i lies between 0 and 1 is always 1. Therefore, we can simplify the expression of Y_i:
Y_i = θX_i^(θ-1)
Now, we can see that Y_i follows a distribution with a known form. This distribution is called the Pareto distribution with parameters θ and θ-1. The Pareto distribution has a power-law tail and is commonly used in statistical modeling for heavy-tailed data.
b) To determine if the parameter θ is identifiable, we need to consider if different values of θ would yield different probability distributions for Y_i.
In this case, since Y_i follows a Pareto distribution with parameters θ and θ-1, changing the value of θ would lead to different probability distributions for Y_i. Therefore, the parameter θ is identifiable in the statistical model associated with this experiment.