Dear colleagues
Good Day;
Let T be a random variable with Gamma distribution:
T~Gamma (n,θ)
What is the probability density function of θ ̂, where
θ ̂= v1 / v2
v1 = a0*T/((n-1) ) + a1*T2/((n-1)(n-2))+ a2*T3/((n-1)(n-2) (n-3))
v2= a0+a1*T/(n-1) + a2*T2/((n-1)(n-2)) , where, a0, a1, a2 are constants.
Many thanks in advance
With Best Regards
Huda
Hi Huda,
The ratio of two random variables does not in general have a well-defined variance. You can try to re-normalize θ ̂ or use approximations or Taylor series expansions about the expected values of v1 and v2. In terms of expected values, it looks like for independent random variables, var(x/y) ≈[⟨xx⟩⟨y⟩⟨y⟩−2⟨xy⟩⟨x⟩⟨y⟩+⟨xx⟩⟨x⟩⟨y⟩]/⟨y⟩^4. However, in your case, v1 and v2 are not independent...
Good luck!
Slava
Dear Kirill,
Thank you very much for your answer.
But as you mentioned it difficult to find g-1(θ).
In fact, I need E (θ ̂ ) and var(θ ̂ )
Best regards,
Huda
I think (θ ̂ ) is distributed Non-central F but what are the degrees of freedom
Many thanks
Huda
Dear Huda,
Since your variables are derived from the random variable T with many multiplication/division operations, it might be more prudent to use a geometric mean for the expected value in the first place. In that case, you could simply calculate it using the expected values for v1 and v2 and var(θ ̂ ) using ratio estimator stats.
Useful links are:
http://www.math.montana.edu/~parker/PattersonStats/Ratio.pdf
http://www.stat.cmu.edu/~hseltman/files/ratio.pdf
I don't know if this will help.-Slava
Hi Vyacheslav,
Many thanks for your response. Unfortunately, the two links you attached related with ratio estimator stats it may has beta distribution because it is one of the continuous probability distributions defined on the interval [0, 1], while θ ̂ does not represent ratio estimator stats.
Best Wishes
Hi Huda,
The ratio of two random variables does not in general have a well-defined variance. You can try to re-normalize θ ̂ or use approximations or Taylor series expansions about the expected values of v1 and v2. In terms of expected values, it looks like for independent random variables, var(x/y) ≈[⟨xx⟩⟨y⟩⟨y⟩−2⟨xy⟩⟨x⟩⟨y⟩+⟨xx⟩⟨x⟩⟨y⟩]/⟨y⟩^4. However, in your case, v1 and v2 are not independent...
Good luck!
Slava
Hi Vyacheslav,
I think each of v1 and v2 has Gamma distribution, as a result, the distribution of (θ ̂ ) will be Non-central F, thus, I have to know the distribution of v1 and v2 accurately.
Many thanks for you
Hi Lm,
principal component using for reduce dimensionality (To the exclusion of unimportant variables).
To solve multicollinearity problem, I'd prefer understand the cause of multicollinearity and remove it. Also you can apply Ridge regression to get BLUE estimators. You may also be able to find a way to combine the correlated variables.
I'm sorry,This unrelated with my theoretical question.
Best Wishes
Hi Huda,
If you start with numerical values of parameters c1, c2, a1, a2, a3, then you can compute the density of \theta^ as follows:
1. First, approximate the
Hi Prakash,
Thank you for your response , I think it is not an accurate solution and take a long time.
Best Wishes
One thing to emphasize: saying "T ~ Gamma (c1,c2)" is ambiguous as there are multiple popular parameterizations for the gamma distribution. The Wikipedia article (https://en.wikipedia.org/wiki/Gamma_distribution) lists two, but I've seen even more. You can eliminate the ambiguity by saying what the mean and variance for T are in terms of c1 and c2.
You don't specify any restrictions on the other parameters, a0, a1, a2 or n. Are there any? Is n an integer? If they are all positive (and n>3), then v1 and v2 seem to be perfectly dependent or "comonotonic", as are all the terms in the sums. This makes it easy to handle the dependence. If X ~ F, and Y ~ G, and X and Y are perfectly dependent, then the Z = X + Y has a distribution whose alpha-quantiles are the sums of the alpha-quantiles of X and the alpha-quantiles of Y. See https://en.wikipedia.org/wiki/Comonotonicity.
Numerical examples suggest the distribution of the ratio can be approximated by a gamma distribution.
Hi Scott,
Thanks alot for your a valuable comments.
I'm sorry there is an a mistake!
I have been changed c1 and c2 to be n ,θ ( i.e.,T~Gamma (n,θ)) and a0, a1, a2 are constants.
Thank you again
Dear colleagues,
Indeed, v1 and v2 are polynomials, for simplicity, I put them as mentioned in the question.
I would derive E(θ ̂ ) and var(θ ̂ ) for minimax estimation.
Many thanks for all
Huda
Hello. I think this problem is too complicated to admit an explicit answer, although it may be possible to write down an integral representation of the PDF. Clearly you need to get the PDF of va and v2 first. Without explicit expressions for these there is little hope for getting the PDF of their ratio. It looks like you need to do a sample-based approach for each value of the parameters. At least the variables are 1-dimensional. Good luck!
Thank you Graham, as you said this problem is too complicated especially, v1 and v2 are kth order polynomials and for simplicity, I put them as mentioned in the question.
Best wishes
Huda
Dear Dr Huda
a probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability of the random variable falling within a particular range of values is given by the integral of this variable’s density over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
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