Dear Amir,

As I mentioned previously, I need the mth moment for reliability function of Erlank distribution when the shape and scale parameters are unknown, which is the solution of the ratio of double integration. I 'll try to use different values of Alpha.

Regards

This is an interesting question.

I suggest that you use Mathematica to evaluate the ratio of double integrals in the expression that have given. A very good introduction, showing how to do this is given at

https://mathematica.stackexchange.com/questions/116311/double-integral-over-the-region-enclosed-by-several-lines

The attached image (from the stackexchange page) shows an example of what is possible.

Dear James,

Thank you so much.

I'm so gratful for your suggestion and I'll try evaluate the ratio of double integrals by using Mathematica .

I wish you all the best

Hi Aden,

may be Lindley’s approximation is useful for such ratio of double integrals but I'll need some complex derivations. Thanks.

Please use software

Hi Enkuahone,

Do you know a suitable software for solving such this ratio of double integrals.

Regards,

Python, Matlab, Mathematica, or Wolfram.

You are right Aden Handasyde , the use of computational software will be of great benefits.

Try Maple software. It is much more easier to use. Other good softwares have been recommended for you by Aden Handasyde.

Thanks Philip, I'll try to use Maple.

Regards,

Dear Dr Huda, Try Matlab software.

Kind regards

Kamel Saoudi

Dear Dr Huda, Tray R language

by library

cubature

by function

adaptIntegrate

Kind regards

Ehab Almetwally

My dears colleagues, Mervat , Kamel and Ehab

Thanks a lot for your answers. But unfortunately I do not have a complete idea of using Matlab or R softwares in integrations.

Regards,

Dear friend

I have a license to use Mathematica software. If you did not get the solution, I can try.

One tip is you use numerical approximations that, in general, work well in practice.

Best regards,

Thiago

I'm 100% sure it's precisely zero, because the rate of change of a maximum limit of alpha-1 in the very first sum (capital sigma) is 0, which kills the whole thing instantly. A trick question.

Hi Aden,

No, it isn't 0 . This question is not A trick. It is related with deriving the reliability function . You may not have enough idea about the reliability functions.

But that first integer sum, has a component in it's maximum limit, and the change in that must be an integer, which makes it proportionally infinitesimal by being discrete in respect to the continuous outer integral.

https://www.wolframalpha.com/input/?i=integral+from+0+to+inf+of+the+sum+from+i%3D1+to+a+of+x%5Ei+da

You can use MATLAB

Hi Aden ,

I would like to estimate the reliability function for one of the important distributions, I need the mth moment for reliability function (E (R )m) which is the result for that ratio of double integration.

The solution may be approximate using Lindley's approximation technique , but I found it is difficult to apply to this integration

Thank you for your participation

It is difficult to solve analytically, so approximations should be used such as Lindley's approximation or Laplace's approximation.

Dear Mahmoud, I agree with you. But Lindley's approximation is also difficult in this integration. I don't know how can I apply Laplace's approximation.

Best Regards,

Huda

In order to approximate the model, you can use some interpolation methods (also called metamodels or surrogates). You just need a few number of sample points. Then can fit a metamdoel (e.g. radial basis function) over whole input/output data, and easily use this approximation instead of the original model.

Regsrds

Amir

Please tell me the variability of the supremum of the first sum. Clearly you have missed this, because Alpha can only be greater than or equal to 1, and is in Z+ not R+.

That function comes from the Taylor-MacLaurin's series, right?

Dear Amir,

As I mentioned previously, I need the mth moment for reliability function of Erlank distribution when the shape and scale parameters are unknown, which is the solution of the ratio of double integration. I 'll try to use different values of Alpha.

Regards

Huda A. Rasheed

First: As Aden also indicates, your expression does not look quite right. The sum over j in the numerator integrand must run over a range of integers. Hence α-1 must be a non-negative integer. That is not consistent with the instruction that α should be integrated over from 0 to ∞. Is this perhaps a misprint?

Second: Consider the integrands as α -> 0. Due to the Γ(α)-factors in the denominators, the whole integrand behaves like αa-n-1 in this limit. This makes the α-integral convergent only when a-n > 0. How is this assured? For a properly formulated problem, like the one you indicates, this requirement should be automatically fulfilled. Is there a relation between a and n (which you don't even mention in your parameter description)?

Third: Your expression depends on n parameters xi, but only through the combinations X = ∑i xi and L = ∑I log(xi), plus the parameter x. Is is possible to assume something about X and/or L, which may allow approximate evaluation of the integrals? If not, it seems you will be stuck with numerical evaluations for a 5-dimensional space of parameters, for each value of m and n. It may be hard to extract useful information out of that.

Finally I must say that I don't understand the problem you want to solve. Could you perhaps explain it a little more in layman's language? By the Erlank distribution you presumably mean the Erlang distribution(?), but what is the reliability function?

Dear Aafaq,

Thank you very much for your answer, I'll try using MATLAB .

Regards

Kåre Olaussen

First: This integration is quite quite right but it is difficult to obtain the solution of the ratio of two integrals by numerical methods. Therefore, the solution can be approximately (by using the Lindley’s approximation for example)

Second: Reliability function (R(t)) gives the probability of an item operating for a certain amount of time without failure. As such, the reliability function is a function of time, in that every reliability value has an associated time value. In other words, one must specify a time value with the desired reliability value, i.e. 95% reliability at 100 hours.

https://weibull.com/hotwire/issue7/relbasics7.htm

R(t) = p (T> t) = 1- F (t) , where F (t) is the cumulative distribution function

Third: The Erlang distribution is a specific case of the Gamma distribution. It is defined by two parameters 1. shape parameter ( α ). This must be a positive integer (an integer is a whole number without a fractional part). In the Gamma distribution, α can be any real number, including fractions. 2. scale parameter (Beta) which must be a positive real number. Where alpha and beta were unknown parameters.

There are a large number of references on the reliability function and Erlang distribution that you can review.

Finally, I would like to estimate R(t) using Bayesian estimation, Which assume that the two parameters are random variables. I need mth moment of R(t) which is the double integration of ( R(t)m * Joint Posterior of α and Beta. as I mentioned.

I agree with Dr. Huda, "This integration is quite quite right but it is difficult to obtain the solution of the ratio of two integrals by numerical methods. the solution can be approximately.

Regards

Huda A. Rasheed and Wafaa S. Hasanain > "This integration is quite quite right"

Not really. And why didn't you consider, and address, my (and Aden's) specific point?

According to your explanation, your reliability function is

R(x) = [1/Γ(α)] ∫βx∞ tα e-t dt/t = Γ(α, βx)/Γ(α),

where Γ(α, βx) is the incomplete Gamma function. This reduces to your finite sum only when α is a positive integer, https://en.wikipedia.org/wiki/Incomplete_gamma_function

I take back the second point of my first post; 1/Γ(α) has unproblematic behaviour when α → 0.

Kåre,

For Erlang distribution, ( α ) must be a positive integer., while in the Gamma distribution, α can be any real number.

Huda A. Rasheed > For Erlang distribution, ( α ) must be a positive integer

You should make that explicitly clear in your opening post. Moreover, then you no longer have a double integration, but one integral over β and one sum over α. That makes numerical work simpler (but not analytic approaches).

A considerable progress can be made if one can assume that n is large, which anyway is required for any relieable parameter estimation. Is large n a realistic assumption in your case(s)?

The integral to infinity with the d-alpha, and alpha in the sum within the integral means that your integration is point-wise, because d-alpha can only be unitary.