I'm searching for the first order moment of a truncated Rice distribution.

More precisely, i'm searching for the result of $$\frac{1}{F(l|\nu,\sigma)}\int_{0}^{l}xf(x|\nu, \sigma)dx,$$

where $$f(x|\nu, \sigma)=\frac{x}{\sigma^2}exp\left(\frac{-(x^2+\nu^2)}{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right),$$ and $$F(x;\nu,\sigma)=1-Q_1(\nu/\sigma, x/\sigma),$$ with $Q_1$ the marcum Q-function and $I_0$ the modified Bessel function of the first kind and order 0.

Integrating by parts the first formula, it seems to me that this resumes to integrating $$\int_0^lQ_1(b,ax)dx.$$

I've came accros this document and the formula (60) is close from what i need, but for $l=\infty$.

I'm stuck there. Any clue ?

Thanks for your help : )


One imperfect answer here, based on an approximation of the Marcum Q-function.

Starting with an integration by part, we get $$\int_{0}^{l}xf(x|\nu, \sigma)dx=\left[x(1-Q_1(\nu/\sigma, x/\sigma))\right]_0^l-\int_0^l(1-Q_1(\nu/\sigma,x/\sigma))dx\\ =lQ_1(\nu/\sigma,x/\sigma)+\int_0^lQ_1(\nu/\sigma,x/\sigma).$$

All the hard work is now to integrate $\int_0^lQ_1(\nu/\sigma,x/\sigma)$.

I did not find any analytical solution to this. But in this paper, the Marcum Q-function is approximated using : $$Q_1(a,b)\approx\tilde{Q}_1(a,b)=\exp(-e^{p}b^{q})$$

where $p$ and $q$ are polynomials depending on $a$ (how to find the polynomials is described in the reference). It turns out that this function now integrates well :

$$\int_0^l\tilde{Q}_1(a,x)dx=\frac{(e^p)^{-1/q}(\Gamma(\tfrac{1}{q})-\Gamma(\tfrac{1}{q},l^qe^p))}{q},$$ with $\Gamma(a)$ the gamma function, and $\Gamma(a,x)$, the incomplete gamma function.

Taking care of the change of variable (dividing with $\sigma$), we now have an approximate solution to $\int_{0}^{l}xf(x|\nu, \sigma)dx$.

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