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Consider $Z\sim N(0,1)$ and the moments: $$ E\left[ \left(C_1(Z+\sqrt{\lambda})^2 - C_2\right)^t\right]. $$ Here, $C_1$, $\lambda$ and $C_2$ are arbitrary constants which are all positive. $t$ is a nonnegative integer. Is there an analytical form for these moments in terms of $t$, $\lambda$, $C_1$ and $C_2$?

So far, I have identified that $C_1(A+\sqrt{\lambda})^2 + C_2$ follows a generalized chi-squared distribution. Where, following the notation on the wiki, $k=1$, $\lambda = \lambda$, $w = C_1$, $m=C_2$, $s=0$. However, I do not see any analytical expression for the moments on the wiki.

I see there are analytical moments for the noncentral chi-squared distribution. I also know how the moment generating function of the noncentral chi-squared distribution can be used to obtain moments for the generalized chi-squared distribution. However, I am able to obtain an analytical expressions using that approach.

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    $\begingroup$ I imagine lambda is positive, is t as well, or either C for that matter? $\endgroup$ Jul 29 at 19:27
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    $\begingroup$ For what it's worth, Mathematica could not solve this integral, even with the assumptions that t and lambda > 0. (Edit: Integral t doesn't help either, unfortunately). $\endgroup$ Jul 29 at 19:29
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    $\begingroup$ Assuming $t$ is a non-negative integer, expand the argument as a polynomial in $Z$ and apply linearity of expectation. $\endgroup$
    – whuber
    Jul 29 at 20:03
  • $\begingroup$ Thank you for that last comment. I will post an answer below. I feel that this is the correct method. $\endgroup$
    – SeanBrooks
    Jul 29 at 20:49

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Following the suggestion provided by @whuber: $$ \left(C_1(A + \sqrt{\lambda})^2-C_2\right)^t = \sum_{k=0}^t\text{nCr}(t, k)\left(C_1(A + \sqrt{\lambda})^2\right)^k\left(-C_2\right)^{t-k} $$ Then apply the expectation operator: $$ \text{E}\left[\left(C_1(A + \sqrt{\lambda})^2+C_2\right)^t\right] = \sum_{k=0}^t\text{nCr}(t, k)C_1^k\text{E}\left[\left((A + \sqrt{\lambda})^2\right)^k\right]\left(-C_2\right)^{t-k} $$

Note that $\text{E}\left[\left((A + \sqrt{\lambda})^2\right)^k\right]$ is the $k$-th moment of a noncentral chi-squared distribution with analytical form reported here.

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