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Question:

$y_0 = z^d$ is computed from the sum of some recordings by a sensor. Let, there be $k$ sensor nodes. This parameter is calculated by each sensor node and then transmitted to the base station which is the receiver where the channel noise is an Additive White Gaussian noise. So, at each $t$ time instant, a new value of this parameter is received, but is a noisy version of the true parameter.

$y(t) = y_0 + \eta(t)$ for $t=1,2,..,T$ are the observations

$d<T$ and is a positive known integer.

$\eta(t) = N(0,\sigma_\eta^2)$

The distribution of $z^d$ is generalized Gamma (How to compute the pdf analytically for the sum of generalized gamma).

Statistical Information about the parameter:

$$z^d = \sum_{i=1}^k X_i^d$$ and $$\sum_{i=1}^k X_i^d\sim\text{G}(kn,\lambda)$$. $Z$ has a Generalized Gamma distribution and

$z^d = \sum_i^{k} X_i^d = G(kn, \lambda)$

Assuming, $\lambda =1$ and $k,n,d$ are known.

I need to find an estimator for the parameters $y_0, \sigma_\eta^2)$ from the observations $y(t)$,in order to get an estimator for the parameter $z^d$.

Assuming, $\lambda =1$.

This is how I proceeded.

Formulating a Maximum likelihood estimator:

So, the likelihood will be

$p(y;\mathbf{h}) = \frac{1}{{(2 \pi \sigma_\eta^2)}^{T/2}} \exp{(-(y - y_0(t))^2)/ 2\sigma_\eta^2}$. $ = \frac{1}{{(2 \pi \sigma_\eta^2)}^{T/2}} \exp{(-(y(t) - z^d)^2)/ 2\sigma_\eta^2}$

$ = \frac{1}{{(2 \pi \sigma_\eta^2)}^{T/2}} \exp{(-(y(t) - \sum_i x_i^d)^2)/ 2\sigma_\eta^2}$

Then, I need to differentiate w.r.t the unknowns and equate to zero.

The MLE of $x$ is the value that minimizes the likelihood function or equivalently the value that minimized

$J(x) = \sum_{t=0}^{T-1} {((y(t) - z^d)}^2$

Differentiating $J(x)$ and setting it to to zero produces

$\sum_{t=0}^{T-1} ((y(t) - z^d) dz^{d-1} = 0$

This is a nonlinear equation in $x$ and cannot be solved directly. Newton-Raphson is a method but it works only for very close initial guess to $x$ and $d$. So, Was thinking how to apply Expectation - Maximization.

I don't know if my approach is correct or not. After this I am stuck because the answer is zero. So, how can I apply expectation maximization

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  • $\begingroup$ As described the problem does not make sense. If you observe the $y(t)$'s and only the $y(t)$'s, you can estimate $(y_0,\sigma_\eta)$ by regular MLE, no need of EM. Since $z^d=y_0$, you can derive immediately the MLE of $z$. The information that $z$ is a generalised Gamma variate is not useful to estimate $z$. $\endgroup$ – Xi'an Apr 12 '15 at 8:38
  • $\begingroup$ @Xi'an: I tried the MLE for $z$, but the derivative when evaluated to zero, makes the expression zero. So, eitehr we go for numerical optimization like Newton Raphson or EM. Can you please show the MLE estimates; maybe I did something wrong in the steps that I have put down in the Question. Thank you for your time and effort. $\endgroup$ – Srishti M Apr 13 '15 at 17:27
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I need to find an estimator for the parameters $(y_0,σ^2_η)$ from the observations $y(t)$,in order to get an estimator for the parameter $z_d$.

In your formulation, $$y(t) = y_0 + \eta(t)$$with $\eta(t)\stackrel{\text{iid}}{\sim}N(0,\sigma^2_\eta)$, i.e.$$y(t) \stackrel{\text{iid}}{\sim}N(y_0,\sigma^2_\eta)\qquad t=1,\ldots,T$$Therefore, if you are only interested in estimating $(y_0,σ^2_η)$, this is a standard normal model with MLEs$$\hat y_0=\frac{1}{T}\sum_{t=1}^T y(t)\quad \hat{σ^2_η}=\frac{1}{T}\sum_{t=1}^T (y(t)-\hat y_0)^2$$ If you assume further that$$y_0=z^d$$then$$\hat z=\hat{y_0}^{1/d}$$since the MLE of the transform is the transform of the MLE (assuming $\hat y_0>0$).

As I mentioned in a comment, the information that $z$ is a generalised Gamma variate is not useful to estimate $z$.

Update

Now, if repeated observations are available, with $y_0(t)=z(t)^d$, and $y_0(t)\sim \text{Ga}(kn,\lambda)$, this item of information does not contain anything about $d$. Once $y_0$ is observed or estimated, you cannot derive an estimator of $d$ because the assumption for the Gamma distribution is on $y_0(t)$, not $z(t)$. The parameter $d$ is not identifiable for this experiment.

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  • $\begingroup$ Thank you for your reply. Can you please clarify the following things since I am unaware of the theory behind them? (1) You mentioned MLE of the transform = transform of MLE - I could not find background for this in Steven Kays book - Statistical Signal Processing. I was under the impression that to find estimator $\hat{z}$ I need to take $\log (y_0) = d \log z^d$ (2) Can you share some ideas if anything will change when $y_0$ changes to y_0(t) then how to find the estimator when $d$ is also unknown? So, for the case $y(t) = y_o(t) + \eta(t)$ $\endgroup$ – Srishti M Apr 15 '15 at 19:33
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    $\begingroup$ (1) There are other textbooks than this one, this result is by all means standard; (2) if $d$ is unknown and you have a single observation $y_0$, $d$ cannot be estimated. $\endgroup$ – Xi'an Apr 15 '15 at 20:04
  • $\begingroup$ For (2) I do not have a single observation; now if $y(t) = y_0(t) + \eta(t)$; can $z^d(t)$ and $d$ be estimated where for every time instant a new value of $y_0$ is obtained. Then $y_0(t)$ is no longer a deterministic parameter. This is where I asked your insights about. Can you please help? Thank you. $\endgroup$ – Srishti M Apr 15 '15 at 20:27

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