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Let $X_1,X_2,...,X_n$ be a random sample from a uniform distribution on $(\mu-\sqrt 3\sigma,\mu+\sqrt3\sigma)$.

Here the unknown parameters are two, namely $\mu$ and $\sigma$, which are the population mean and standard deviation.

Find the point estimator of $\mu$ and $\sigma$.

I have tried to do that by the Method-of-Moments(MOM). The procedure is,

$M\prime_j=\mu\prime_j(\theta_1,\theta_2,...,\theta_k); j=1,2,...,k$

where $M\prime_j$ is the $j^{th}$ sample moment about zero & $M\prime_j=\frac{1}{n}\sum_{i=1}^n X_i^j$

& $\mu\prime_j$ is the $j^{th}$ moment about zero ,ie, $j^{th}$ raw moment.

Now,

$M\prime_1=\mu\prime_1=\mu\prime_1(\mu,\sigma)=\mu$

And

$M\prime_1=\frac{1}{n}\sum_{i=1}^n X_i=\bar X$

Again,

$M\prime_2=\mu\prime_2=\mu\prime_2(\mu,\sigma)=\sigma^2+\mu^2$

$\Rightarrow M\prime_2=\sigma^2+\mu^2$

$\Rightarrow \sigma^2=M\prime_2-\mu^2$

$\Rightarrow \sigma=\sqrt{\frac{1}{n}\sum_{i=1}^n(X_i-\bar X^2)}$

see https://math.stackexchange.com/questions/416581

Hence Method-of-Moment estimators are $\bar X$ for $\mu$

and $\sqrt{\frac{1}{n}\sum_{i=1}^n(X_i-\bar X^2)}$ for $\sigma$.

But the procedure seems illogical to me for the following reason:

$\bullet$ I haven't considered the pdf of uniform density. so this procedure is also applicable for normal density. Then where is the difference?

What is the correct process of finding the point estimators for the above situation?

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the procedure seems illogical to me ... (because) ... I haven't considered the pdf of the uniform density, so this procedure is also applicable for normal density. Then where is the difference?

That's how the method of moments works; if you set your location-scale family of distributions up so it's parameterized in terms of its population mean and variance, you'll use the sample mean and variance to estimate them.

What is the correct process of finding the point estimators for the above situation?

Method of moments isn't 'incorrect' as such. It's more a matter of 'what properties do you want?'.

The more common method to estimate parameters is by maximizing the likelihood (sometimes with adjustments to unbias or nearly unbias the result). Maximizing the likelihood generally results in estimators with a range of nice properties (though unbiasedness is rarely one of them).

However, sometimes maximum likelihood estimators (MLEs) can't be used, or are impractical or ... just difficult. There are many situations where MOM estimators are fairly reasonable and they're often quite easy, even if often less efficient (but in some situations they can be better than ML, even when both are reasonable).

These are not the only possible methods of estimating parameters, but they're the most common ones.

In the case of your uniform example, ML estimators - while biased - have some nice properties compared to MOM estimators:

i) they have much smaller variance as $n$ increases,

ii) they don't produce "impossible" results (it's possible with MOM to produce an estimate of the upper or lower limit that leaves some observations 'outside' those bounds).

However, the estimator for $\sigma^2$ is biased downward.

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