Questions tagged [method-of-moments]
A method of parameter estimation by equating sample and population moments then solving the equations for the unknown parameters.
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Is this correct for (generalised) method of moments?
I am reviewing a MOOC and realised... I didn't grasp the method of moments, so I did try to get it back "from scratch".
Are the above equations/sentences correct ? In the MOOC the analysis ...
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1answer
62 views
How does the information in the problem statement and this solution align with the provided description of the method of moments?
I have the following problem:
Let $Y_1, Y_2, \dots, Y_n$ be i.i.d. $\text{Uniform}(\theta, 1)$ random variables, and let an estimator be $\hat{\theta} = \min\{ Y_1, Y_2, \dots, Y_n \}$.
You may find ...
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1answer
110 views
Negative-Binomial Method of moments with an offset
Given the method-of-moments approach to estimate the parameters of the NB-2 distribution $\mu$ and $\phi$:
$$
\mu = \bar{y}
$$
$$
\phi = \frac{\bar{y}^2}{s^2 -\bar{y}}
$$
How can this be extended to ...
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1answer
39 views
Method of Moment for Gamma Distribution
I know that the empirical $r$-th moment is defined as:
$$\hat E(X^r) = \frac{1}{n} \sum_{i=1}^n x_i^r $$
So for the first moment I did:
$$E_{\lambda,\alpha}(X) = \hat E(X) = \bar X $$
$$\bar X = \...
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advantage of mcmc over method of moments?
When I want to estimation parameter of beta distribution and I have large amount of samples, what would be the advantage of using MCMC over calculating these two parameters (alpha and beta) of beta ...
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53 views
MSPE predictor of MA model
Assume $e_i=(\epsilon_i+\epsilon_{i+1})/2, i=1,...,n$ where $\epsilon_1,...,\epsilon_{n+1}$ are iid with mean zero and variance $\sigma^2$. Then $e_i$ are the moving average errors. Now, consider the ...
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Confidence Interval for Estimator using Delta method
The statement
I am given the following discrete distribution with $\theta>0$
$$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$
I need to ...
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41 views
Fixed repeated sampling
Suppose we are dealing with a linear regression $𝔼[𝑌𝑖|𝑋𝑖]=\beta_0+\beta_1X_𝑖$
The distribution of $X_i$ is fixed in repeated samples.
$\epsilon_i$ follows an unconditional i.i.d. normal ...
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1answer
70 views
Method of moments and MLE
Suppose a random variable follows the logistic distribution, $X ∼ Logistic (\mu, \sigma)$ and we
restrict our attention to random samples drawn from this random variable $X$.
What would be the MoM and ...
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1answer
99 views
Method of Moment, MLE and Information matrix
We have $$\mathbb{E}[Y_i| X_i] = β_0 + β_1X_i$$
What would be the Method of Moments estimator and MLE for this model?
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Why methods of moments is not widely used in machine learning?
If I recall correctly, in most machine learning context I have encountered, only maximum likelihood estimation and maximum a posteriori method are used. Why not method of moments?
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1answer
158 views
MLE and Methods of Moments of Negative Binomial in R
Question
Let's say we define the Negative Binomial as follows:
$$f(x) = {x+r-1 \choose x} p^x (1-p)^r$$
With mean and variance:
$$E(x) = \frac{rp}{1-p} \quad \quad V(x) = \frac{rp}{(1-p)^2}$$
We are ...
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30 views
Finding the Method of Moments in two Parameters
Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n$ from the following distribution
$$f(x;\mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}e^{\frac{-\lambda(x - \mu)^2}{2\mu^2 x}}$$
where $x, \...
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Method of Moments Estimators
Let $X_1, \ldots, X_n$ be a random sample of size $n$ from the following distribution:
$$f(x) = \frac{1}{\theta_2}e^{-\frac{(x - \theta_1)}{\theta_2}}, \;\; x\geq \theta_1, \;\; (\theta_1,\theta_2)\in\...
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Variance of the method of moments estimator for $\mu$ of log-normal distribution
Suppose the data has originated from a log-normal distribution with parameters $\mu$ and $\sigma$ (i.e. the mean and standard deviation of the underlying normal distribution). I have only the sample ...
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1answer
57 views
Relation between OLS, MM and ML
What is the relation between OLS, MM (method of moments) and ML (maximum likelihood)? During my studies, the three concepts got taught completely separated from each other. However, they seem to be ...
2
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1answer
119 views
Poisson Process Method of Moments
Disclaimer: This is a homework problem
A School of Ornithology researcher wants to estimate the number of red-tailed hawks in Ithaca. She radio tags 10 birds, and then sets up a feeding station with ...
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1answer
83 views
Method of Moments for Mixture distribution
The restaurant owner also wants to reconfigure her seating layout, and has asked you for help in modeling her clients.
She gives you a dataset of past reservations, and tells you that she gets a mix ...
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1answer
70 views
Is this estimator biased or unbiased?
A random variable X constructed as follows:
$$X = \sum_{i=1}^{N} Z_i \ $$
where $N$~Poisson$(\lambda)$ with $\lambda > 0,\space$ and$\space$ {${{Z_i}}$}$^N_{i=1}$ is an independent and identically ...
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42 views
Asymptotic distribution of mme of coefficient of variation
For random sample from unif(0,1) distribution, method of moments estimator for coefficient of variation is sample mean divided by sample standard deviation. Here, coefficient of variation theta is mu/...
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1answer
48 views
Weighted sum of negative binomial distributions - approximate fast parameter calculation
Let's suppose we have a convolution (weighted sum) of three negative binomials (parameterised as mean and overdispersion).
...
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78 views
Method of moments as non-parametric model
In Wikipedia article (https://en.wikipedia.org/wiki/Nonparametric_statistics) about nonparametric statistics there is "Method of moments (statistics) with polynomial probability distributions&...
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Can we use pooled ols on dynamic panel without individual fixed effect (but with group fixed effects) and no serial correlation in error
Is it correct that dynamic panel without individual fixed effect (but with group fixed effects), no serial correlation in error can be estimated consistently through pooled ols?
For example, suppose ...
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1answer
174 views
Confusion about method of moments for linear regression
It is known that linear regression estimator can also be viewed as a method of moment estimator derived using the moment condition $E[X\epsilon]=0$. This moment condition follows from exogeneity ...
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24 views
Derivation of the Satterthwaite appproximation for nonnegative estimators
I am reading Statistical Inference, Berger & Casella (Page 314-315) This is for deriving Satterthwaite approximation, where we want to approximate the distribution of $\sum_{i=1}^k a_i Y_i$ where $...
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1answer
113 views
An example of continuous random variable X > 0 with finite second moment but Infinite third moment [duplicate]
Can someone construct an example of this?
i.e., $E[X^2] < \infty$ but $E[X^3] = \infty$. Results could be in terms of pdf, or cdf, or
survival function. Justification would be appreciated
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1answer
35 views
Exhaustive list of techniques used to estimate population mean and variance?
In beginning stats, we were told that:
$\bar{x}$ is an unbiased estimate of $\mu$
$\frac{1}{n - 1}\sum(x - \bar{x})^2$ is an unbiased estimate of $\sigma^2$
As I am reading more, I have learned that ...
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1answer
174 views
Method of moments and MLE estimates for Lomax (Pareto Type 2)
I have this dataset, on which I am supposed to fit Lomax distribution with MM and MLE. Lomax pdf is:
$$f(x|\alpha, \lambda) = \frac{\alpha\lambda^\alpha}{\left(\lambda+x\right)^{\alpha+1}}$$
For MM, ...
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33 views
Higher moments of Lognormal Distribution
I am experiencing a weird problem modeling lognormal distributions and I am quite stuck on this one.
For a normal distributed variable X following a $N(\mu,\sigma^2)$ distribution, we have that $Z=e^...
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Moments method and plug-in method
We have the following discrete distribution on $\{1,2,3\}$:
$$\mathbb{P}(X=1)= \theta^2, \mathbb{P}(X=2) = 2\theta(1-\theta), \mathbb{P}(X=3)=(1-\theta)^2$$
with $\theta \in (0,1)$ and want to use ...
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2answers
122 views
Which estimator is preferred for a random sample from $P_\theta(X=x)=\theta^x(1-\theta)^{1-x}, x=0,1; 0 \le \theta \le \frac{1}{2}$?
Let $X_1,\cdots,X_n$ be an i.i.d sample from $P_\theta(X=x)=\theta^x(1-\theta)^{1-x}, x=0,1; 0 \le \theta \le \frac{1}{2}$. Its the method of moments estimator of the MLE better? Why?
My work:
I ...
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1answer
388 views
Intuition behind Method of Moments estimators of Binomial distribution
The method of moments estimators of the binomial distributions ($x \sim Binom(n, p)$) are a bit weird... I got $\hat p = \bar x + 1 - \frac{\sum x_i^2}{\sum x_i}$ and $\hat n = \frac{\bar x}{\hat p}$. ...
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1answer
159 views
How to estimate parameter using yule-walker/method of moments?
Suppose you observe the first T periods.
X1, X2, · · · , XT of an AR(1) process Xt = µ + φXt−1 + et.
Derive the Yule-Walker/Method of Moment estimate φˆMM for φ.
I thought YW was used to solve for ...
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1answer
81 views
Method of moments when there's no closed form expression
I am trying to code up a method of moments algorithm for parameter estimation. I have a closed form for the moments as a function of the parameters, but these expressions are complicated, so there's ...
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1answer
68 views
Method of moments estimator, $P_\theta(X = x) = \frac{1}{\theta}$
I am struggling with finding a method of moments estimator for (seemingly) simple situation:
pdf is given by $P_\theta(X = x) = \frac{1}{\theta}$,
$x \in$ {1,2,...$\theta$}, where $\theta \in N$.
My ...
3
votes
1answer
314 views
What is the limiting distribution of $Y_n = \sqrt{n}(\bar{X}_n-1)$ as $n \to \infty$?
Let $X_1,\cdots,X_n$ be independently and identically distributed with pdf $f(x)=e^{-x}, 0 < x < \infty$. Let $Y_n = \sqrt{n}(\bar{X}_n-1)$.
What is the limiting distribution of $Y_n$ as $n \to ...
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a question about method of moment estimator
I have a question about method of moment estimator.
Say I have a IID sample $X_1, X_2, ..., X_n$ from an exponential distribution $Exp(\theta$), say I want to find the method of moment estimator of $...
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1answer
216 views
inverse of an exponential distribution
I have a question regarding this.
Say I have $X_1, ..., X_n$ be random sample from an exponential distribution i.e. $Exp(\theta)$, and let $\gamma = \theta^2$. Let denote $\gamma^{mme}$ as the ...
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1answer
206 views
“Appropriate conditions” for method of moments estimator to exist, be consistent, and asymptotically normal?
My statistics text has the following theorem, and alludes to "appropriate conditions on the model", but never specifies what those conditions are. What conditions are necessary?
Let $\hat{\theta}_n$...
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1answer
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Finding method of moments estimator of $\theta$ in $\Gamma(\theta,\theta)$ distribution
Please refer to the question in image
I have tried to find $ E(x) $ but i ended up with $\overline x $ = $\frac{\theta + 1}{\theta} $ which statisfies no option , i also tried to find $ E(x-1)^2 ...
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Method of moment through covariance derivation
Given a Bivariate INAR(1) Poisson Process:
$Y_t^1 = \rho_1 * Y_{t-1}^1+R_t^1$
$Y_t^2 = \rho_2 * Y_{t-1}^1+R_t^2$
Where $R_t^1$ and $R_t^2$ are the innovation terms and follow the bivariate Poisson ...
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1answer
448 views
Method of Moments Bernoulli
We have this pdf for $x_1, x_2,\dotsc, x_n$ :
$$\theta x^{\theta -1 }$$ with indicator variable 1 for $ 0 \le x \le 1$.
We decide not to observe the $x_1,x_2,\dotsc,x_n$ but $y_1,y_2,\dotsc,y_n$ ...
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2answers
4k views
95% Confidence interval of $\lambda$ for $X_1,…,X_n$ IID exponential with rate $\lambda$
I know how how to find the estimation of $\hat{\lambda}$ using the method of moments. I can take the first moment and equate it to the empirical to get,
$E(X) = \frac{1}{\lambda} = \frac{\sum_{i=1}^{...
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1answer
397 views
Deviations of Method of moments estimators for linear regression with constant
I am new to method of moments and want to figure out how to derive the method of moment estimator for $\beta$ in the linear equation with a constant term and three corresponding moments, namely, I ...
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why is method of moments estimates asymptotically normal
I have noticed that a lot of statistics textbooks contain lengthy discussions and detailed proofs on showing that MLE estimates are asymptotically normal (under regularity conditions).
On the other ...
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Approximating standard error that contains a parameter, by replacing the parameter with its estimate
I am a bit confused about the following step I have seen in the stats literature which seems to me a bit circular.
Say you are approximating the standard error of the MoM estimate of an exponential ...
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24 views
How to derive the distribution of OLS starting from the sample moments?
I know I am supposed to start from
$N^{1/2}[N^{-1}\sum x_{i}u_{i}]$
Then by central limit theorem that that it is asymptotically
$ N(E(x_{i}u_{i}),var(x_{i}u_{i})) $
and $E(x_{i}u_{i})=0$
so $ ...
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1answer
182 views
Estimate Information Entropy from Moments
I hope I'm using the right terminology below.
I have access to the moments statistics of a large sample.
That is, I have $\sum(x)$, $\sum(x^2)$, ..., $\sum(x^k)$.
I also have access to max and min, ...
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1answer
68 views
Moment generating function of binomial distribution
I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomial distribution iwth $p=\frac{1}{2}$. With the standardized test statitics is $S=\frac{S(\theta_0)-(\frac{n}{2})}{...
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33 views
Suggesting a method of moments estimator for the chance that some event happens
Let $X_i$~ $\text{Pois}(\lambda)$ be the number of breakdowns a certain ATM machine experiences in the $i^{th}$ week. $\implies$ Let $ \{X_i\}_{i=1}^n$ be iid of the number of breakdowns the machine ...
