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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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 ...
2
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1answer
22 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 ...
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1answer
74 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
54 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
43 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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15 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
26 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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32 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
98 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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18 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
57 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
29 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
54 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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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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21 views
How to generate numbers from Beta distribution in R if one of the parameters is $\theta$?
I have to estimate ( using method of moments) $\theta$ with an estimator $\hat{\theta}$ for a function with pdf given as $$\theta x^{\theta-1}, \ \ \ \text{ for 0 < x < 1, and } 0 \ \text{ ...
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43 views
Moments method and plug-in method
We have the following discrete distribution on $\{1,2,3\}$:
$$\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
99 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
58 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
88 views
How to estimate parameter using yule-walker/method of moments?
Suppose you observe the first T periods.
X1, X2, Ā· Ā· Ā· , XT of an AR(1) process Xt = Āµ + ĻXtā1 + et.
Derive the Yule-Walker/Method of Moment estimate ĻĖMM for Ļ.
I thought YW was used to solve for ...
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1answer
44 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
41 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 ...
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0answers
37 views
Approximating distribution by moment matching?
I am going to compare distributions by moment-matching (expected value, standard deviation, skewness, kurtosis etc). The question is simple: As the moment-matching relates to Taylor expansion, would ...
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26 views
Checking unbiasedness of an estimator
After finding the method of moments estimator, how do we check for unbiasedness?
For an exponential distribution, I found the method of moments estimator as (theta + 1), the book concluded that ...
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1answer
155 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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63 views
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
122 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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37 views
Tau2 in random-effect multiple meta-regression
I'm doing random effect multiple meta-regression with two predictors and using method of moments to estimate between-studies variance (tau2). Tau2 formula for one predictor case is:
with F being:
Q -...
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1answer
38 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
281 views
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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64 views
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
301 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
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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
241 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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39 views
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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37 views
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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23 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 $ ...
4
votes
1answer
145 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
53 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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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 ...
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2answers
2k views
Method of moments for linear regression?
I have been reading about the method of moments, and now I understand how to obtain the method of moments estimator for a random sample $x_1,...,x_n$ from a distribution $f(x;\theta)$, in the ...
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0answers
125 views
Check Computation of MME and MLE
Let $X_1$, . . . , $X_n$ be i.i.d random variables having pdf
$$f(x\mid\theta) = (\theta+ 1)x^{\theta}I_{(0,1)}(x)$$
where $\theta \gtā1$
(a) Give a MME of $\theta$ based on the first ...
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0answers
25 views
Estimated Standard Error of a Method of Moment Estimator, Poisson Example
For X randomly sampled from a Poisson(lambda) population, the method of moment estimate of lambda is the sample mean:
lambda_hat = X_bar
Now, say we are interested in the variance of lambda_hat (...
3
votes
2answers
329 views
Reasons for different parameters via MoM and MLE
I got a study of 210 samples and I tried fitting gamma distribution to them. I used method of moments and maximum likelihood estimation to calculate the parameters, but parameters came out quite ...
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21 views
Method of moments giving super sensitive estimates
I'm trying to study a process that produces, in theory, an equilibrium distribution where the $i$th raw moment is given by:
$$ \mu_i = \exp(-\theta_1 \sum_{j=0}^{i-1}(1 + j\theta_2)^{-\theta_3}) $$
...
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170 views
Method of Moment Estimator — Uniform Dist
Find the two method of moment estimators for $\theta$ given that $Y_i | \theta$ is distributed i.i.d U(0,$\theta$).
We know that E($Y_1$) = $\frac{\theta}{2}$ and Var($Y_1$) = $\frac{\theta^2}{12}$ ....
3
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1answer
2k views
Confidence Interval for a Uniform Distribution based on Method of Moments
Let $X_1,..,X_n$ be a random sample of $X$~$U[\theta,\theta+1]$. Given a sample $n=100$ from that distribution, the following statistic was calculated:
$\sum\limits_{i=1}^n X_i = 350.492$
I need to ...
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0answers
257 views
Solving a system of equation by moment condition reports error but minimum distance works
I do not understand the following error message I get using the gmm function in R. The code below creates two moment conditions (...
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1answer
776 views
By conditioning on $N$, show that the moment generating function of $Y$ is given by $m_Y(t)=m_N(\ln(m_X(t)))$
I am having a difficult time using moment generating function properties to prove this: (any direction or key properties will be very helpful)
Let $X_1$, $X_2$, . . . be independent and identically ...
