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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Estimation of variance in GMM
In GMM, the efficient weight matrix minimizes the asymptotic variance of the GMM estimator by setting:
$$ W_T^{opt} = S_T^{-1}$$
where $S_T$ is an estimator of the asymptotic variance of the moments,...
3
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1answer
16 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
48 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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0answers
27 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
58 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 $y_1,y_2,\dotsc,y_n$ ...
3
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2answers
141 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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21 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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26 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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0answers
36 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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22 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 $ ...
2
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1answer
94 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
30 views
Moment generating function of binomial distribution
I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomail distribution iwth $p=\frac{1}{2}$. With the standardized test statitics is $S=\frac{S(\theta_0)-(\frac{n}{2})}{...
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31 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 ...
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2answers
460 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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84 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 $\theta \gt−1$
(a) Give a MME of $\theta$ based on the first ...
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0answers
21 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
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2answers
107 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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0answers
40 views
Interpreting weighted MLE, MoM estimates with stochastic weights
I deal with the weighted probability density $h(x)=w*f(x, \theta)$, where x is the stochastic variable, f is the unweighted probability density, w is stochastic weight depending on x and also measured ...
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18 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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110 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
803 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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110 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
163 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 ...
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1answer
408 views
Two Parameter Method of Moments Estimation
I know this is a very basic question, but I am getting a bit confused due to some variation between resources I'm using for a statistics course.
Say you have some $iid$ random samples $X_1$,$X_2$,......
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1answer
81 views
MME for exponential family
Let $X_1, X_2,...,X_n$ be iid random variables having pdf
$$f(x|\theta) = \frac{1}{x \sqrt{2\pi\theta}}e^{(-[\log x]^2/[2\theta])} I_{(0,\inf)}(x)$$ where $\theta > 0$. Determine the MME of $\theta$...
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1answer
39 views
Find the Method of moments estimate
So here i am getting E(X)=1/2,E(X^2)=0 and E(X^3)=Theta squared/4
How do i proceed now?How do i use the given x values ?
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1answer
323 views
Deriving method of moments estimator for AR(1) process
The method of moments estimator for AR processes can be had with the Yule-Walker equations. But how is it derived?
The equation for AR(1):
$$Y_t =aY_{t-1}+\epsilon_t$$
Where $\epsilon $ ~ $N(0,\...
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1answer
171 views
How can we determine the parameters of a Beta-Binomial Distribution for given mean and variance?
Given a beta-binomial random variate $X$ with $N$ known, how can I choose $\alpha$, $\beta$ such that the distribution's mean matches a chosen quantity $\mu$ and its variance matches a chosen quantity ...
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1answer
125 views
It is required to obtain the method of moment estimator and maximum likelihood estimator of a exponential distribution with two parameters
I have the following density:
$f(x)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}$ where $x>\theta$. It is required to obtain the maximum likelihood estimator and method of moment estimator for $\...
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0answers
42 views
How does a method of moments estimate compare with a true value of the parameter?
To cut the long story short, I've been given this information:
pdf:$$f_X=\tau xexp\left({\frac{-\tau x^2}{2}}\right)$$
$x,\tau>0$ I went ahead and found the cdf:$$F(X)=1-exp\left(\frac{-\tau x^2}{2}...
3
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1answer
246 views
Truncated Beta parameters - method of moments
Context
Let me first introduce some context.
The probability density function and cumulative distribution function of a Beta random variable with parameters $\alpha>0$, $\beta>0$ and support ...
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1answer
60 views
Some questions about the Generalized Method of Moments
Here are three (somewhat related) questions about the (Generalized) Method of Moments. I have only just today started studying this method.
Concerning the following statement by Greene:
Why is $...
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52 views
Can I use the method-of-moments estimator to get the adjustment factor for a generic transformation of the regressand?
When the regressand is in logarithmic form, the model looks like this:
$$log(y)=X\beta+u$$
When we are ultimately interested in predicting $y$, we cannot simply use $exp(X_0\hat\beta)$ as a ...
3
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0answers
79 views
Discrete, finite L-moment problem
Suppose that we have a real-valued discrete random variable, whose probability distribution has finite support on some set $S$ of real numbers. Then if $N = |S|$, we know that we can construct the ...
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2answers
219 views
Is independence assumption needed for Method of Moments estimator
I read about Method of Moments estimator (MOM) in Statistical Inference by Casella and Berger. In MOM description, I do not see the requirements that the sample should be iid. However, in the examples,...
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1answer
185 views
Empirical Bayes: method of moments
The model for the data is $X_{i}$ ~ $Bin(n_{i},\theta_{i})$ (iid, $i=1,...,k$).
The prior distribution is $\theta_{i}$ ~ $Beta(\alpha,\beta)$.
How do we choose (and deduct) moment estimators for $\...
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0answers
60 views
Numerical method to compress empirical probability distribution
I am trying to grapple with the following problem.
I have an application that develops empirical distributions. In essence, I end up with a histogram of equally spaced $x$ values, with both a $max$...
4
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1answer
500 views
Hyper-parameter estimation for Beta-Binomial Empirical Bayes
I am reading a paper Illustrating empirical Bayes methods and in the paper the author uses method of moments to derive the value of an estimate. In equation 17 the author gives the following marginal ...
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1answer
345 views
Solving log-logistic distribution parameters from moments
Let's say we have a log-logistic random variable $X$ with probability density function:
$$f(x)=\frac{(\beta/\alpha)(x/\alpha)^{\beta-1}}{(1+(x/\alpha)^{\beta})^{2}}$$
where $\alpha>0$, $\beta>...
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1answer
751 views
Beta distribution parameter estimation: method of moments
In a paper: Topics over time, method of moments was applied to estimate $\alpha$ and $\beta$ for a Beta distribution.
My question is that how $\alpha$ or $\beta$ should be calculated if there are no ...
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1answer
93 views
Hawkes processes: interpretation of maximum likelihood estimates disagreeing with moment estimator estimates?
So, I have some data.. and a parameterized Hawkes process which I estimate parameters for via maximum likelihood... the residuals ( the compensator aka the dual-predictable projection) are good in ...
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1answer
2k views
How to use method of moment to find Pareto distribution estimator?
I have $f_{\alpha, \beta}(y)=\frac{\alpha}{\beta}(\frac{\beta}{y})^{\alpha +1}, y\ge\beta,\ \ \alpha,\beta\gt 0$. Both $\alpha, \beta$ unknown.
To find estimators using the method of moment, we ...
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2answers
3k views
Real life uses of Moment generating functions
In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in ...
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1answer
76 views
Estimating variance of dgp that measured as Bernoulli Parameter
I have some parameters $p_1,\ldots,P_n$ distributed iid. with mean and variance $\mu, \sigma^2$.
Then, for each $t\in\{1,\ldots,n\}$ I have $\{X_{1, t}, \dots X_{N, t}\}$ realizations of the ...
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1answer
55 views
Method of moments to estimate a parameter
Suppose that $Y_1, Y_2, ..., Y_n$ is a random sample from a
distribution with density function
$$ f(y) = \begin{cases} \theta y^{\theta - 1}\ \ \ \ 0 < y < 1, \\ 0\
\ \ \ \ \ \ \ \ \ \ ...
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1answer
123 views
Method of Moments for $\nu$ of standard t-distribution: what if true $\nu=2$?
Note I am considering the standard $t$ distribution $(\mu=0,\sigma=1)$
The method of moments for $\nu>2$ is derived in this question
My question is, if the true (population) value of $\nu$ is $2$,...
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1answer
200 views
method of moments for skew normal distribution
I have a random Variable $X$ is $ SN(\lambda)$ and is pdf is given by:
$f(x)=2\phi(x)\Phi(\lambda x)$.
The model of the variable X is given by:$X=\frac{1}{\sqrt{1+\lambda^2}}Z_1+\frac{\lambda}{\sqrt{...
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1answer
288 views
Matching moments in Gaussian Mixtures
Consider $$q^{\backslash n}(\theta) = \mathcal{N}(\theta | m^{\backslash n},v^{\backslash n}I)$$ and $$f_n(\theta) = (1-w)\mathcal{N}(x_n | \theta,I) + w\mathcal{N}(x_n|0,aI)$$ Then let $$\hat{P}(\...
4
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1answer
1k views
Principle of Analogy and Method of Moments
I am studying method of moments and GMM in the context of econometrics.
Can someone explain on intuitive level, what does it mean to match moments? And how does this differ from the classical linear ...
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2answers
2k views
Method of moment estimates for n Bernoulli trials
Let $X_1, X_2 \dots X_N$ be the indicators of $n$ Bernoulli trials with probability of success $p$.
What is the method of moments estimate of $p$?
Exhibit method of moments estimates for $p \cdot (1 ...
