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2
votes
2answers
12 views

unbiased estimator in a random sample

I Have a statistic statement here which I need to decide if it's true or false Statement: "When the sample size is random, there is no way to get an unbiased estimator for the population average." ...
2
votes
0answers
31 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
0
votes
0answers
17 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
0
votes
0answers
19 views

Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
0
votes
1answer
32 views

Find a 10% likelihood interval

The function is: (n choose x)$[(1-y)^{k}]^{x}[1-(1-y)^{k}]^{n-x}$ Suppose n = 100, k = 10, x = 89 I found the maximum likelihood of y-hat to be 0.0116 Now I need to find a 10% likelihood interval. ...
0
votes
0answers
18 views

How could you find the probability that the estimator is within 0.03 of the mean?

p = fraction of large population that smokes n = sample size y = # in sample that smoke The maximum likelihood estimate of p is p-hat = y/n Consider the random variable Y and estimator F = Y/n ...
0
votes
1answer
18 views

Kalman filter innovation residual inversion

I'm trying to implement a Kalman filter in a computationally efficient way. The main issue is the inversion of the innovation residual: $$S=HPH^T+R$$ $$K=PH^TS^{-1}$$ My question is, can one assume ...
0
votes
0answers
17 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
-1
votes
0answers
18 views

problem in finding PDF of estimator

estimate of variance ~(σ^2) = (σ^2)y/2 PDF of estimate of variance is given by PDF of y / (|d~(σ^2)/dy| how comes this PDF, why we divide by differential?
0
votes
1answer
29 views

Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$

Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...
1
vote
0answers
20 views

Improving Schauder estimate for a linear elliptic PDE with oblique boundary

Let $\Omega \subset \mathbb R^n$ a $C^{2,\alpha}$ domain, $f \in C^{0,\alpha}(\overline{\Omega})$, $g \in C^{1,\alpha}(\overline{\Omega})$, $h \in C^{1,\alpha}(\overline{\Omega};\mathbb{R}^n)$ such ...
0
votes
1answer
114 views

why is the answer 21,845 and not 218,450?

How can you tell whether $$\frac{(250,000)(5.47)}{6.26}$$ is closer to 21,845 or 218,450 without calculating it exactly? Thank you.
1
vote
1answer
43 views

Combining statistical distributions

I have a situation where a distribution is dependent on 2 variables, one of which follows the poisson distribution, and the other the normal distribution, and I want to establish the method of ...
0
votes
0answers
13 views

regression problem

Regression was estimated using OLS. We get y=a0 + a1x1 + a2x2 + error term. We know covariance matrix ∑ of our estimator. 1. How to get confidence interval for a1/a2 ratio? 2. In what case would ...
0
votes
1answer
43 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
0
votes
0answers
22 views

How to find the constant for sets of numbers to have a specific relation

Lets say I have two columns A B 1 6 2 7 3 8 4 9 The above numbers have a specific relationship. A is in seconds and B multiplied by a constant k is in ...
0
votes
0answers
9 views

Lp estimates from Elliptic Equation

Using the theorem: Let $f \in L^{p}(\Omega)$, $1<p<\infty$, and let $w$ be the Newtonian potential of $f$, $w(x)=\int_{\Omega}\Gamma(x-y)f(y)dy$. Then $w\in W^{2,p}(\Omega), \Delta w=f$ a.e and ...
0
votes
0answers
10 views

Optimal combination of multiple estimates of a random variable

For the following estimation problem: y = hx + n, x is the sent data, y is the observation (received data), h is a scaling factor (known), n is an AWGN random variable with zero mean ...
1
vote
1answer
18 views

Probability that a sample comes from one of two distributions

Let's say I have two normal distributions with means $\mu_1$, $\mu_2$ and standard deviations $\sigma_1$, $\sigma_2$ (which I know). I am handed a random variate from one of the distributions (I don't ...
0
votes
0answers
30 views

Approximation of optimum for two linear programs

Suppose you got two linear programs. They are the same except that one has a shifted objective by a positive constant (1) $$\min c^Tx$$ (2) $$\min c^Tx + d$$ For (2) there exists a ...
2
votes
0answers
90 views

Is it compulsory to make transformation to the econometric model in order to have only diagonal elements on variance-covariance matrix of errors?

I need some sharped and advanced advices for the following issue ... Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $y_{jis} = ...
0
votes
1answer
102 views

Test if estimator is unbiased

I'm having problems with the following question for my econometrics homework. Is $\ \ \hat \beta_2 = (y_n - y_1)/(n - 1)\ $ an unbiased estimator of $\beta_2$ for $\ \ y_t\ =\ \beta_1\ +\ \beta_2 \ ...
1
vote
1answer
79 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
0
votes
0answers
20 views

Estimating “task duration” variance through questions in a questionnaire

I am working to estimate the average task duration for complex processes - a total of 19 tasks / processes (each task made of several subtasks of varying duration). I will be organising a ...
0
votes
1answer
21 views

Show that $\hat{\delta}_1=\hat{\beta}_1+(X_1^T X_1)^{-1} X_1^TX_2\hat{\beta}_2$

Let $\hat{\beta}=(\hat{\beta}_1,\hat{\beta}_2)^T$ be the least squares estimator in the regression model $Y=X_1\beta_1+X_2\beta_2+u$. Let $\hat{\delta}_1$ be the least squares estimator of the ...
0
votes
0answers
32 views

Linear model: Show that $\hat{\theta}$ and $\hat{e}$ are independent

Show that under the assumptions $Y\sim N(X\theta,\sigma^2I_n)$and $\text{rang}(X)=\text{rang}(\theta)$ the residual vector $\hat{e}$ and the least squares estimator $\hat{\theta}$ are ...
0
votes
1answer
36 views

Estimating the Growth Rate of Worms

I would like to know how much worms I'd have on hand given an initial amount and period of time. Here are some metrics regarding the growth rate of these worms. ...
5
votes
1answer
91 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
2
votes
1answer
30 views

Estimating sum of n elements by throwing away half of elements

I've got a task where i need to proove the asymptotic big-Theta equation: $$ \log n! = \Theta(n \, \log n) $$ $ \ $ Since $f(\mathit{n}) \in \Theta(g(\mathit{n}))$ means that $g(n)\cdot k{_1} \leq ...
0
votes
0answers
12 views

Estimating the accuracy of precipitation odds.

Modern weather forecasts often include a percentage chance of rain. If this estimate was a constant it would be a simple matter of calculating the ratio. But the reality is quite another. This is the ...
2
votes
1answer
46 views

Estimate for integral of sine to the power of $-(1+a)$ where $a>0$

I'm trying to solve or estimate this integral $$ I=\int\limits_{\arcsin{k}}^{\pi/2}\dfrac{1}{(\sin{x})^{1+a}}\mathrm{d}x, $$ where $0<k<1/2$ and $a>0$. The estimate should depend on $k$. I ...
1
vote
3answers
45 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
0
votes
1answer
28 views

Maximum-Likelihood Estimator: What problems occur if data is not i.i.d.?

This is a question from an exam: You want to estimate the parameters for a gaussian distribution using the Maximum-Likelihood Method for an i.i.d. set of data. What role does the property ...
1
vote
1answer
22 views

show that the sample median of normal distribution is median unbiased

$X_i \sim N(\mu,\sigma^2)$, $i = 1,2,\ldots,n$. Show that the sample median is unbiased in median for $\mu$. I have obtained the pdf of sample median for $n=2m+1$ as: $$f(x_{(m+1)}) = \frac ...
0
votes
0answers
14 views

OLS estimators in stationary process

Given a stationary process xt=a+b*t+et with et a white noise, how can I find the OLS estimators for a and b? Cheers
1
vote
0answers
59 views

Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Do: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + ...
0
votes
1answer
17 views

Confidence/Tolerance interval for a percentage of a population

I have a problem I'm not sure how to solve. It goes something like: ...
1
vote
1answer
21 views

How to estimate the lower bound of a given Toeplitz matrix's eignvalue?

A given Toeplitz matrix is $$\left( \begin{matrix} 1 & a & a^2 & \cdots & a^n \\ a &1 &a & \cdots & a^{n-1} \\ a^2&a & 1 & \cdots& \cdots ...
1
vote
1answer
69 views

Estimation solving for binomial k?

Hello all trying to do an estimation problem at work and wondering if I'm on the right track! I'm running a study and its on the internet. I'm trying to determine how many people I need to show an ...
1
vote
1answer
118 views

What initial guess is used for finding n-th root using Newton-Raphson method?

I would like to know what is an optimal initial guess for use with Newton-Raphson method when finding n-th root. I develop some program which uses GMP C++ library. GMP manual says: The initial ...
1
vote
2answers
31 views

Estimate of L2 norm of a product of functions

Assume that $f,g, f\cdot g\in L^2(\mathbb{R}^n)$. Is the estimate $\|fg\|_{L^2(\mathbb{R}^n)}\leq \|f\|_{L^2(\mathbb{R}^n)}\|g\|_{L^2(\mathbb{R}^n)} $ correct? Thank you!
1
vote
0answers
20 views

Derive Maximum Likelihood Estimator of a Generalised Linear Regression Model

I understand how to find the MLE estimator for $b$ if it is a simple linear regression model. However, when $u\sim N(0,\sigma^2\Omega)$ where $\Omega\ne I$. I am getting confused. The model is: ...
0
votes
0answers
20 views

Is it possible to calculate the width of this table

Is it possible to calculate the length of x (width of the table) using the given values and any information that can be inferred from the image. If not, what is the best estimate that can be found.
0
votes
0answers
12 views

Parameter estimation of matrix-valued processes

I have a hard question for the probability theorist under you. Suppose we have the following process as defined by $dX_t = [-M(\bar{X} - X_t) - (\bar{X} - X_t)M^T] dt + \sqrt{X_t}dB_tQ + ...
2
votes
0answers
34 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
1
vote
0answers
44 views

How to find the MLE of the mean of Gamma distribution

If I parameterize Gamma distribution in the way as $\Gamma(\alpha,\frac{\mu}{\alpha})$, am I able to find the maximum likelihood estimator of $\mu$. Here, $\alpha$ is the shape parameter, ...
0
votes
0answers
19 views

Find the covariance of estimation error

Define the linear operator $O_T: \mathbb{R}^n \to L_2[0,T] $ \begin{equation} O_Tx = Ce^{At} x, \end{equation} where $t \in [0, T]$, $ C$ and $A$ are matrices with compatible dimentions, and $x \in ...
2
votes
2answers
84 views

What's the best strategy to count the eggs in the jar?

It's Easter time, and in my workplace we have a "Count the eggs in the jar!" kind of game. What would be the best mathematical strategy to get as close as possible to the correct count? Update: ...
1
vote
2answers
66 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
0
votes
0answers
51 views

Can I estimate Variance of Gamma from Negative Binomial distributed data, given NB is Poisson-Gamma mixture

I believe the data I have follows Negative Binomial distribution (over-dispersed Poisson). We know Negative Binomial is a mixture of Poisson and Gamma. The variance of this Gamma distribution is ...