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2
votes
2answers
76 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: ...
2
votes
0answers
24 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
27 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, ...
-1
votes
0answers
16 views

Unbiased estimator exponential distribution [on hold]

Is the reciprocal of the sample mean an unbiased estimator of the exponential distribution parameter? How can we get it unbiased? Hint: use out, that the sum (convolution) of independent exponential ...
0
votes
1answer
790 views

Fisher Information for Geometric Distribution

Find the Cramer-Rao lower bound for unbiased estimators of $\theta$, and then given the approximate distribution of $\hat{\theta}$ as $n$ gets large. This is for a geometric($\theta$) distribution. ...
0
votes
0answers
15 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 ...
3
votes
2answers
2k views

Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$

I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, ...
0
votes
0answers
5 views

WLS: Finding minimum variance linear estimator of 3 observations

I am given three observations from a distribution with different variances (ie heteroskedastic errors). \begin{align} Y_i = \mu + u_i \end{align} The summary statistics of the three observations, ...
1
vote
2answers
49 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
36 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 ...
1
vote
2answers
41 views

how to estimate the phase parameter of a complex function

There is a complex serie: $f(t_n)=\alpha_n+\beta_n i$, for $n = 1,...,N$,$t_n,\alpha_n$ and $\beta_n$ are known.When we have know that $f(t)$ has the following form: $$f(t)=Ae^{-iBt}$$ with unknown ...
0
votes
1answer
21 views

Uniform distribution unbiased estimator

Let xi be iid observations in a sample from a uniform distribution over [0,θ]. Now I need to estimate θ based on N observations and I want the estimator to be unbiased. I thought about simple ...
0
votes
1answer
22 views

How to calculate the initial approximation in Newton - Raphson division algorithm.

I would like to know how to calculate first approximation in N-R division algorithm. I want to find the inverse of R. Here is the formula: $x_{i+1} = x_{i}(2-R*x_{i})$ I'm trying to implement it in ...
-1
votes
0answers
34 views

Derivation of the unbiased sample variation

I was wondering if someone could explain the last step in the derivation of the unbiased sample variance in the attached screenshot of my lecture notes. I don't quite get why in the last step an ...
1
vote
1answer
47 views

Find an unbiased estimator

Let $X$ be an r.v defined by $P(X=0)=p$ and $P(X=1)=1-p$. Find an unbiased estimator for $2p$. My solution: $E(X)=1-p$ so $2-2E(X)$ is unbiased. Is this correct?
0
votes
0answers
19 views

How to compute MAP estimate of y?

Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both ...
2
votes
1answer
40 views

Expectation of the MLE $e^{-\frac{1}{\overline{X}}}$

I am having a bit of a problem with examining the properties of a maximum likelihood estimator. I feel like I am missing something simple, but I have been unable to find someone doing an example quite ...
0
votes
0answers
43 views

Probabilities and Estimation of average and standard deviation

I've done a good bit of this number but I have trouble with part 2. I'll show you my work and the questions I can't figure in bold. A guy has a machine that scans his apples. The machine rules are : ...
0
votes
1answer
23 views

How to call $(E\hat{x} - x)^2$?

Let $\hat{x}$ be an estimation of $x$. Quantity $E(\hat{x} - x)^2$ is called Mean Squared Error. How one would call $(E\hat{x} -x )^2$?
1
vote
0answers
35 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
2
votes
1answer
18 views

Why is the MLE a special case of the minimum contrast estimator?

In my statistics lecture, we had two definitons, namely Let $X_1,\ldots.X_n$ be iid random variables, each with density $p_{\Theta_0}(x)$. Furthermore, let $\varrho$ be a real function such that ...
0
votes
0answers
55 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...
0
votes
1answer
43 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
1
vote
0answers
33 views

Parametric transition matrix in Markov Chains

I am trying to model a discrete-time MC with transition probabilities that depend on some function of parameters i.e $p_{ij} = f(X_0,X_1)$. Suppose we take a log-linear model where $p_{ij} = ...
0
votes
0answers
5 views

EKF to fuse gyroscope and accelerometer readings

I found it interesting to implement EKF for fusing gyroscope and accelerometer data. Trying reach my goal i discovered a lab with some theory explaned, also it has nice app for the phone to stream ...
1
vote
0answers
24 views

ML estimate of sum of guassian variables?

consider the sum $z=x_{1}+...+x_{k}$, where the scalar variables $x_{i}$ are statistically independent and Gaussian, each having the same mean $0$ and variance $\sigma^2_{x}.$ how can I construct the ...
0
votes
0answers
8 views

Maximum Likelihood parameters

I have a generative model with class conditional probability distribution $\Bbb P(x | C_k)$ and class priors $\Bbb P(C_k)$. I am having trouble with deriving the Likelihood function and hence the ...
2
votes
0answers
17 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
1
vote
0answers
30 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
0
votes
0answers
35 views

Unbiased estimators in an exponential distribution

We have $Y_{1}, Y_{2}, Y_{3}$ a random sample from an exponential distribution with the density function $ f(y) = \left\{ \begin{array}{ll} (1/\theta)\mathrm{e}^{-y/\theta} & y \gt 0 \\ 0 ...
0
votes
0answers
22 views

Parabolic regression with restricted shape

How can I calculate the parabolic regression with vertex at minimum. Is it possible? I have a set of points from which I estimate the parabola using the (I believe) standard equation (from ...
0
votes
0answers
21 views

Problem on population estmation

Can anyone help with this question.I can show my progress provided you need it
1
vote
0answers
7 views

Determining the liklihood in Baye's rule for parameter estimation

I have used Bayesian statistics in classes but what I am trying to do now is different than anything I have done in class. Previously, I was given information and certain numbers adn I could ...
3
votes
1answer
59 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
1
vote
0answers
34 views

Convergence in Probability of an estimator

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
0
votes
1answer
15 views

Generate function from discrete data (time-series)

How to transform discrete data into continous function ? I am working extensively with time series data and I would like to reduce amount of data in our frontend application. It would be cool to ...
0
votes
1answer
64 views

Determine the Asymptotic Distribution of the Method of Moments Estimator of $\theta$, $\tilde{\theta}$

I am having difficulty understanding what it means to find the asymptotic distribution of a statistic. I have the correct answer (as far as I know), but I am unconvinced that I understand the process ...
0
votes
1answer
23 views

Analytical solution to fitting two functions

I have two oscillatory functions $f(x)$ and $(k x)^2 g(x)$ where $f$ and $g$ are known and it is also known that the two functions are approximately similar. How can I analytically find the best ...
1
vote
0answers
63 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
0
votes
0answers
39 views

MME of $\theta^2x^2e^{-{\theta}x}$

I need to find the Method Moment Estimator of parameter $\theta$ based on a random sample $X_1…X_n$ with the following pdf: $f(x;\theta)=\theta^2x^2e^{-{\theta}x}$; $0<x$, zero otherwise; ...
0
votes
0answers
28 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
-1
votes
1answer
37 views

How to estimate the upper bound of y in this situation? [closed]

How to estimate the upper bound of y in this situation? Given 1. a function $y=f(x_1,x_2,x_3,x_4,x_5)$ with 5 parameters ($y=f(...)$ can be any function). 2. for each $x_i$ there are $k_i$ possible ...
2
votes
4answers
115 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
0
votes
0answers
52 views

Covariance of $(\bar{X}, \bar{Y})$ under Simple Random Sampling

$\newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Var}{\operatorname{Var}}$ Under Simple Random Sampling without replacement, and two variables of interest $X$ and $Y$, want to estimate "$r=Y/X$". ...
3
votes
0answers
48 views

Upper bound for area of polygons

is there a formula for an upper bound for the area of a polygon, knowing the length of its edges? In the ideal situation, the answer would be a function $f_n(l_1,l_2,\dots,l_n)$ of the edges lengths ...
1
vote
0answers
40 views

Calculating a metric to compare multiple posterior probability distributions

I am beginner in mathematics/statistics and apologise in advance for my faulty use of language. Especially because I assume this to be a simple problem. I am working on a problem in statistical ...
1
vote
1answer
54 views

Rolling a die 100 times and adding results

Simple problem. We role a die 100 times and we add the results. What is the probability of getting sum between 330 and 380 ? I got this: $P(330 \le X \le 380) = P\left( \frac{330 - n * ...
0
votes
0answers
33 views

Commulative degree distribution of nodes in a scale-free network

In a Barabasi-Albert model, which is a special kind of scale-free graphs, the degree distribution of each node is $$P(k) \sim k^{-3}$$ Given $\| V \|$ (number of nodes), how can I compute "number of ...
1
vote
1answer
191 views

error of the composite trapezoidal rule

Let $f\in C^2[a,b]$. The composite trapezoidal rule is given by $$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$ First, I've ...
1
vote
1answer
37 views

Some mean value limiting result

Let $\phi$ be continuous in a neighborhood of $0\in\mathbf{R}^3$ (you may assume it to be uniformly continuous, if you like). Do we have that $$\lim_{\epsilon\rightarrow ...