For questions about estimation and how and when to estimate correctly

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6
votes
1answer
47 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
0
votes
0answers
17 views

Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
-1
votes
0answers
21 views

Can anybody help with this state space model for filtering

I need an urgent help in an issue with a state space model for filtering. My state model is like: $\mathbf{d}_k = \mathbf{d}_{k-1} + \boldsymbol{\eta}_k$ with $\boldsymbol{\eta}_k \sim \mathcal{N}(0, ...
2
votes
3answers
48 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
1
vote
1answer
23 views

construct confidence interval from proportions

Suppose you have a population of count data, i.e., $1,2,3, \dots, k$, you have a sample of the population of size $n$, and you have a confidence interval for the proportion of $1$'s , $2$'s,\dots$n$'s ...
0
votes
0answers
16 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
2
votes
0answers
29 views

MLE for CTMC parameters

Let the data set be $$D = \{(s_0, t_0), (s_1, t_1), ..., (s_{N-1}, t_{N-1})\}$$ where $N=|D|$. Each $s_i$ is a state from the state space $S$ and during the time $[t_i,t_{i+1}]$ the chain is in state ...
0
votes
1answer
23 views

Solve the system of equations by variable estimation

Solve the system of equations: $\left\{\begin{array}{l}(x-1)\sqrt{x-y^2}=y(x-2y+1)\\y\sqrt{x-1}+3\sqrt{x-y^2}=2x+y-1\end{array}\right.$ I guess there is only one solution $(x;y)=(2;1)$. This is my ...
1
vote
2answers
55 views

Estimate how many times to flip a coin to get at least 30 heads with probability of 80%

Im completely stumped by this problem. It goes as follows: Estimate how many times a fair coin must be thrown in order to obtain at least 30 heads with a probability of 0.80. Ive tried playing with ...
5
votes
1answer
109 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
votes
1answer
20 views

Bayes estimator under squared error loss

Consider one random variable X from the Bernoulli distribution with parameter θ. Let p, the prior density, be equal to 6θ(1 − θ), for θ ∈ (0, 1). Under squared error loss, L(t, θ) = (t − θ)$^2$, the ...
2
votes
0answers
20 views

Can I adjust linear growth of a a subpopulation to a linear decay of the general population?

I need to estimate the amount of CF patients in Poland in the next four years. I have: estimations of the Polish population for the future years a CF patients' register for the last couple of years ...
-1
votes
0answers
15 views

Sample variance estimator OLS without intercept

In the classical OLS regression model the estimator of the variance is $s_n^2=\frac{1}{n-k}\sum_{i=1}^n \hat{e}_i$ where $k$ is the number of regressors without the constant. What happens to this ...
0
votes
1answer
15 views

Functions of polynomial growth and the Schwartz space

A smooth function $m \in \mathcal C^\infty(\mathbb R^n)$ is said to be slowly increasing if for all $\alpha \in \mathbb N^n_0$ there exists $C_\alpha, k_\alpha$ such that $|\partial_\alpha f(x)| \leq ...
1
vote
2answers
32 views

Maximum a Posteriori (MAP) Estimator of Exponential Random Variable with Uniform Prior

What would be the Maximum a Posteriori (MAP) estimator for $ \lambda $ for IID $ \left\{ {x}_{i} \right\}_{i = 1}^{N} $ where $ {x}_{i} \sim \exp \left( \lambda \right), \; \lambda \sim U \left[ ...
3
votes
1answer
52 views

What are the bounds (upper and lower) for $|A+A|$?

Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is ...
-1
votes
0answers
40 views

Solving $Ax=b$ with $x = [sin(\theta) cos(\theta)]'$

I am trying to solve a computer vision problem where the next step involves solving an equation of the form $Ax = b$ where A is a 2x2 matrix and $x = [sin(\theta), cos(\theta)]'$. So as I understand ...
2
votes
0answers
39 views

Sampling with no duplicates

I am sampling a population of unknown size and unknown distribution. The sample will be taken over distinct time intervals, but I have to reject any duplicates in the given time interval. The sample ...
1
vote
0answers
47 views

Polynomial roots finding algorithm

My initial problem is a parameter estimation problem that is solved by minimining a least-square criterion with the Gauss-Newton algorithm. However finding a good initial iterate is very tedious. ...
3
votes
3answers
57 views

Bounding $\sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$

I'm looking for a bound depending on $N$ of $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$. The following holds $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}} = \sum_{k=1}^N ...
1
vote
1answer
60 views

Variance of sample estimator [closed]

I have a sample estimator that I want to calculate the variance of. The estimator is \begin{align*} \hat\sigma_1=\sqrt{\frac{1}{n}\sum_{i=1}^nx_i^2}\\ \end{align*} How do I calculate ...
2
votes
0answers
9 views

Decay of reciprocal gamma function and similar functions

It is known that the reciprocal gamma function $1/\Gamma$ is entire of order 1 meaning, in particular, that for any $\varepsilon > 1$ there exists $C_\varepsilon > 0$ such that $$ \left| ...
0
votes
1answer
66 views

Deriving Point Estimate Based on Sample Mean with λ

This is a review question I'm trying to solve. I didn't receive a direct answer, only a few tips from my professor, and I want to see if I'm moving in the right direction. It's a very general ...
1
vote
0answers
35 views

Is there an exact solution for this resampling (synchronization) problem?

I want to know if there is an exact solution for the following problem and how to approach solving it: I have a discrete-time signal where the Nyquist theorem is satisfied: $$ r_k = \sum_i ...
1
vote
2answers
34 views

Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
0
votes
0answers
19 views

Estimating a population in multiple locations

I have a set of sequential data of unknown size randomly spread across $n$ locations. I am trying to estimate the population size of all $n$ locations and provide a confidence interval as well. This ...
3
votes
1answer
35 views

Estimate number of songs a radio station has [duplicate]

Imagine the following problem: You listen to a radio station and take notes how often was each song played. How can you estimate based on your notes (e.g. 30 songs played once, 2 played twice, one ...
3
votes
2answers
40 views

Difficult to understand difference between the estimates on E(X) and V(X) and the estimates on variance and std.dev. on lambda-hat

I'm having a very hard time to separate estimates on population values versus estimates on sample values. I'm struggling with this exercise (not homework, self-study for my exam in introductionary ...
0
votes
0answers
32 views

estimations in the birthday paradox?

The birthday paradox is the famous following problem: What is the probability $p_n$ that at least $2$ persons amongst $n$ persons chosen at random have the same birthday? Leap years are not taken ...
0
votes
0answers
6 views

Deriving conditional distributions for a normally distributed change point problem

Considering the change point problem of $y_i \left\{ \begin{array}{ll} y_i \tilde{~} N(u_1, \sigma) & i=1,..,t \\ y_i \tilde{~} N(u_2,\sigma) & i= t+1,...,n \\ \end{array} ...
0
votes
0answers
20 views

multivariate interval estimation

I have several samples of probabilistic vectors, i.e, each sample is of the form $(x_1, \cdots, x_n)$ such that $\sum_{i=1}^n x_i\leq 1$ (they are sub-probabilistic vectors), how can I obtain a ...
0
votes
0answers
20 views

Expectation Maximization Algorithm and low number of observations

I am observing a strange behaviour with EM algorithm and I would like to see whether anyone else has had a similar experience or not or if I could be guided through a reference or anything like that ...
1
vote
1answer
110 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
1
vote
0answers
22 views

Estimating compound growth

I have a compound interest function with the following parameters: Value at time 0 = 13.8 Interest rate = 0.05 time interval = 10 I need to check quickly, (without a calculator, only pen and paper) ...
1
vote
0answers
29 views

Estimating the size of my population

I have a following problem: Imagine you have a hat with many different balls in it and you want to estimate, how many balls are totally in the hat. The only think you are allowed to do is to take one ...
-2
votes
1answer
20 views

variance and sample confused

when solving (b) Is variance $$V(\frac{1}{2}(x_1+x_2)) = \frac{1}{4}V(x_1+x_2)= \frac{1}{4}(v(x_1)+v(x_2))= \frac{1}{2}\sigma^2$$ or should I divide variance by the sample size so that ...
2
votes
1answer
43 views

student's $t$-distribution

Random sample of $457$ Sample mean = $3.59$ Sample standard deviation $1.045$ Confidence interval from $3.49$ to $3.69$ What is the confidence level? How can I get the answer when sample size is ...
6
votes
1answer
88 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
2
votes
1answer
32 views

Conditional expectation and rao-blacwell

I am studying on UMVUE, and I'm struggling to find that conditional expectation Let $X_1,\ldots,X_n$ random sample of $X\sim U[0,\theta]$. i) Show that $2X_1$ is a unbiased estimator for $\theta$ and ...
3
votes
1answer
41 views

Kurtosis of sum of Independent Random Variables

Suppose that $X$ and $Y$ are independent random variables with different expected values and variances. Suppose we define kurtosis as $$Kurt(X)=\frac{E[(X- \mu)^4]}{E[(X- \mu)^2]^2}$$ My question is ...
4
votes
1answer
68 views

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
0
votes
1answer
24 views

Is the event is plausible or not?

An atlete specialized in long jump events jumps an average of $\bar x=7.91m$ in $12$ trials. The standard error of the mean jump distance in these trials is $0.2m$. Is it plausible that when the ...
0
votes
1answer
25 views

Defining bias function for n trial

Let a point estimate for the sample variance be given as $\hat{\sigma}^2 = \frac{1}{n}\sum\limits_{i=1}^n(X_i- \bar{X})^2$ where $n$ is the number of samples. What is the bias in this estimate as a ...
1
vote
1answer
46 views

comparing MSE of estimations of binomial random variables

$X$ is a binomial random variable defined over 12 Bernoulli trials with a success probability of $p$ in each (i.e. $X\sim\operatorname{Bin}(12,p)$. Consider $\hat p=\frac X{10}$ Determine the range ...
-2
votes
2answers
40 views

Integral with logarithm is positive

Given the following integral: $$I(f) = \int_\mathbb{R} f(x) \log \left(f(x) \sqrt{2\pi} e^{\frac{x^2}{2}}\right) dx,$$ where we assume $\int_{\mathbb{R}} f(x)\, dx =1$ and $f\geq 0$ a.e. Assume for ...
0
votes
1answer
26 views

Estimation of the integral

I am trying to compute, or find a good estimate from above the following integral $$ \frac{1}{\pi}\int_{-\infty}^{\infty}|t|^{-1/p}\left|\frac{|t|^{\nu}-1}{t-1}\right|dt, $$where $0<1/p<1$ and ...
3
votes
2answers
38 views

Show that MLE of $\lambda = \frac{n-T_n}{S_n+cT_n}$

$X_i$ are i.i.d exponential, mean $\lambda^{-1}$ for $1 \leq i \leq n$ and, the values are measured such that $X_i = c$ if $X_i \geq c$ and $X_i$ otherwise. Show that MLE of $\lambda = ...
2
votes
2answers
45 views

Simpson's rule to estimate distance traveled given velocity at certain points

Problem: A boat drives a steady course with a variable speed for 4 hours. The speed is registered at regular intervals in meters per second. The registration shows $2.4, 4.4, 7.6, 8.4, 8.6, 7.9, ...
1
vote
1answer
27 views

How to find the estimator using random variables in statistics

I'm doing an assignment for homework in my statistics class. I'm having trouble really understanding what is going on when it comes to estimators, and what the estimator of something is given a random ...
3
votes
0answers
43 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...