0
votes
0answers
35 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 ...
0
votes
1answer
17 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 ...
-1
votes
0answers
33 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
46 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
42 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 : ...
2
votes
1answer
17 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
31 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 ...
1
vote
0answers
33 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
59 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 ...
-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 ...
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 "$r=Y/X$". ...
0
votes
2answers
341 views

Moment Estimate of theta

Consider a random variable $X$ whose pdf is $f(x;θ)=θx^{θ−1}$ for $0<x<1$ and zero otherwise. i) Show this is a density function ii) determine the moment estimate of theta on the basis of a ...
0
votes
0answers
186 views

95% confidence interval around sum of random variables

Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
1
vote
0answers
33 views

what is the bias of an estimator

The point estimator $\hat\theta$ of a parameter $\theta$ is some function of the sample $D=\{x_1,...,x_n\}$, $$\hat\theta=g(D)$$, since $\hat\theta$ depends on the sample $D$ we're using, so ...
2
votes
2answers
120 views

Finding the variance of a statistic.

$X_1,\cdots,X_n$ are independent random variables from $N(\mu,\sigma^2)$ distribution. Define $$T=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$ I have shown that it is an unbiased estimator of the ...
0
votes
1answer
36 views

proving unbiasedness of an estimator

Question given independent random variable $X_{1},X_{2},...,X_{n}$ from a geometric distribution with parameter $p$. we have an estimator for $p$, mainly $T=Y/n$ where Y is number of $i$ that ...
1
vote
1answer
76 views

Variance of maximum likelihood estimator for discrete distribution

Lets say we have a discrete distribution with following probabilities: $P(X=0)=\frac{1}{3}\theta, P(X=1)=\frac{2}{3}\theta, P(X=2)=\frac{2}{3}(1-\theta), P(X=3)=\frac{1}{3}(1-\theta)$ Estimating ...
0
votes
2answers
22 views

Calculating a sample's representativeness to confirm/refute a given hypothesis?

Why hello! I'm fairly new to statistics, which is why I'm somewhat confused as to how I can approach this problem in a scientific way. The problem: Experiments are conducted to find the probabilities ...
3
votes
1answer
43 views

Unbiased Estimator Question and Understanding

I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test ...
0
votes
0answers
40 views

Variance of a difference in estimated proportions with trivariate discrete distributions

Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the ...
1
vote
1answer
40 views

Why is it called the score of the log likelihood function?

Since the score of the log likelihood function is just the gradient of the log likelihood function, why give it a special name? Why not just call it the gradient?
1
vote
0answers
39 views

Finding an unbiased estimator for function of Poisson

Let $X_1,...,X_n \sim Poi(\lambda)$ then unbiased estimator for $\lambda$ is obviously $\bar{X}$. What about $\tau(\lambda)=\sqrt{\lambda}$? Also how would one derive UMVUE for this lambda?
0
votes
1answer
48 views

How to estimate mean from sanples of multiple correlated random variables?

Suppose we have $n$ normal random variables with variance $1$ and unknown mean. Suppose we have $n$ samples of size 1 from those random variables. Suppose also we know the correlations between the ...
0
votes
0answers
12 views

When would you use a triangular or box kernel instead of gaussian?

Just a conceptual question regarding density estimation. Empirically, the gaussian kernel gives me lower MISE values than triangular or box. Epanechnikov gives me the best MISE values if the ...
0
votes
0answers
63 views

unbiased estimator of the area of the circle

the radius of a circle is measured with an error of measurement which is distributed normal with mean $0$ and variance $\sigma^2$,$\sigma^2$ unknown.Given $n$ independent measurements of the radius , ...
1
vote
0answers
19 views

How do I compute the variance (or confidence interval) of a Maximum Spacing estimator?

I am trying to solve a problem using a Maximum Possible Spacing estimator (see Maximum spacing estimation on wikipedia for links). Details on what I am trying to do can be found in the following ...
0
votes
0answers
66 views

Using confidence interval

Suppose each time a base event B occur (the trials), there's a fixed probability p that it will trigger another event E (the successes). We are interested to know the chance p of E happening, thus ...
1
vote
1answer
333 views

How can I show that sample mean has the smallest variance?

Let the population distribution is $N(\mu,1)$. Sample mean: $\bar{X_n}=\frac{\sum_{i=1}^{n} X_i}{n}$ Then $E(\bar{X_n})=\mu$ and $V(\bar{X_n})=\frac{1}{n}$ It is an unbiased estimator, and as $n ...
0
votes
1answer
24 views

Curve Fitting and Multiple Experiments

Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the ...
0
votes
1answer
119 views

Method of Moments, MLE, and Estimation Question

This is just a practice question. Not a take-home exam or a homework or an extra credit. It is not related with course work at all. Can anyone please give me detailed solution? Thank you
2
votes
1answer
122 views

Likelihood of a Uniform Distribution

I have been looking at this solution for two days and still can't understand the solution. The question is as follows: Given $w[i], i = 1, 2, \ldots, N$ are IID following a distribution of $U[0, ...
1
vote
2answers
102 views

Maximum likelihood function (MLE) for Levy distribution

I am a student who is writing a little thesis belonged in the applied mathematics category. I choose a "Levy distribution" defined as, \begin{equation} \lambda(t;u,c) = \begin{cases} ...
0
votes
1answer
84 views

Finding parameters for curve fitting

I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is where Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed ...
1
vote
2answers
428 views

Biased/Unbiased estimator

I'm trying to solve a statistic exam and i got lost with this exercise. 1) Consider a sample from a continuos probability distribution with density: $$ f(x) = \begin{cases} (1+\theta x)/8 & ...
0
votes
1answer
125 views

square root estimator

Let's say we want to do an estimation using iid samples $X_i, i=1,2,3,..., N$ the following formula, $$\hat{X}_1 = \frac{1}{N}(\sum_i\sqrt{X_i})^2$$ square sum of square roots. This form also seems ...
1
vote
1answer
150 views

Weibull Scale Parameter Meaning and Estimation

Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the ...
0
votes
1answer
196 views

MLE of Poisson Variable

Consider a random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Find the MLE of $\theta$ and show that it is a consistent estimator. --We have ...
1
vote
2answers
62 views

Poisson Estimators

Consider a simple random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Let $T=\sum X_{i}$. Show that $\tilde{\theta}=[(n-1)/n]^{T}$ is an unbiased estimator of ...
1
vote
1answer
164 views

Gaussian Curve Fitting - Parameter Estimation

I was redirected here because someone in SO pointed out this is more of a math question than a programming question: I have to fit a Gaussian curve to a noisy set of data and then take it's FWHM for ...
0
votes
1answer
107 views

What is $\sum\ln{(x_i!)}$?

I started learning statistics and in my homework i should find the Maximum Likelihood Estimate. The function is $f_x(x)=e^{-\lambda n}\prod_{i=1}^n \frac{\lambda^{x_i}}{x_i!}$ Now i take the ...
2
votes
1answer
261 views

Method of Moments and Maximum Likelihood estimators?

The random variables $X_1,...X_n$ are independent draws from continuous unifirm distribution with support $[0,\theta]$. Derive a method of moments and maximum likelihood estimators of $\theta$. Your ...
1
vote
1answer
166 views

$\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$

$f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x<\infty,\quad-\infty<\theta<\infty$ $\log f(x;\theta)=\log (\frac{1}{\pi[1+(x-\theta)^2]})$ $\Rightarrow \log ...
0
votes
1answer
41 views

MLE estimation for number of customers.

A clerk in a shop has noticed two customers arrived at the shop between 12:00 and 12:45. Another clerk noticed only one customer between 12:15 and 13:00. Assuming a Poisson distribution on the number ...
2
votes
2answers
38 views

Estimating Poisson $\theta$ only from which percentage of intervals have events

Radioactive particles are emitted randomly over time from a source at an average rate of per second. In $n$ time periods of varying lengths $t_1,t_2,\dots,t_n$ (seconds), the numbers of particles ...
3
votes
1answer
82 views

Maximum likelihood estimation - why is $\mathcal{L}$ not the joint pdf?

Here's an excerpt from my notes: Define the likelihood function: $$\mathcal{L}(\vec{x};\theta)=\prod_{i=1}^{n} f(x_i;\theta)$$ Where $f$ is the pdf of the distribution we're sampling the $x$'s ...
1
vote
1answer
56 views

Statistical inference, estimation, conceptual trouble

I've just begun learning about statistical inference and I'm having a bit of trouble understanding the concepts at hand. The exercises I've done and lectures I've read kind of gloss over the details ...
0
votes
1answer
938 views

Finding an unbiased estimator for the negative binomial distribution

Consider a negative binomial random variable Y as the number of failures that occur before the r th success in a sequence of independent and identical success/failure trials. The pmf of $Y$ is ...
0
votes
0answers
21 views

Kernel distribution estimation

Due to an assignment I need to implement a algorithm based on KDE to schedule an input data in different servers. So far, I studied statistics in my bachelor but we did not go that far and they did ...
2
votes
1answer
80 views

Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?

We know that $\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and $\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ ...
1
vote
2answers
40 views

Expected value of total accumulated lifetime (understanding gap in proof)

Problem: I understand the first line $E(T) = ...$ However, I don't get the next two steps. I feel like I almost get it. It's like we are factoring out a $\sum_{j=1}^{20}$ but how did he ...