The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Statistics, least square method

I am having problems with an exercise. I have some observations of the random variable $Y$: $0.17, 0.06, 1.76, 3.41, 11.68, 1.86, 1.27, 0.00, 0.04,$ and $2.10$. I know that $Y = X^2$ and that $X \sim ...
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1answer
16 views

About regression model and assumptions

I have the following general regression model $$y=E_{Y|X}[y|x]+u.....(1)$$ Where $u$ is understood as the error. In the basic model there is a common basic assumption about avoid endogeneity, i.e. ...
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21 views

testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...
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29 views

The four assumptions on linear regression

It is clear that the four assumptions of a linear regression model are: Linearity, Independence of error, Homoscedasticity and Normality of error distribution. My question is does any of these four ...
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1answer
28 views

Maximum Likelihood Estimator of $\theta$

I have the following question I tried to answer I got answer that same like this answer Is this true answer? (Note that: in the question $0<p<\frac{1}{2}$, but in this answer ...
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2answers
29 views

Linear regression model assumptions

Linear regression models have to follow 2 key assumptions: (1)error terms are iid and each follows normal distribution with zero mean and variance sigma^2 (2)the matrix X has to be non-random and full ...
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1answer
37 views

Negative Mean Square Error

For simple random sampling, I have calculated somemean square errors for ratio-type estimators such as Isaki estimator, and Prasad Singh estimator. But, Mean Square Errors i obtained are negative. ...
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5 views

Finding confidence interval in an experiment

I conducted an experiment by weighting 20 boxes with same kind of item each with known quantity. My aim is to know the confidence interval of the weight of one specific item. How should I solve this ...
2
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1answer
39 views

How to prove that $\hat\sigma^2$ has $\chi^2_{n-p}$ distribution (linear regression)

Consider the linear regression model: $$Y_i=r(x_i)+\varepsilon_i\equiv\sum_{j = 1}^p x_{ij} \beta _j + \varepsilon _i,\quad i=1,\ldots,n.$$ where $x_1,\ldots,x_n\in \mathbb{R}^p$ are fixed, ...
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15 views

Find the correct expression for this squared error in linear models

Problem Let $S_1=\sum_{i=1}^nU_i(V_i-\bar{V})$, $S_2=\sum_{i=1}^n(U_i-\bar{U})^2$, $S_3=\sum_{i=1}^n(V_i-\bar{V})^2$. If ...
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1answer
47 views

Variance of relative frequency

Just began studying statistical inference and I'm hoping someone could help clarify an example in the literature. If we perform $10$ trials and note that the event $A$ happens in x = 4 of these, then ...
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41 views

Point estimate and margin of error

The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 39 for a sample of size 19 and standard ...
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20 views

Statistics application problem - estimate number of items by weight

An experiment was conducted by weighting 20 sets of items with known quantity and the weight of items in each trail were obtained. We also know the weight of each item is supposed to be $W$ kg ...
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1answer
16 views

Find the Expectation of uniform random variables

I need to find the expectation of the following problem, but now assuming that U is uniformly distributed from a to b i.e U~(a,b). My problem is that I don't understand how to obtain the limits of ...
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0answers
24 views

How to do Modified Mann Kendall test for trend analysis?

In mann Kendal trend analysis if auto-correlation in the time series exist we do pre-whitening to remove the affect of lag-1 correlation. However, it has been reported that pre-whitening reduces the ...
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19 views

100% confidence interval

In taking a confidence interval of a sample mean, is it possible to derive a 100% interval? Say a machine produces 200g boxes and infinity samples are taken and the means calculated, would it be ...
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8 views

Treatment Effect Approach and Selection Bias

Suppose to have the National Health Interview Survey data (NHIS), the health status of the observed people as outcome $y_i$, which has got different potential outcomes on the basis of a treatment ...
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1answer
80 views

The Wald test with Poisson distribution

Let $X_1\ldots X_n\sim \operatorname{Poisson}(\lambda)$. Let $\lambda_w>0$ be given, I am trying to find the size $\alpha$ Wald test for $H_0$: $\lambda=\lambda_w$ vs $H_1$: $\lambda\neq ...
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28 views

Question about method of moment

According to genetic theory, blood types MM, NM and NN should occur in a very large population with relative frequencies $\theta^{2},\ 2\theta(1-\theta),$ and $(1-\theta)^{2}$, where $\theta$ is ...
2
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1answer
32 views

Fisher Information for Exponential RV

Let $X \sim exp(\lambda_0)$; i.e, an exponential random variable with true parameter $\lambda_0 > 0$. The density is then $f(x;\lambda_0) = \lambda_0 e^{-\lambda_0 x}$. For a given $\lambda > ...
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1answer
18 views

For what value of $k$ is $f(x)=\frac{k}{(1+x^3)}$a distrib. function and what is its variance?

Let $X$ be a random variable with the folowing distribution function: $f(x)=\frac{k}{(1+x^3)}$ for all $x>0$. Find a value for the constant $k$ for which $f$ will be a distribution ...
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20 views

Problem on finding estimator using MM and MLE

I have a problem on solving the following kind of question $X_{1},X_{2},......,X_{n}$ are sample from distribution of $p(x_{1})=f(a)$, $p(x_{2})=g(a)$, $p(x_{3})=1-f(a)-g(a)$ Then find estimator of ...
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34 views

A Bayesian exercise

I have encountered the following problem in a book I am reading: Suppose you are offered to participate in the following game: Two fair dies are thrown untill '1' will apear (in one of them at ...
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36 views

Rank of a random matix

This arises in Time-Series modelling. Suppose $Y_i \sim N_p(0,\Sigma_i)$ and they are not necessarily independent (but assuming $\Sigma_i$ to be p.d.). Then for any ${a}\neq 0 \:\:\:\:$ $Y_i'a\neq0$ ...
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20 views

Conditioning multivariate Gaussian on a function of coordinates

I have a pretty general question and I would really appreciate if you give me any hints or point me towards some relevant literature. Suppose $X$ is an $n$-dimensional Gaussian vector. What is the ...
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67 views

Estimating Parameters using Method of Moments and Maximum Likelihood and Finding expected values/variance

Let's say we have a dataset $(x_i, Y_i)$ on each randomly n chosen non cities in a country where $x_i$ i=1,...,n is the known population size in city i with cancer. Say $Y_i$ has a Poisson ...
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15 views

Why can't you multiply Interval scale variables?

As in the title, one is allowed to sum up Interval scale variables, but one isn't allowed to multiply Interval scale variables. One both can sum up and multiply Ratio scale variables. Why isn't it ...
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25 views

Using an unbiased estimator to deduce relationship of Mean Squared Error

Suppose $\theta_{hat}$ is an unbiased estimator of $\theta$ and Var$\theta_{hat}$=$\lambda \theta^2$ for constant $\lambda$. Find the MSE of c$\theta$, where c is a constant. From there, find the ...
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60 views

Find maximum error of function given errors with respect to individual arguments

So, I know the error of the function $F = f(a_0, a_1, ..., a_n)$ where $a_0, ..., a_n$ are all variables with respect to time. Now, the function $f$ mixes $a_0, a_1, ...$ non-linearly. The problem is ...
0
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1answer
28 views

Statistical Tests Would Reject Existence of Rare Diseases? Why?

I was wondering the other day. Given a rare disease that occurs 1 in every thousand. That would infect 7 million people in the world. Yet, most statistical tests, testing for the existence of this ...
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15 views

Special case of the inverse Ising problem with equal correlations

Let $s_1,\dots,s_N\in \{-1,1\}$ be $N$ binary spins. The problem of finding a symmetric interaction matrix $J=(J_{i,j})_{i,j=1}^N$ with zero diagonal and an external magnetic field $h=(h_i)_{i=1}^N$ ...
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25 views

Identifying an appropriate statistical treatment

A College Algebra teacher would like to find out whether the possession of a textbook enhances achievement score $X$. From an engineering freshmen class of 50 students, she took the scores in a 20 – ...
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24 views

Determening the best estimator for the mean value.

You have two testing methods A and B. Let X equal the measured values with method A and Y equal the measured values with method B. X and Y are both normally distributed and have a mean value of μ. ...
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21 views

Assumptions of Exponential Family with possibly a Counterexample

In U.Küchler "Exponential Families of Stochastic Processes" 1997 [p.19-20] one consider a class of probability measures $P:=\{P_{\theta}:\theta \in \Theta\subset \mathbb{R}^{k}\}$ on ...
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Penalty function of multi-peak fit?

The question I have is about the answer from here by @Silvia: http://mathematica.stackexchange.com/questions/26336/how-to-perform-a-multi-peak-fitting I can only understand some of the code but the ...
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1answer
22 views

Calculating the probabilities of a certain random variable

If I have an exponential random variable with a mean of $0.5$ and we consider a sample of length $10$; How can I calculate if its minimum value is bigger than a certain probability or how can I ...
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37 views

Consistency of an Estimator in continous Time

there are many general definitions of the consistency of an estimator in discrete sampling theory. But i haven't found any for estimation in continous time, like for stochastic processes. Even, if the ...
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28 views

Constrained Maximum Likelihood

Assume, that we consider a multi-parameter ML estimator: \begin{eqnarray}\nonumber L_n(q_{1}, ...,q_{s}\vert\mathbf{X}) &=&\prod_{j=1}^n f(x_{j} \vert q_{1}, ...,q_{s})\\\nonumber ...
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1answer
26 views

Find a confidence interval using as pivotal quantity a function of the MLS

Let $X_1,...,X_n$ be a random sample from $f(x|\theta)=\theta x^(\theta -1)$ for $0<x<1$. Find a confidence interval for $\theta$ using as pivotal quantity a function of the maximum likelihood ...
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2answers
26 views

R.V. transformation without using the transformation theorem

Let $X$ have the pdf $f(x) = \frac{1}{2} (1+x), \quad -1 < x < 1$. Find the pdf of $Y = X^2$. I want to solve this question without using the transformation theorem, but I always arrive ...
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2answers
35 views

Interpreting OLS Regression Coefficients with High Multicolinearity

I am having trouble understanding the interpretation of OLS coefficients when predictors are highly correlated. My understanding of OLS coefficients is that they estimate a change in the expected ...
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0answers
7 views

How to find MLE of a function if we have frequency or group data using R

Suppose we have the following function: \begin{equation} f_Y(y)=\frac{\alpha\exp\{-\beta y\}[\gamma+\beta(\alpha+\gamma y)]}{(\alpha+\gamma y)^{2}}. \end{equation} We are interested to find its MLE of ...
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1answer
199 views

How do I calculate a point estimate of the largest 10%?

Here is the provided data: The question asked is: Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state ...
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1answer
60 views

Empirical Kullback-Leibler divergence of two time series

I have an two vectors (time series) with the same length (1200 elements) $x$ and $y$. Further both time series are stationary. I don't know the theoretical distribution of $x$ and $y$. I would like to ...
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1answer
47 views

Confidence Interval problem

Which of the following will result in a wider confidence interval? Check all that apply. ...
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1answer
15 views

Inferential Statistics problem about a poll and samples

The wording of the problem is a bit odd, but I'll try to keep true to the translation: ...
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1answer
19 views

Time Series with increasing dimension

I am stuck with the following problem from research. I am not sure how to model this situation. I have a vector time series whose dimension increases with time, $t$. Specifically, let ...
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40 views

MSE of an estimator as sum of bias and variance

I am reading that how the MSE of an estimator $\hat{\theta}$ of $\theta$ can be expressed as $E(\hat{\theta} - \theta)^2$. Then this can be further simplified to $ (E[\hat{\theta}] - \theta)^2 + ...
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62 views

UMVUE for $\theta^2$

Let $X_1,...X_n$ be a random sample with distribution $\text{Normal}(\theta,1)$. Find the UMVUE for $\theta^2$ What I´ve done so far: I have already shown that $T=\sum_{i=1}^nX_i$ is a complete ...
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1answer
25 views

Calculating the Standard Error for a one sample T-test: σ/sqrt(n) or s/sqrt(n)?

Consider a case when you are looking to test whether a small sample deviates significant from the population (normally distributed) whence it is drawn. Both the population standard deviation (σ) and ...