Inference, in a statistical context, refers to drawing conclusions from data containing an element of randomness introduced by e.g. measurement error, sampling variation, or assignment of experimental treatments. A common inferential paradigm is drawing conclusions about population parameters from ...

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transitive property in statistical comparisons

I want to know if the exposure to a given dose of a toxic chemical alters the concentration of a given substance in blood in my animals. I make this super general, bear with me, because I think this ...
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2answers
25 views

$\chi^2$ tabulated value

I noticed that the critical $\chi^2$ value increases as the degrees of freedom increase in a $\chi^2$ table. Why is that?
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1answer
25 views

Statistical Significance

Our teacher told us If a test is rejected at 1% level of significance , then it will be rejected at 5% , 10% level of significance . I don't understand how does the rejection at 1% level of ...
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9 views

Find maximum joint distribution in bayesian network that is not singly connected

I am puzzled the last days because i am trying to find the maximum of a joint distribution represented by a bayesian network. My network is not singly connected. Let's assume that my distribution is ...
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72 views

Karlin-Rubin theorem and exponential family

Let $X_1,...X_n$ a random sample of $X$ with density $$f(x;\alpha,\beta)=\frac{\alpha}{x^{\alpha+1}}\beta^\alpha I_{[\beta,\infty]}(x)\space\alpha>0,\beta>0$$ with $\beta$ known. Find ...
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1answer
27 views

Is there a method to form a CI based on an MLE which is not the mean

I know how to find confidence intervals for parameters based on sample means. I want to know whether there exists a method for finding confidence intervals based on MLE's which are not sample means. ...
3
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1answer
48 views

$E_{\theta_1}[\ell(\theta_2;X)] $

I am faced with the following in my "Statistical inference book" $E_{\theta_1}[\ell(\theta_2;X)] $ where $\ell(\theta_2;X)$ (loglikelihood) is $\log [P(\theta_2;X)]$, X is a random variable. What ...
3
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1answer
72 views

Ways to find a UMP test

I'm studying for my final exams and the subject of proof will basically test hypotheses, I will try to summarize here my doubts. For found the UMP test the ways are 1) Use Neyman–Pearson lemma ...
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42 views

How to combine normal distributions to have a mixture with specified kurtosis

I want to generate random samples from Normal Distributions $N(\mu_i,\sigma_i)$ by fixing kurtosis parameters ($\beta$s), as I need to simulate data by varying $\beta$ for my problem. I am trying to ...
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1answer
27 views

Multiple comparisons with normality tests

Suppose I do $1000$ normality tests, and set $\alpha = 0.05$. When testing for normality, should I correct for multiple comparisons? So if $p>\frac{0.05}{1000}$, I should keep the null hypothesis ...
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2answers
20 views

How to choose the sample size for a pilot study for making a power analysis?

Suppose one would like to test some new hypothesis, for which there are no previous data available. To estimate the needed sample size, one should do a power analysis. Since there are no previous data ...
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1answer
38 views

Neyman-Pearson Lemma and hypothesis-testing

Consider testing $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1$, where the pdf or pmf corresponding to $\theta_i$ is $f(x|\theta_i)$, $i=0,1$ using a test with rejection region R that satisfies ...
2
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1answer
38 views

Likelihood ratio for normal distribution with known variance

Let $X_1,...,X_n$ random sample of $X$~$N(\mu,\sigma^2)$ with known $\sigma^2$.Take $a=.05$ find the expression for power function of the likelihood-ratio test $$H_0:\mu\leq 0\space vs\space ...
2
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1answer
47 views

Likelihood-ratio test

I was studying on this subject and I got some questions. Lets take the test $$H_0:\theta\in\Theta_0 \space vs\space H_1:\theta\in\Theta_0^c$$ where $\Theta$ is the parametric space and ...
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1answer
48 views

Testing for Significance - Should t-test be used or paired t-test

Let's say there are 100 stock exchanges on which I trade stocks and the cost of me trading has the averages below ...
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6 views

How to compute Potentials in Junction Tree from Set Chain using LS also?

I was going through the original LS algorithm paper. I was not able to compute the potentials from the set chains shown on the page 171 (Table 3). Apart from that, I was able to compute all the ...
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1answer
41 views

Maximum Likelihood, normal distribution

Let $Z$~$N(\mu,\sigma^2+1)$, find the maximum likelihood estimator for $\mu$ and $\sigma^2$. I did but I want to check that this right actually ...
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0answers
11 views

sadists package/ Sum of (non-central) chi-squares to a power

I try use "sadists" package in R to compute quantiles and probabilities on sum of non-central chi-squares distribution, but there are some issues in this package. I give an example: wts = ...
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0answers
32 views

chi square test for large data sets

I use the Chi-square test for feature selection. I use it only when all entries in the contingency table are greater then 5. Is that the correct approach statistically? What happens for example, if ...
3
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1answer
28 views

Confidence interval for exponential distribution

Let $X_1,...,X_n$ random sample of $X$~$exp(\theta)$. i) Find a exact confidence interval for $\theta$ with coefficient of confidence equal to $\gamma$ ii)Find a asymptotic confidence ...
3
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0answers
26 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
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2answers
57 views

Find the confidence interval, uniform distribution

Let $X_1,..,X_n$ a random sample of $X$~$U[-\theta,\theta]$, $\theta>0$. Find the confidence interval for $\theta$. I'm trying to find a pivotal quantity with the maximum and minimum, but I ...
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0answers
26 views

Sufficient Statistic for non-exponential family distribution

Question: Let $X_1,X_2,....X_n$ be an iid sample from $N(\theta , 4 \theta^2 )$. I want to show that this model is not a member of the exponential family and to find a sufficient statistic for ...
3
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2answers
82 views

Difference of two squared normal dependent variables

I need to find the distribution of the random variable $Z$ $Z = \frac{(X - \mu_0)^2}{\sigma_0^2} - \frac{(X - \mu_1)^2}{\sigma_1^2}$, where $X \sim \mathcal{N}(\mu_0, \sigma_0)$. We can find the ...
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18 views

Testing if means are different and reporting on that usng R t.test

Lets say I have the 4 means that are the average response time for some variable: Response Time A .12 B .16 C .65 D -.35 I want to test ...
2
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2answers
75 views

Lagrange multipliers and confidence interval

Suppose $X_1,...X_n$ is a random sample from $X$~$N(\mu,\sigma^2)$, with $\sigma^2$ unknow. If $$[\overline{X}+z_{a_2}\frac{\sigma}{\sqrt{n}};\overline{X}-z_{a_1}\frac{\sigma}{\sqrt{n}}]$$ ...
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17 views

Marginal likelihood and coordinate ascent

When updating posterior distributions in Bayesian inference using coordinate ascent, is the marginal likelihood of the data guaranteed to increase after each update?
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1answer
35 views

Recommended Books on Advanced Statistical Inference [duplicate]

Can anyone suggest a nice book on advanced classical statistical inference? Thanks.
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1answer
55 views

How to extrapolate my sample results to population

We have tested randomly selected bacteria ($k = 198$) for antibiotic resistance over a period of 3 years (total isolates $n = 444$) and observed $117$ resistant strains. I would like to extrapolate ...
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13 views

How do I model chapter-verse references?

Context: I am part of an 8-person group in which each person posts a Bible verse every day. For those who don't know, that is of the format "Psalm 30:1" where first we reference the chapter, then the ...
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38 views

Inference with sample statistics

If I have mean and std deviation for two samples, can I comment on if the difference is statistically significant.
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1answer
53 views

Applying the T-test [closed]

So far I have calculated the sample means $\overline{A} = 0.75$ and $\overline{B} = 2.33$. Using these I computed the sample variances: $S_A^2 \approx \frac{28.805}{9} = 3.2005$ and $S_B^2 = ...
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1answer
28 views

What's my population? Is this Descriptive or Inferential?

I have a list of employees at a company. I want to show some comparison of some demographics (say, %female) of those who have left the company and those who have stayed with the company. However, I ...
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45 views

Rao-Blackwell exponential distribution

Let $X_1,..,X_n$ random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$ i) Find a unbiased estimator of $\lambda$ based ...
2
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1answer
45 views

UMVUE for a function

Let $X_1,...,X_n$ random sample $X$~$Bernoulli(p)$. For $n\geq 4$ show that the product $X_1X_2X_3X_4$ is a unbiased estimator for $p^4$, and use this fact for find the best unbiased estimator of ...
2
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0answers
43 views

Proper test for comparing two means from different distributions? (with limited data)

I have some data on frog weights for a small sample ($n \approx 30$). I need to compare the mean frog weight from this sample $\mu_s$ to the mean of another, much larger sample of frog weights $\mu_r$ ...
4
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30 views

Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals (CIs) for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fitted parameters and after that to ...
1
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1answer
58 views

Testing a power-law hypothesis from the averaged distribution

A way to test the existence of power-law in the distribution is given in the following paper: http://arxiv.org/abs/0706.1062 The gist of the whole procedure is as follows: Let $\textbf{X}$ be the ...
2
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0answers
76 views

Confidence interval and probability

Suppose that $T_1$ is $100\gamma$ percent lower confidence limit for $\tau(\theta)$ and $T_2$ is $100\gamma$ percent uper confidence limit for $\tau(\theta)$. Further assume that ...
1
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1answer
39 views

Confidence Interval, uniform distribution

Let $X_1,...X_n$ random sample from $f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$. i) Show that $(X_{(1)},X_{(n)})$ is a confidence interval for $\theta$.ii) find your ...
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1answer
41 views

UMVUE explanations

Let $X_1,...,X_n$ a random sample where $X$~Poisson$(\theta)$. i)Find UMVUE for $\theta$ ii)Exists UMVUE for $\frac{1}{\theta}$ For i) I found that $T=\overline{X}$ is UMVUE for ...
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0answers
21 views

UMVUE for pareto distribution

Let $X_1,..X_n$ random sample with $f(x;\theta,a)=\frac{\theta}{a}(\frac{a}{x})^{(\theta+1)}I_{(a,\infty)}(x),a>0,\theta>0$. Find the UMVUE for $\theta$ when $a$ is fixed. My attempt ...
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0answers
34 views

What is Frequentist Inference?

Frequentist Inference is defined as (according to the tag wiki) : In the frequentist approach to inference , statistical procedures are assessed by their performance over a hypothetical long run ...
2
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1answer
55 views

I want to take a decision on two gaussian distributions, what approach can I take?

I observe a one dimensional random source, which could be any of two Gaussian distributions with a different set of parameters that do not change over time. They have a the same variance and a ...
2
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1answer
35 views

Cramer-Rao Lower Bound

Let $X_1,..,X_n$ be an iid sample of $N(0,\sigma^2)$. Find an unbiased estimator of $\sigma^2$ and its lower bound. I found that $$\hat{\sigma}^2 = \sum_{i=1}^{n} X_i^2$$ is an unbiased ...
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1answer
16 views

How to scale up mean/variance and confidence interval of clinical trial for whole population

In a clinical trial I have 1000 patients with disease A. Their mean treatment cost is 1000$ per patient with a 95% confidence interval of [900;1100] for example. The whole population of country X is ...
2
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1answer
68 views

Unbiased estimator and sufficient statistics

Let $X_1,..,X_n$ be a random sample of $f(x;\theta)=\theta x^{\theta-1}I_{[0,1]}(x)$ Find a sufficient statistic for $\theta$ and construct a unbiased estimator for $\theta$ as a function ...
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1answer
49 views

is an unbiased estimator better than an efficient estimator?

I've learnt that the efficiency of an estimator, say $\theta$, is defined by $Var(\theta)$. The closer to 0 this value is, the more efficient the estimator is. Let's say I have two estimators: ...
0
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1answer
73 views

Find a complete sufficient statistic,

Let $X_1,...,X_n$ be iid observations.Find a complete sufficient statistics for i)$f(x|\theta)=\frac{\theta}{(1+x)^{1+\theta}}I_{[0\infty)}(x), \theta>0$ What I did ...
2
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1answer
35 views

Problems due to analyzing variables from different levels at one single level

Please ease the following paragraph from the first chapter , Introduction to Multilevel Analysis , p.3 of the book: Historically , multilevel problems have led to analysis approaches that moved ...