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As @Suvrit wrote in a comment, there's not always a solution. We have $$\frac{L'(s)}{L(s)} = \sum_{i\le m} \frac{t_i}{s^2} - \sum_{i>m} \frac{t_i}{s^2}\frac{p_i}{1-p_i} $$ $$=\sum_{i\le m} \frac{t_i}{ …

Note that the authors are discussing different things (QMLE vs. QLR) when discussing consistent estimator vs. consistent test. The quasi-likelihood of the QLR is discussed on page 1 of the paper. Also …

The basic observation is that $$P(\#\ge k)=P(Y_k:=\sum^k X_i\le T)$$ where $X_i$ are your independent Gaussians $N(\mu,\sigma^2)$ (not really an appropriate assumption since they can be negative, but …

Geometrically, completeness means something like this: if a vector $g(T)$ is orthogonal to the p.d.f. $f_\theta$ of $T$ for each $\theta$, $$\mathbb E_\theta g(T) = \langle g(T),f_\theta\rangle=0$$ th …

One limitation of the von Neumann trick is that you don't know in advance how many samples will be needed. If you limit the number of samples in advance, we can get a negative answer. That is, suppos …

I don't know why that is a common practice, but it makes sense in the case of a Brownian motion with drift $X_t=\sigma W_t+\nu t$. Then we have $$ \mathbb E X_{t+1}=\nu(t+1) $$ which is not time invar …

Assume they are four independent beta random variables $X_i$, with means $\mu_i$. Note that the density functions would depend on the observed samples. We could then test the hypothesis that $\mu_ …

I think you would have to assume something about how often the system switches between down and up. If all the downtime occurs in a single block then the probability of the user hitting the system w …

As you point out, the colors observed will have the same distribution with each of these models. Statistical tests involve asking "what is the probability of what is observed according to various …

Well, the uniform distribution on $\{(0,1,1), (1,0,1), (1,1,0)\} $ does not factor like that, since it would imply $$0=f(1,1,1)=\prod_{i, j} f_{i, j}(1,1)\ne 0,$$ a contradiction. To get a positive e …

You're right, it seems. Suppose $X=\{1, -1\}$; $\mu(\{1\})=\mu(\{-1\})=1/2$; $S$ is the power set of $X$; our data sets have just one element each, so that $f(\{1\})$ is either 0 or 1; and $\epsilon …

This answer is referring to version 1 of the question. What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniform …

You can use maximum likelihood estimation: https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation

As a counterexample let $X$ and $Y$ be independent with $E(X)=E(Y)=0$ and let $n$ be even. For each $i$ let $X_{2i}=X$ and $X_{2i+1}=-X$; similarly $Y_{2i}=Y$ and $Y_{2i+1}=-Y$. Then $\sum_{i\ne j}\ …

It seems that Wolfram Alpha understands this integral. Its answers are given in terms of modified Bessel functions of the first kind $I_k$. It seems that the result is, for $A(k,c):=I(n,c)$, with $n …