5

The series converges to a distribution function. It can be evaluated in closed form. Upon identifying the terms varying with $n,$ write your function in a simpler form as $$F_{U_i}(y)=\frac{2(R^{\alpha}y/\theta)^k}{\Gamma(k)}\sum_{n=0}^\infty \frac {(-R^\alpha y / \theta)^n}{n!(k+n)((k+n)\alpha+2)} = \frac{2x^k}{\Gamma(k)}\sum_{n=0}^\infty \frac{(-x)^n}{n!...


3

The density function of the Benini distribution is strictly quasi-concave, and is strictly decreasing if and only if $\beta \leqslant \alpha (1+\alpha)/2$. The mode of the distribution has the explicit form: $$\text{Mode} = \hat{x} = \sigma \cdot \exp \bigg( \bigg\{ \frac{\sqrt{1+8\beta} - (1+2\alpha)}{4 \beta} \bigg\} \bigg),$$ where the brackets $\{ \...


2

You can resort to the bootstrap version of Student's t-test. It works as follows: Compute the sample mean and standard deviation for each group and label the results $X_1$ and $s_1$ for group 1, and $X_2$ and $s_2$ for group 2. Set $d_1=\frac{s_1^2}{n_1}$ and $d_2=\frac{s_2^2}{n_2}$, where $n_1$ and $n_2$ are the sample sizes. Generate a bootstrap sample ...


2

After having done some homework on the subject, I think I've got a better handle on the proof that I find in [1]. I wanted to take an opportunity to set down my understanding for pedagogic purposes. Scope: I'm going to limit this answer to the case of a strictly monotonic cumulative distribution function. Its my understanding that, in his answer to this ...


1

The old rule of thumb was that the sample size is too small when the expected count of one of the cells of the contingency table is lower than five. Recall that the expected count of a cell is the ratio $$\frac{row\_count \times column\_count}{total}.$$ In http://www.biostathandbook.com/small.html and http://www.biostathandbook.com/fishers.html, owing ...


1

The result can be found in the 2012 paper Some results on the truncated multivariate $t$ distribution by Ho et al. First, let $T(\cdot\,|\,\nu)$ be the cdf of the (untruncated) standard $t$ distribution with $\nu$ degrees of freedom. In the following exposition, $a$ and $b$ are the lower and upper truncation limits, respectively. Before we can give the ...


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