This paper deals with the likelihood ratio test (LRT) for testing hypotheses on the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRT statistics under some conditions that are proved to be almost necessary. A detailed solution is given for two testing problems: the test of a single distribution against any mixture, with application to Gaussian, Poisson and binomial distributions; the test of the number of populations...

We study the LRT statistic for testing a single population i.i.d. model against a mixture of two populations with Markov regime. We prove that the LRT statistic converges to infinity in probability as the number of observations tends to infinity. This is a consequence of a convergence result of the LRT statistic for a subproblem where the parameters are restricted to a subset of the whole parameter set.

Let ${\theta }^{*}$ be a biased estimate of the parameter $\vartheta$ based on all observations $x_1$, $\cdots$, $x_n$ and let ${\theta }_{-i}^{*}$ ($i=1,2,\cdots ,n$) be the same estimate of the parameter $\vartheta$ obtained after deletion of the $i$-th observation. If the expectation of the estimators ${\theta }^{*}$ and ${\theta }_{-i}^{*}$ are expressed as $$\begin{array}{cc}\hfill \mathrm{E}\left({\theta }^{*}\right)& =\vartheta +a\left(n\right)b\left(\vartheta \right)\hfill \\ \hfill \mathrm{E}\left({\theta }_{-i}^{*}\right)& =\vartheta +a(n-1)b\left(\vartheta \right)i=1,2,\cdots ,n,\hfill \end{array}$$ where $a\left(n\right)$ is a known sequence of real numbers and $b\left(\vartheta \right)$ is a function of $\vartheta$, then this system of equations can be regarded as a linear model. The least squares method gives the generalized jackknife estimator. Using this method, it is possible to obtain the unbiased...

The present paper deals with the extension of the likelihood estimation to the situation where the experimentation does not provide exact information but rather vague information.The extension process tries to achieve three fundamental objectives: the new method must be an extension of the maximum likelihood method, it has to be very simple to apply and it must allow for an interesting interpretation.These objectives are achieved herein by using the following concepts: the fuzzy information (introduced...

If a symmetric distribution is ε-contaminated and the contaminants have finite first moments, the median may cease to be the most robust estimator of location.

The moving average (MA) chart, the exponentially weighted moving average (EWMA) chart and the cumulative sum (CUSUM) chart are the most popular schemes for detecting shifts in a relevant process parameter. Any control chart system of span $k$ is specified by a partition of the space ${ℝ}^k$ into three disjoint parts. We call this partition as the control chart frame of span $k.$ A shift in the process parameter is signalled at time $t$ by having the vector of the last $k$ sample characteristics fall out of the...

We propose a novel approach to introducing hypothesis testing into the biology curriculum. Instead of telling students the hypothesis and what kind of data to collect followed by a rigid recipe of testing the hypothesis with a given test statistic, we ask students to develop a hypothesis and a mathematical model that describes the null hypothesis. Simulation of the model under the null hypothesis allows students to compare their experimental data...

The existence of the shortest confidence interval for the probability of success in a negative binomial distribution is shown. The method of obtaining such an interval is presented as well. The interval obtained is compared with the Clopper-Pearson shortest confidence interval for the probability in the binomial model.

Zieliński (2012) showed the existence of the shortest confidence interval for a probability of success in a negative binomial distribution. The method of obtaining such an interval was presented as well. Unfortunately, the confidence interval obtained has one disadvantage: it does not keep the prescribed confidence level. In the present article, a small modification is introduced, after which the resulting shortest confidence interval does not have that disadvantage.

Recently, the parameter estimations for normal fuzzy variables in the Nahmias’ sense was studied by Cai [4]. These estimates were also studied for general $T$-related, but not necessarily normal fuzzy variables by Hong [10] In this paper, we report on some properties of estimators that would appear to be desirable, including unbiasedness. We also consider asymptotic or “large-sample” properties of a particular type of estimator.