Methods for studying the stability over time of regression relationships are considered. Recursive residuals, defined to be uncorrelated with zero means and constant variance, are introduced and tests based on the cusum and cusum of squares of recursive residuals are developed. Further techniques based on moving regressions, in which the regression model is fitted from a segment of data which is moved along the series, and on regression models whose coefficients are polynomials in time are studied. The Quandt log-likelihood ratio statistic is considered. Emphasis is placed on the use of graphical methods. The techniques proposed have been embodied in a comprehensive computer program, TIMVAR. Use of the techniques is illustrated by applying them to three sets of data.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1975.tb01532.x

Techniques for Testing the Constancy of Regression Relationships Over Time

The Markov-modulated Poisson process (MMPP) cookbook

On etudie les proprietes generales de la classe des modeles de dispersion exponentiels qui est la generalisation multivariable des modeles de distribution d'erreur de Nelder et Wedderburn (1972)

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1987.tb01685.x

Exponential Dispersion Models

Tests concerning random points on a circle

In this paper, Markov Renewal processes having a finite number of states are studied. Explicit expressions are derived for the distribution functions of first passage times, as well as for the marginal distribution function of the corresponding Semi-Markov process. Double generating functions are obtained for the distribution functions of the $N_j$-processes. The limiting behavior of a Markov Renewal process is discussed, the stationary probabilities being derived completely. General Markov Renewal processes are introduced, and a related stationary process is determined. Several examples are given.

/pdf/markov-renewal-processes-with-finitely-many-states-54awyzxr8n.pdf

Markov Renewal Processes with Finitely Many States

Several renewal processes related to the Type I and Type II counter models are defined and studied. The distribution and characteristic functions for the secondary (or output) process of the Type I counter model are obtained explicitly. Both the non-stationary and stationary probabilities of the state of the counter, (locked or unlocked), are derived. Integral equations determining the distribution and characteristic functions for the secondary process of the Type II counter model are obtained. Also it is shown that a more general model proposed by Albert and Nelson [1] may be solved explicitly in terms of a corresponding Type II counter model. An example of this general model is given. Related with each model is a discrete renewal process which is also studied.

/pdf/on-renewal-processes-related-to-type-i-and-type-ii-counter-mpz0fri6i3.pdf

On Renewal Processes Related to Type I and Type II Counter Models

Let $X_1 0$ define the random variable $i^{\ast}$ as that value of $i$, determined uniquely with probability 1, for which the maximum in (2) is reached, i.e., such that \begin{equation*}\tag{3}D^+_n = i^{\ast}/n - U_{i^{\ast}},\end{equation*} and write \begin{equation*}\tag{3.1} U_{i^{\ast}} = U^{\ast}.\end{equation*} The main object of this paper is to obtain the distribution functions of $(i^{\ast}, U^{\ast})$, of $i^{\ast}$ and of $U^{\ast}$. The asymptotic distribution of $\alpha_n = i^{\ast}/n$ is also investigated, and bounds are obtained on the difference between the exact and the asymptotic distribution. A number of general identities, which are not commonly known, have been verified and used in proving the above-mentioned results. Since these identities may be helpful in other problems of this type, they are separated from the main proofs and appear in the next section.

/pdf/on-some-distributions-related-to-the-statistic-d-n-2i15uo2lao.pdf

On Some Distributions Related to the Statistic $D^+_n$

/pdf/on-centering-infinitely-divisible-processes-emsym16fl5.pdf

On Centering Infinitely Divisible Processes

/pdf/the-supremum-and-infimum-of-the-poisson-process-2740xhd00r.pdf

The supremum and infimum of the poisson process

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