Modeling of gradual epidemic changes

. The article is devoted to analysis of epidemic changes, when transition between regimes is gradual. The consistency of CUSUM, Uniform Increments (UI) and Dyadic Increments (DI) statistics is shown. The comparison of the size-adjusted power of the tests is presented graphically.


Introduction
Many articles analyze change with epidemic alternative, when switching between regimes is instant. Our purpose is to expand the theory in cases, where regime switching is gradual or of another form. We consider the following model under H 0 : whereas under H 1 : µ + ε i , 1 i k * , a i + ε i , k * + 1 i k * + l * , µ + ε i , k * + l * + 1 i n (1) where ε i , i 1, are i.i.d. random variables with Eε i = 0, Eε 2 i < ∞. The paper is organized as follows. In Section 2, consistency of CUSUM, UI and DI statistics is shown. In Section 3, graphical representation of power analysis is provided.

Consistency
We start with consistency of classical CUSUM statistics Convergence of T n under H 0 is proved by W. Ploberger and W. Krämer [2]. We consider conditions for divergence of CUSUM statistics under H 1 . By we denote convergence in probability.

Theorem 1.
Assume that for model (1) the following condition is satisfied: Proof. If k * > n − k * − ℓ * , we have By the classical Donsker invariance principle V n = O P (1). If k * n − k * − ℓ * , we have Finally, we complete the proof by using the assumption of Theorem.

To define dyadic increments statistics let
Then the dyadic increments statistic DI (n, α) is defined by Convergence of both uniform increments and dyadic increments statistics under H 0 is shown in [4]. Here we consider conditions for consistency of these statistics under H 1 .
1. r − n k * rn k * + ℓ * r + n; 2. k * r − n k * + ℓ * rn; 3. rn k * r + n k * + ℓ * ; Due to dyadic structure the interval [r − j , r + j ] is a half of the interval [r − j−1 , r + j−1 ] and the configuration No. 1 becomes one of previously analysed configuration of the dyadic level j − 1. The level of dyadic number is reduced as follows: ℓ or ℓ − 1 is odd number, we take this odd number and rewrite in such form: 2k − 1.
The same approach applies for the configuration No. 2. We may need apply several times the idea of the decreasing dyadic level for configurations No. 2 until we reach such j that for r ∈ D j is valid r − n k * < k * + l * rn. This level j is always reachable as for j = 0 (r ∈ D 0 ), [r − n, rn] or [rn, r + n] covers the whole interval. The configuration No. 3 are reduced to configuration No. 2 by changing dyadic interval of the same level.
The configuration No. 3 becomes the configuration No. 2 in respect of dyadic number r 0 . This completes the proof.

Graphical representation of power
We use graphical representation of the size-adjusted power of the tests for comparison. We explore the idea of visualization of power analysis developed by R. Davidson and J.G. MacKinnon [1]. It consists of plotting two empirical distribution functions: one empirical distribution function under H 0 another one under H 1 . These distribution functions are plotted on a [0, 1] × [0, 1] square. When the curve of one test is higher than the curve of another, it shows that the size-adjusted power of first test is higher than the power of another test.
We will analyse two computer-based modelling examples. The first example represents the gradual change that consists of three parts: increase period, stable period and decrease period. The second example represents gradual change that consists only of two parts: increase period and decrease period.The graph of the size-adjusted power of the examples is presented below in Fig. 1. Now we will describe these examples in details: We have generated random values from N (1, 1). The length of total sample is 1024 (2 10 ). The length of epidemic change is 18.75% of the total sample in the first example and 12.5% in the second one. The epidemic change consists of three equal parts in the first example: a. gradual (45-degree line) increase of a mean from 1 to 1.35; b. stable period with a mean equal to 1.35; c. gradual (45-degree line) decrease of a mean from 1.35 to 1. The epidemic change consists of two equal parts in the second example: a. gradual (45-degree line) increase of a mean from 1 to 1.5; b. gradual (45-degree line) decrease of a mean from 1.5 to 1. The computer based modeling shows that the size-adjusted power of the CUSUM test is smaller than the power of the UI and the DI tests in both cases. The sizeadjusted power of the UI and the DI tests is almost equal, but the DI test requires less operations and due to this performs much faster.