Outlier Detection in Adaptive Functional-Coefficient Autoregressive Models Based on Extreme Value Theory

Abstract

This paper proposes several test statistics to detect additive or innovative outliers in adaptive functional-coefficient autoregressive (AFAR) models based on extreme value theory and likelihood ratio tests. All the test statistics follow a tractable asymptotic Gumbel distribution. Also, we propose an asymptotic critical value on a fixed significance level and obtain an asymptotic 𝑝-value for testing, which is used to detect outliers in time series. Simulation studies indicate that the extreme value method for detecting outliers in AFAR models is effective both for AO and IO, for a lone outlier and multiple outliers, and for separate outliers and outlier patches. Furthermore, it is shown that our procedure can reduce possible effects of masking and swamping.

1. Introduction

Outlier detection and analysis play important roles in practical applications. For instance, outlier detection can be applied to anomaly detection in computer networks, financial time series, and data series in geosciences, as can be seen from [1, Chap. 1] and [2]. Other examples include the study on loss of customers in the commercial field and the detection and tracking of financial crime as credit card fraud, all of which involve and exploit the useful information provided by the presence of outliers. On the other hand, outliers in dynamic systems or engineering time series [3] can have adverse effects on model identification and parameter estimation, where eliminating outliers is necessary in the statistical modeling of time series for the purpose of preprocessing data, see, for example, [4]. Some studies have even shown that the emerging of outliers generates certain nonlinear time series. Several procedures are available in the literature to deal with problems related to outliers. Chen [5] developed a method for detecting additive outliers in bilinear time series, which belong to the family of fractal time series models [6]. Cai et al. [7] studied the functional-coefficient regression models of nonlinear time series. Battaglia [8] discovered a way to identify and estimate outliers in functional autoregressive time series. Battaglia and Orfei [9] addressed the issue of outlier detection and estimation in nonlinear time series. Chen et al. [10] and Chen et al. [11] discussed the detection of outlier patches, change point, and outliers in bilinear models.

Extreme value theory and likelihood ratio tests have been used in the detection of outliers and time series analysis. For instance, Martin [12] conducted extreme value analysis on the optimal level cross-prediction of linear Gaussian processes. Zhu and Ling [13] performed likelihood ratio tests on the structural change from an AR(p) model to a threshold AR(p) model. Furthermore, based on extreme value theory Chareka et al. [14] proposed an alternative test method for the detection of additive outliers in Gaussian time series. On the other hand, some scholars such as Fung et al. [15] and Río [16], focused their studies on special cases of outliers detection. It is commonly agreed that the key of outlier detection lies in determining whether the test statistic exceeds a critical value, that is, the threshold under a given significance level. However, explanations for the selection of threshold in many literatures are ambiguous, and the threshold itself can hardly be controlled under a certain level of significance. In this paper, we propose an asymptotic critical value on a fixed significance level, which is used to detect additive and innovative outliers in adaptive functional-coefficient autoregressive (AFAR) models (see, e.g., Fan and Yao [17]).

This paper is structured as follows. In Section 2, we consider AFAR models with additive or innovative outliers and their estimation. In Section 3, several procedures are proposed to detect outliers in the AFAR models based on extreme value theory and likelihood ratio tests. In Section 4, we present some simulation studies and demonstrate the effectiveness of the proposed method through an empirical way. Concluding remarks are summarized in Section 5.

2. Outliers Models and Test Statistics

We now consider the AFAR model: 𝑥_𝑡=cos𝑎₁𝑥_𝑡−2𝑎₂+𝑎₃𝑥²_𝑡−2𝑥_𝑡−1+𝑎₄+𝑎₅exp𝑎₆𝑥²_𝑡−2𝑥_𝑡−2+𝜀_𝑡, (1) which can be written as 𝑥_𝑡=²_𝑗=1𝑓_𝑗𝑥_𝑡−2𝑥_𝑡−𝑗+𝜀_𝑡, (2) where 𝜀_𝑡 is a Gaussian white noise with mean zero and variance 𝜎² and 𝑓₁(𝑥_𝑡−2)=cos(𝑎₁𝑥_𝑡−2)/(𝑎₂+𝑎₃𝑥²_𝑡−2), 𝑓₂(𝑥_𝑡−2)=𝑎₄+𝑎₅exp(𝑎₆𝑥²_𝑡−2).

Suppose that {𝑥_𝑡}^𝑛_𝑡=1 is a zero-mean stationary process following model (2) description, and there is an outlier at time 𝑡=𝑞, whose influence magnitude is 𝜔_𝑞, then the observed series {𝑦_𝑡} may be presented as follows.

(1) Innovative outlier (IO) model: 𝑦_𝑡=𝑥_𝑡𝑦,1≤𝑡<𝑞_𝑡=²_𝑗=1𝑓_𝑗𝑦_𝑡−2𝑦_𝑡−𝑗+𝜂_𝑡𝜂,𝑞≤𝑡≤𝑛,_𝑡=𝜀_𝑡+𝜔^IO_𝑞⋅𝛿𝑡−𝑞. (3)

(2) Additive outlier (AO) model: 𝑦_𝑡=𝑥_𝑡+𝜔^AO_𝑞⋅𝛿𝑡−𝑞,𝑡=1,…,𝑛, (4) where 𝛿(𝑡) is the Kronecker symbol: If 𝑡=0, then 𝛿(𝑡)=1, else 𝛿(𝑡)=0.

It is obvious that the detection for IO or AO may be transformed to test whether the null hypothesis 𝐻₀∶𝜔^IO_𝑞=0 or 𝐻^′₀∶𝜔^AO_𝑞=0 is true under a certain level of significance. This can be solved by knowing the distributions of 𝜔^IO_𝑞 and 𝜔^AO_𝑞 under the null hypothesis. We estimate 𝜔_𝑞 by using the maximum likelihood method. The maximum likelihood function of observations is the positive ratio of the maximum likelihood function of residuals. For a given 𝑞, assume that the 𝑓_𝑗 and corresponding parameters are known, then we can obtain the estimation of 𝜔_𝑞 by maximizing the likelihood function. For convenience, suppose that there is no outlier in the two former observations. Denote the residual of observations by 𝜂_𝑡, for any 𝑘≤0, and let 𝜀_𝑘=0, 𝑥_𝑘=0, and 𝑓_𝑘(⋅)=0. 𝜀₁, 𝜀₂ be obtained by letting 𝜀_𝑡=𝑥_𝑡−∑²_𝑗=1𝑓_𝑗(𝑥_𝑡−2)𝑥_𝑡−𝑗. Under initial conditions mentioned above, the conditional likelihood function of 𝜔_𝑞 is given by 𝑙𝑦∣𝜔=𝑙(𝑥)=𝑙(𝜀)=^𝑛_𝑡=32𝜋𝜎^2−1/2−𝜀exp^2𝑡2𝜎²=2𝜋𝜎^{2−(𝑛−2)/2}−∑exp^𝑛𝑡=3𝜀²_𝑡2𝜎². (5)

We can obtain the maximum likelihood estimation of 𝜔_𝑞 at the minimum of ∑^𝑛_𝑡=3𝜀²_𝑡. It is an accurate estimation for the linear model. However, it is just an approximation for the nonlinear model. For 𝑡>2, the residual of observations is given by 𝜂_𝑡=𝑦_𝑡−∑²_𝑗=1𝑓_𝑗(𝑦_𝑡−2)𝑦_𝑡−𝑗. We then discuss the estimations of 𝜔^IO_𝑞 and 𝜔^AO_𝑞 for model (2).

(1) Assume that there is an IO at 𝑡=𝑞 and its influence magnitude is 𝜔^IO_𝑞. From (3), it follows that 𝜂_𝑞=𝜀_𝑞+𝜔^IO_𝑞 and 𝜂_𝑡=𝜀_𝑡 (𝑡≠𝑞), where ∑^𝑛_𝑡=3𝜀²_𝑡=∑_𝑡≠𝑞𝜂²_𝑡+(𝜂_𝑞−𝜔^IO_𝑞)². So, the maximum likelihood estimation of 𝜔^IO_𝑞 is given by 𝜔^IO_𝑞=𝜂_𝑞=𝑦_𝑞−²_𝑗=1𝑓_𝑗𝑦_𝑞−2𝑦_𝑞−𝑗. (6) For known 𝑓_𝑗 and 𝜎², we have 𝜔^IO_𝑞∼𝑁(𝜔^IO_𝑞,𝜎²), which is similar to Battaglia [8]. Under the null hypothesis 𝐻₀∶𝜔^IO_𝑞=0, 𝜔^IO_𝑞∼𝑁(0,𝜎²), we obtain likelihood radio test statistic 𝜆^IO_𝑞=(𝜔^IO_𝑞/𝜎)∼𝑁(0,1) by standardizing 𝜔^IO_𝑞. When 𝑓_𝑗 and 𝜎² are unknown, they can be replaced by their consistent estimations 𝜎²_IO=(𝑛−2)⁻¹∑^𝑛_{𝑡=3,𝑡≠𝑞}𝜂²_𝑡 and 𝜂_𝑡=𝑦_𝑡−∑²_𝑗=1𝑓_𝑗(𝑦_𝑡−2)𝑦_𝑡−𝑗. Similarly, we also have that 𝜆^IO_𝑞=𝜔^IO_𝑞𝜎_IO∼𝑁(0,1). (7)

(2) Assume that there is an AO at 𝑡=𝑞 and its influence magnitude is 𝜔^AO_𝑞. From (4), it follows that 𝜂_𝑞=𝑦_𝑞−²_𝑗=1𝑓_𝑗𝑦_𝑞−2𝑦_𝑞−𝑗=𝑥_𝑞+𝜔^AO_𝑞−²_𝑗=1𝑓_𝑗𝑥_𝑞−2𝑥_𝑞−𝑗=𝜀_𝑞+𝜔^AO_𝑞. (8) When 𝑡<𝑞, 𝜂_𝑡=𝑦_𝑡−∑²_𝑗=1𝑓_𝑗(𝑦_𝑡−2)𝑦_𝑡−𝑗=𝑥_𝑡−∑²_𝑗=1𝑓_𝑗(𝑥_𝑡−2)𝑥_𝑡−𝑗=𝜀_𝑡, if 𝑡>𝑞, that is, 𝑘>0, then 𝜂_𝑞+𝑘=𝑦_𝑞+𝑘−∑²_𝑗=1𝑓_𝑗(𝑦_{𝑞+𝑘−2})𝑦_{𝑞+𝑘−𝑗}, if 𝑘=1, we have that 𝜂_𝑞+1=𝑦_𝑞+1−²_𝑗=1𝑓_𝑗𝑦_𝑞−1𝑦_{𝑞+1−𝑗}=𝑥_𝑞+1−²_𝑗=1𝑓_𝑗𝑥_𝑞−1𝑥_{𝑞+1−𝑗}+𝜔^AO_𝑞𝛿1−𝑗=𝜀_𝑞+1−²_𝑗=1𝑓_𝑗𝑥_𝑞−1𝜔^AO_𝑞𝛿1−𝑗=𝜀_𝑞+1−𝑓₁𝑥_𝑞−1𝜔^AO_𝑞=𝜀_𝑞+1−𝑓₁𝑦_𝑞−1𝜔^AO_𝑞, (9) and if 𝑘=2, we have that 𝜂_𝑞+2=𝑦_𝑞+2−²_𝑗=1𝑓_𝑗𝑦_𝑞𝑦_{𝑞+2−𝑗}=𝑥_𝑞+2−²_𝑗=1𝑓_𝑗𝑥_𝑞𝑥_{𝑞+2−𝑗}+²_𝑗=1𝑓_𝑗𝑥_𝑞𝑥_{𝑞+2−𝑗}−²_𝑗=1𝑓_𝑗𝑦_𝑞𝑦_{𝑞+2−𝑗}=𝜀_𝑞+2−𝑓₁𝑦_𝑞−𝑓₁𝑥_𝑞𝑦_𝑞+1+𝑓₂𝑥_𝑞𝑦_𝑞−𝜔^AO_𝑞−𝑓₂𝑦_𝑞𝑦_𝑞=𝜀_𝑞+2−𝑓^′₁𝑥_𝑞+𝜃₁𝜔^AO_𝑞𝜔^AO_𝑞𝑦_𝑞+1−𝑓₂𝑦_𝑞−𝑓₂𝑥_𝑞𝑦_𝑞−𝑓₂𝑥_𝑞𝜔^AO_𝑞=𝜀_𝑞+2−𝑓^′₁𝑥_𝑞+𝜃₁𝜔^AO_𝑞𝜔^AO_𝑞𝑦_𝑞+1−𝑓^′₂𝑥_𝑞+𝜃₂𝜔^AO_𝑞𝜔^AO_𝑞𝑦_𝑞−𝑓₂𝑥_𝑞𝜔^AO_𝑞=𝜀_𝑞+2−𝜔^AO_𝑞⋅𝑓₂𝑥_𝑞+²_𝑗=1𝑓^′_𝑗𝑥^∗_𝑞,𝑗𝑦_{𝑞+2−𝑗}, (10) where 𝑥^∗_𝑞,𝑗=𝑥_𝑞+𝜃_𝑗𝜔^AO_𝑞, 0<𝜃_𝑗<1, and 𝑗=1,2. If 𝑘>2, then 𝜂_𝑞+𝑘=𝜀_𝑞+𝑘, from which it follows that 𝜂_𝑞+𝑘=𝑐_𝑘𝜔^AO_𝑞+𝜀_𝑞+𝑘,𝑘=1,2, (11) where 𝑐₁=−𝑓₁(𝑦_𝑞−1) and 𝑐₂=−[𝑓₂(𝑥_𝑞)+∑²_𝑗=1𝑓^′_𝑗(𝑥^∗_𝑞,𝑗)𝑦_{𝑞+2−𝑗}], so we have that ^𝑛_𝑡=3𝜀²_𝑡=^𝑞−1_𝑡=3𝜀²_𝑡+𝜀²_𝑞+^𝑞+2_𝑡=𝑞+1𝜀²_𝑡+^𝑛_𝑡=𝑞+3𝜀²_𝑡=^𝑞−1_𝑡=3𝜂²_𝑡+𝜂_𝑞−𝜔^AO_𝑞²+²_𝑘=1𝜂_𝑞+𝑘−𝑐_𝑘𝜔^AO_𝑞²+^𝑛_𝑡=𝑞+3𝜂²_𝑡=^𝑞−1_𝑡=3𝜂²_𝑡+^𝑛_𝑡=𝑞+3𝜂²_𝑡+1+𝑐²₁+𝑐²₂𝜔^AO_𝑞²−2𝜂_𝑞+𝑐₁𝜂_𝑞+1+𝑐₂𝜂_𝑞+2𝜔^AO_𝑞+𝜂²_𝑞+𝜂²_𝑞+1+𝜂²_𝑞+2. (12) By minimizing the above expression, we obtain 𝜔^AO_𝑞=𝜂_𝑞+𝑐₁𝜂_𝑞+1+𝑐₂𝜂_𝑞+21+𝑐²₁+𝑐²₂, (13) where 𝑐₁ may be obtained by estimation of −𝑓₁(𝑦_𝑞−1), while 𝑐₂ is complicated and difficult to be confirmed. Nevertheless, Battaglia [8] indicated that we could estimate 𝑐₂ by −𝑓₂(𝑦_𝑞), which is convenient and effective. If all 𝑓_𝑗(⋅) and 𝜎² are known, then we have 𝜔^AO_𝑞=𝜂_𝑞−𝑓₁𝑦_𝑞−1𝜂_𝑞+1−𝑓₂𝑦_𝑞𝜂_𝑞+2Δ_𝑞,Δ_𝑞=1+𝑓²₁𝑦_𝑞−1+𝑓²₂𝑦_𝑞. (14) Similar to Battaglia [8], we obtain 𝜔^AO_𝑞∼𝑁𝜔^AO_𝑞,𝜎²Δ_𝑞. (15) Under the null hypothesis 𝐻^′₀∶𝜔^AO_𝑞=0, we obtain likelihood radio test statistics 𝜆^AO_𝑞=(𝜔^AO_𝑞/(𝜎/Δ_𝑞))∼𝑁(0,1) by standardizing 𝜔^AO_𝑞. In practice, if 𝑓_𝑗(⋅) and 𝜎² are unknown, which can be replaced by their consistent estimations 𝜎²_AO=(𝑛−2)⁻¹[∑^𝑛_𝑡=3𝜂²_𝑡−(𝜔^AO_𝑞)²⋅∑²_𝑗=1𝑓²_𝑗(𝑦_{𝑞+𝑗−2})], we likewise obtain 𝜆^AO_𝑞=𝜔^AO_𝑞𝜎_AO/Δ_𝑞∼𝑁(0,1). (16)

It is indicated by (6) and (13) that the best estimate of 𝜔 is 𝜂_𝑞 at time 𝑞 for the case of IO. the best estimate of 𝜔 is the linear combination of errors 𝜂_𝑞, 𝜂_𝑞+1, and 𝜂_𝑞+2 at time 𝑞 in the case of AO. If the location 𝑞 of outlier is known, the distributions of likelihood ratio test statistics 𝜆^IO_𝑞 and 𝜆^AO_𝑞 are given by (7) and (16). However, in practice the location of outlier is unknown, thus we should inspect the magnitude of test statistic at every time point. By replacing 𝜆^IO_𝑞 and 𝜆^AO_𝑞 with 𝜆^IO_𝑡 and 𝜆^AO_𝑡, the following theorem indicates that the processes of {𝜆^IO_𝑡} and {𝜆^AO_𝑡} are both Gaussian process, respectively.

Theorem 1. (1) Under the hypothesis 𝐻₀∶𝜔^IO_𝑡=0, that is, there is no IO at any time, {𝜔^IO_𝑡} is a stationary zero-mean Gaussian process with variance 𝜎², and its autocovariance function is 𝑟^IO(ℎ)=𝜎²⋅𝐼(ℎ=0), where 𝐼_𝐴(𝑥) is the index function. If 𝑥∈𝐴, then 𝐼_𝐴(𝑥)=1, and else 𝐼_𝐴(𝑥)=0.
(2) Under the hypothesis 𝐻^′₀∶𝜔^AO_𝑡=0, that is, there is no AO at any time, {𝜔^AO_𝑡} is a stationary zero-mean Gaussian process with variance 𝜎²/Δ_𝑡, and its autocovariance function is 𝑟^AO(𝑡,ℎ)=𝜎²Δ_𝑡⁻¹Δ⁻¹_𝑡+ℎ²_𝑗=ℎ+1𝑓_𝑗𝑦_{𝑡+𝑗−2}𝑓_𝑗−ℎ𝑦_{𝑡+𝑗−2}−𝑓_ℎ𝑦_𝑡+ℎ−2𝐼(1≤ℎ≤2). (17)

Proof. The Gaussian character is shown by (7) and (16). We derive their autocovariance function as follows.(1)Under the null hypothesis 𝐻₀∶𝜔^IO_𝑡=0, we have 𝜂_𝑡=𝜀_𝑡 and 𝑟^IO(ℎ)=cov𝜔^IO_𝑡,𝜔^IO_𝑡+ℎ=cov𝜂_𝑡,𝜂_𝑡+ℎ=cov𝜀_𝑡,𝜀_𝑡+ℎ=𝜎²⋅𝐼(ℎ=0). (18)(2)Under the null hypothesis 𝐻^′₀∶𝜔^AO_𝑡=0, we have 𝜂_𝑡=𝜀_𝑡 and 𝑟^AO(𝑡,ℎ)=cov𝜔^AO_𝑡,𝜔^AO_𝑡+ℎ=covΔ_𝑡⁻¹𝜂_𝑡−²_𝑗=1𝑓_𝑗𝑦_{𝑡+𝑗−2}𝜂_𝑡+𝑗,Δ⁻¹_𝑡+ℎ𝜂_𝑡+ℎ−²_𝑗=1𝑓_𝑗𝑦_{𝑡+ℎ+𝑗−2}𝜂_{𝑡+ℎ+𝑗}=Δ_𝑡⁻¹Δ⁻¹_𝑡+ℎcov𝜂_𝑡,𝜂_𝑡+ℎ−²_𝑗=1𝑓_𝑗𝑦_{𝑡+ℎ+𝑗−2}cov𝜂_𝑡,𝜂_{𝑡+ℎ+𝑗}−²_𝑗=1𝑓_𝑗𝑦_{𝑡+𝑗−2}cov𝜂_𝑡+𝑗,𝜂_𝑡+ℎ+cov²_𝑗=1𝑓_𝑗𝑦_{𝑡+𝑗−2}𝜂_𝑡+𝑗,_2𝑗=1𝑓_𝑗𝑦_{𝑡+ℎ+𝑗−2}𝜂_{𝑡+ℎ+𝑗}=𝜎²Δ_𝑡⁻¹Δ⁻¹_𝑡+ℎ²_𝑗=ℎ+1𝑓_𝑗𝑦_{𝑡+𝑗−2}𝑓_𝑗−ℎ𝑦_{𝑡+𝑗−2}−𝑓_ℎ𝑦_𝑡+ℎ−2𝐼(1≤ℎ≤2). (19)

We know that {𝜔^IO_𝑡} is not only Gaussian, but also a stationary process by Theorem 1. Also, it is serial independent under the null hypothesis 𝐻₀∶𝜔^IO_𝑡=0. However, despite that {𝜔^AO_𝑡} is Gaussian, it is not stationary due to the fact that its autocovariance function 𝑟^AO(𝑡,ℎ) is dependent on 𝑡. Besides, {𝜔^AO_𝑡} is truncated in finite steps, which means that its maximum correlation is within two steps; that is, 𝜔^AO_𝑡 is independent with 𝜔^AO_𝑡+ℎ when ℎ>2. By standardizing {𝜔^IO_𝑡} and {𝜔^AO_𝑡}, we can obtain {𝜆^IO_𝑡} and {𝜆^AO_𝑡}. It is obvious that {𝜆^IO_𝑡} is a standardized stationary Gaussian process under the hypothesis 𝐻₀∶𝜔^IO_𝑡=0. Also, {𝜆^AO_𝑡} is a standardized Gaussian process under the hypothesis 𝐻^′₀∶𝜔^AO_𝑡=0, but it is not stationary, whereas its autocovariance function is truncated in finite steps.

3. Detection of Outliers Based on Extreme Value Theory

As similar to Chareka et al. [14] and Leadbetter and Rootzén [18], we obtain the following.

Lemma 2 (see [18]). Assume that {𝑋_𝑛} is a stationary zero-mean Gaussian time series and its autocorrelation function is 𝜌(ℎ). Let 𝑍_𝑛=max{𝑋₁,𝑋₂,…,𝑋_𝑛}, and if the Berman condition lim_ℎ→∞𝜌(ℎ)⋅logℎ=0 is satisfied, then one has that 𝑃(((𝑍_𝑛−𝑐_𝑛)/𝑑_𝑛)≤𝑥)→Λ(𝑥)=exp{−𝑒^−𝑥}, 𝑛→∞, where 𝑑_𝑛=(2log𝑛)^−1/2, and 𝑐_𝑛=(1/𝑑_𝑛)−(𝑑_𝑛/2)(loglog𝑛+log(4𝜋)).

Lemma 3 (see [18]). Assume that {𝑋_𝑛} is a stationary zero-mean Gaussian time series, and let 𝐴_𝑛=max^𝑛_𝑡=1{|𝑋_𝑡|} and 𝑍_2𝑛=max^2𝑛_𝑡=1{𝑋_𝑡}, and if 𝑋_𝑛 satisfies the Berman condition, then 𝐴_𝑛 and 𝑍_2𝑛 have the same distribution.

Theorem 4. Let 𝜂_IO=max_{𝑡=1,2,…,𝑛}{|𝜆^IO_𝑡|} and 𝜂_AO=max_{𝑡=1,2,…,𝑛}{|𝜆^AO_𝑡|}, and under the hypothesis 𝐻₀∶𝜔^IO_𝑡=0 and 𝐻^′₀∶𝜔^AO_𝑡=0, one has that 𝑃𝜂_IO−𝑐_2𝑛𝑑_2𝑛(𝑥),𝑃𝜂≤𝑥⟶Λ_AO−𝑐_2𝑛𝑑_2𝑛≤𝑥⟶Λ(𝑥),𝑛⟶∞, (20) where 𝑑_2𝑛=[2log(2𝑛)]^−1/2 and 𝑐_2𝑛=(1/𝑑_2𝑛)−(𝑑_2𝑛/2)[loglog(2𝑛)+log(4𝜋)].

Proof. It follows from Theorem 1 that the autocorrelation function of {𝜆^IO_𝑡} is 𝜌^IO(ℎ)=𝐼(ℎ=0) under the hypothesis 𝐻₀∶𝜔^IO_𝑡=0, so lim_ℎ→∞𝜌^IO(ℎ)⋅logℎ=0; that is, the Berman condition is satisfied. Again, {𝜆^IO_𝑡} is a stationary zero-mean Gaussian time series. Hence, from Lemma 2, we have that 𝑃(((max^𝑛_𝑡=1{𝜆^IO_𝑡}−𝑐_𝑛)/𝑑_𝑛)≤𝑥)→Λ(𝑥). Again from Lemma 3, we know that the distribution of 𝜂_IO=max^𝑛_𝑡=1{|𝜆^IO_𝑡|} is the same as that of max^2𝑛_𝑡=1{𝜆^IO_𝑡}, so we have that 𝑃(((𝜂_IO−𝑐_2𝑛)/𝑑_2𝑛)≤𝑥)→Λ(𝑥). Similarly, under the hypothesis 𝐻^′₀∶𝜔^AO_𝑡=0, from Theorem 1 we know that the autocorrelation function of 𝜆^AO_𝑡 is 𝜌^AO(𝑡,ℎ)=Δ_𝑡^−1/2⋅Δ^−1/2_𝑡+ℎ⋅²_𝑗=ℎ+1𝑓_𝑗𝑦_{𝑡+𝑗−2}𝑓_𝑗−ℎ𝑦_{𝑡+𝑗−2}−𝑓_ℎ𝑦_𝑡+ℎ−2𝐼(1≤ℎ≤2), (21) where lim_ℎ→∞𝜌^AO(ℎ)⋅logℎ=0; that is, the Berman condition is satisfied. Therefore, similar to the statements above, it must hold that 𝑃(((𝜂_AO−𝑐_2𝑛)/𝑑_2𝑛)≤𝑥)→Λ(𝑥).

Lemma 5 (see [14]). Assume that {𝑋_𝑛} is a stationary zero-mean Gaussian time series with variance 1 and autocorrelation function 𝜌(ℎ), and let 𝑀_𝑛=max{𝑋²₁,…,𝑋²_𝑛}, and if 𝜌(ℎ) satisfies the Berman condition, then one has 𝑃(((𝑀_𝑛−𝑒_𝑛)/2)≤𝑥)→Λ(𝑥), 𝑛→∞, where 𝑒_𝑛=2log𝑛−log(log𝑛)−log𝜋.

Theorem 6. Let 𝜗_IO=max^𝑛_𝑡=1{(𝜆^IO_𝑡)²} and 𝜗_AO=max^𝑛_𝑡=1{(𝜆^AO_𝑡)²}, under the hypothesis 𝐻₀∶𝜔^IO_𝑡=0 and 𝐻^′₀∶𝜔^AO_𝑡=0, and if the corresponding Berman conditions for {𝜆^IO_𝑡} and {𝜆^AO_𝑡} are satisfied, then one has 𝑃𝜗_IO−𝑒_𝑛2(𝑥),𝑃𝜗≤𝑥⟶Λ_AO−𝑒_𝑛2≤𝑥⟶Λ(𝑥),𝑛⟶∞, (22) where 𝑒_𝑛=2log𝑛−log(log𝑛)−log𝜋.

Proof. We know that the Berman conditions for {𝜆^IO_𝑡} and {𝜆^AO_𝑡} are satisfied under the hypothesis 𝐻₀∶𝜔^IO_𝑡=0 and 𝐻^′₀∶𝜔^AO_𝑡=0 by the inferential process of Theorem 4. Thus, the conclusion is also obtained by Lemma 5.

Let 𝛼 be the test significance level and 𝑦_𝑛=(1+(1−𝛼)^1/𝑛)/2. We denote the 𝑡-distribution function with 𝑛 degrees of freedom by 𝐻_𝑛(𝑡). Let 𝑥_𝑛=𝐻⁻¹_𝑛−2(𝑦_𝑛) and 𝑒_𝑛=(𝑛/(𝑛−1))𝑥²_𝑛−2Λ⁻¹(1−𝛼), where Λ⁻¹(1−𝛼) is the 1−𝛼 quantile of the Gumbel distribution. Similarly, we have the following.

Theorem 7. Under the conditions above, one has that 𝑃𝑀_𝑛−𝑒_𝑛2(𝑥),𝑃𝜗≤𝑥⟶Λ_IO−𝑒_𝑛2(𝑥),𝑃𝜗≤𝑥⟶Λ_AO−𝑒_𝑛2≤𝑥⟶Λ(𝑥),𝑛⟶∞. (23)

At this point, it is convenient to introduce two more pieces of notation: 𝐴^IO_𝑡=((|𝜆^IO_𝑡|−𝑐_2𝑛)/𝑑_2𝑛) and 𝐴^AO_𝑡=((|𝜆^AO_𝑡|−𝑐_2𝑛)/𝑑_2𝑛), which we call absolute value test statistics; 𝑆^IO_𝑡=(((𝜆^IO_𝑡)²−𝑒_𝑛)/2) and 𝑆^AO_𝑡=(((𝜆^AO_𝑡)²−𝑒_𝑛)/2), which we call square test statistics; 𝐶^IO_𝑡=(((𝜆^IO_𝑡)²−𝑒_𝑛)/2) and 𝐶^AO_𝑡=(((𝜆^AO_𝑡)²−𝑒_𝑛)/2), which we call adjusted square test statistic. We likewise obtain 𝑃m^𝑛ax_𝑡=1𝐴^IO_𝑡(𝑥),𝑃≤𝑥⟶Λ_𝑛max_𝑡=1𝐴^AO_𝑡(𝑥),𝑃≤𝑥⟶Λ_𝑛max_𝑡=1𝑆^IO_𝑡(𝑥)𝑃≤𝑥⟶Λ,𝑛⟶∞,_𝑛max_𝑡=1𝑆^AO_𝑡(𝑥),𝑃≤𝑥⟶Λ_𝑛max_𝑡=1𝐶^IO_𝑡(𝑥),𝑃≤𝑥⟶Λ_𝑛max_𝑡=1𝐶^AO_𝑡≤𝑥⟶Λ(𝑥),𝑛⟶∞. (24)

If there are several outliers, the main idea is to first detect the maximum outlier by using our method and to obtain new test statistics by deleting its effect. Then, we go on to detect the next outlier and repeat the procedure till there is no outlier. When two types of outliers appear, the test is 𝐻₀∶𝜔^IO_𝑡=0,𝜔^AO_𝑡=0versus𝐻₁∶𝜔^IO_𝑡≠0or𝜔^AO_𝑡≠0,𝑡=1,…,𝑛. (25)

For details, we provide the following steps using absolute value test statistics to detect the AO and IO in AFAR model as follows: (a) Let test statistics 𝑇₁=max^𝑛_𝑡=1{𝐴^IO_𝑡} and 𝑇₂=max^𝑛_𝑡=1{𝐴^AO_𝑡}. (b) Calculate 𝜆^IO_𝑡 and 𝜆^AO_𝑡 at every time point, and calculate the maximum 𝑡₁=𝐴^IO_𝑞1=max^𝑛_𝑡=1{𝐴^IO_𝑡} and 𝑡₂=𝐴^AO_𝑞2=max^𝑛_𝑡=1{𝐴^AO_𝑡} of absolute value test statistics. (c) Let 𝐶𝑉=Λ⁻¹(1−𝛼)=−log(−log(1−𝛼)), if 𝑡₁=max{𝑡₁,𝑡₂}>𝐶𝑉, then we believe that the observation is an IO at 𝑡=𝑞₁, and if 𝑡₂=max{𝑡₁,𝑡₂}>𝐶𝑉, then we believe that the observation is an AO at 𝑡=𝑞₂; else, we believe there is no IO or AO. (d) Calculate the 𝑝-value 𝑝₁=𝑃_𝐻0(𝑇₁≥𝑡₁)=1−Λ(𝑡₁) and 𝑝₂=𝑃_𝐻0(𝑇₂≥𝑡₂)=1−Λ(𝑡₂), and if 𝑝₁≤𝛼 or 𝑝₂≤𝛼, then reject the hypothesis 𝐻₀, believing there is an IO or an AO. Furthermore, we decide whether it is an IO or an AO by its minimal 𝑝-value. (e) Delete the effect of detected outliers and detect the next outlier. Repeat the above steps till there is no outlier.

4. Simulation Studies

Example 8. In this simulation, we consider the following model, where there is only one IO. The sample sizes are 𝑛=50, 100, 500, 1000, respectively and 𝜔^IO_𝑛/2=4. We have 𝑦_𝑡=cos1.5𝑦_𝑡−21+𝑦²_𝑡−2𝑦_𝑡−1+−𝑦−exp^2𝑡−22𝑦_𝑡−2+𝜂_𝑡,𝜂_𝑡=𝜀_𝑡+𝜔^IO_𝑛/2𝑛⋅𝛿𝑡−2,𝜀_𝑡∼𝑁(0,0.5). (26)
For 𝑛=50, All the three test statistics detect an outlier at 𝑡=25. The 𝑝-values for believing it is an IO are 𝑝(𝐴^IO)=4.8414𝑒−6, 𝑝(𝑆^IO)=1.8326𝑒−8, and 𝑝(𝐶^IO)=3.3739𝑒−8, respectively. The 𝑝-values for believing it is an AO are 𝑝(𝐴^AO)=3.1509𝑒−4, 𝑝(𝑆^AO)=4.7392𝑒−5, and 𝑝(𝐶^AO)=8.7247𝑒−5, respectively. Because the 𝑝-values for believing it is an IO are smaller than the 𝑝-values for believing it is an AO, we believe it is an IO, and its influence magnitude is 3.1994. The results are similar for 𝑛=100, 500, 1000, respectively, we thus omit the details here. For different test statistics, their 𝑝-values for believing it is an IO are summarized as follows in Table 1.

Table 1 

Example 9. There are four IOs and one AO in the time series. These IOs appear at 𝑡=49, 𝑡=50, 𝑡=51, and 𝑡=52 sequentially. Their sizes are 3, 6, 5, and 4, respectively. One AO appears at 𝑡=70 alone, and its size is 7, 𝑛=100. The model is as follows: 𝑥_𝑡=cos1.5𝑥_𝑡−21+𝑥²_𝑡−2𝑥_𝑡−1+−exp−0.5𝑥²_𝑡−2𝑥_𝑡−2+𝜀_𝑡,𝜀_𝑡∼𝑁(0,0.6), (27) where 𝑦_𝑡 is obtained by (3) or (4).
First, we detect an AO at 𝑡=70, and its size is 6.8904. The three test statistics and their corresponding critical values are shown in Figure 1, where Figure 1(a) corresponds to the absolute value test statistics, Figure 1(b) corresponds to square test statistics, and Figure 1(c) corresponds to adjusted square test statistics. Symbol ‘‘⋄’’ denotes IO, and ‘‘∗’’ denotes AO. Parallel broken line denotes critical value, which is uniform in the figures. Deleting the effect of this AO and then continuing to detect other outliers in the series, we observe an IO at 𝑡=50, whose size is 6.0285; see details in Figure 2. Deleting the effect of the aforementioned two outliers and then continuing to detect other outliers in the series, we observe an IO at 𝑡=51, whose size is 4.8119. Deleting the effect of the aforementioned three outliers and then continuing to detect other outliers in the series, we observe an IO at 𝑡=52, whose size is 4.0024. Deleting the effect of the aforementioned four outliers and then continuing to detect other outlier in the series, we detect an IO at 𝑡=49, whose size is 2.9046; see Figure 3. Deleting the effect of the aforementioned five outliers and then continuing to detect other outliers in the series, we have not detected any outlier; see Figure 4. Considering the length of paper, we omit some figures here. The result is consistent with the advance enactment.

(a)

(b)

(c)

Figure 1 

Example 9, first detection.

(a)

(b)

(c)

Figure 2 

Example 9, second detection.

(a)

(b)

(c)

Figure 3 

Example 9, fifth detection.

(a)

(b)

(c)

Figure 4 

Example 9, sixth detection.

5. Conclusions

The FAR model is mainly featured by the model-dependent-variable, which in one way or another limits the scope of its applications. As a generalization of the class of models, AFAR model clearly covers a larger range of objects than the FAR model, which makes it possible to reduce modeling biases [17] via choosing a proper model-dependent direction. This paper is concerned with detecting AO and IO in AFAR models using extreme value methods. We derive the asymptotic distribution of test statistics and provide a control for significance level, which serves as an extension and improvement of existing methods. Based on several simulation studies, we give conclusion remarks as follows. (a) The extreme value method for detecting outliers in AFAR models is tractable and effective not only for IO and AO, but also for separate outliers and outlier patches. Furthermore, it is shown that our method can reduce possible effects of masking and swamping. (b) When applying extreme value theory to detect outliers with a relatively small size samples at hand, the employment of square test statistics works better than that of adjusted square test statistics as well as absolute value test statistics. While following the increments in the samples size, the detecting effect of adjusted square test statistics also increases against that of square test statistics and absolute value test statistics (e.g., see Table 1). (c) Selection of model parameters and the magnitude of outliers have huge influences on the effect of detection.

Acknowledgments

The authors sincerely wish to thank the editor and the three referees for their insightful suggestions which have led to improving the early version of the paper. The research is supported by the National Natural Science Foundation of China (11171065), the Natural Science Foundation of Jiangsu Province (BK2011058), and Research Fund for the Doctoral Program of Higher Education of China (20120092110021).