The stopped clock model

1 Introduction

Let { X n } n ∈ Z and { U n } n ∈ Z be stationary sequences of real random variables on the probability space ( Ω , A , P ) and P ( U n ∈ { 0 , 1 } ) = 1 . We define, for n ≥ 1 ,

Sequence { Y n } n ≥ 1 corresponds to a model of failures on records of { X n } n ∈ Z replaced by the last available record, which occurs in some random past instant, if we interpret n as time. Thus, if, for example, it occurs { U 1 = 1 , U 2 = 0 , U 3 = 1 , U 4 = 0 , U 5 = 0 , U 6 = 0 , U 7 = 1 } , we will have { Y 1 = X 1 , Y 2 = X 1 , Y 3 = X 3 , Y 4 = X 3 , Y 5 = X 3 , Y 6 = X 3 , Y 7 = X 7 } . This constancy of some variables of { X n } n ∈ Z for random periods of time motivates the designation of “stopped clock model” for sequence { Y n } n ≥ 1 .

Failure models studied in the literature from the point of view of extremal behavior do not consider the stopped clock model (Hall and Hüsler [4]; Ferreira et al. [3] and references therein).

The model we will study can also be represented by { X N n } n ≥ 1 , where { N n } n ≥ 1 is a sequence of positive integer variables representable by

We can also state a recursive formulation for { Y n } n ≥ 1 through

Under any of the three possible representations (failures model, random index sequence or recursive sequence), we are not aware of an extremal behavior study of { Y n } n ≥ 1 in the literature.

Our departure hypotheses about the base sequence { X n } n ∈ Z and about sequence { U n } n ∈ Z are as follows:

1. { X n } n ∈ Z is a stationary sequence of random variables almost surely distinct and, without loss of generality, such that F X n ( x ) ≔ F ( x ) = exp ( − 1 / x ) , x > 0 , i.e., standard Fréchet distributed.

2. { X n } n ∈ Z and { U n } n ∈ Z are independent.

3. { U n } n ∈ Z is stationary and p n 1 , … , n s ( i 1 , … , i s ) ≔ P ( U n 1 = i 1 , … , U n s = i s ) , i j ∈ { 0 , 1 } , j = 1 , … , s , is such that p n , n + 1 , … , n + κ − 1 ( 0 , … , 0 ) = 0 , for some κ ≥ 1 .

Figure 1

Sample path of 100 observations simulated from { Y n } defined in (1) based on independent standard Fréchet { X n } and on { U n } given in (2) where we take random variables { W n } standard exponential distributed, A = ] 0 , 1 / 2 ] , and thus, p = 0.3935 and considering κ = 3 .

In the next section, we propose an estimator for probabilities p n , … , n + s ( 1 , 0 , … , 0 ) , 0 ≤ s < κ − 1 . In Section 3, we analyze the existence of the extremal index for { Y n } n ≥ 1 , an important measure to evaluate the tendency to occur clusters of its high values (see, e.g., Kulik and Solier [6], and references therein). A characterization of the tail dependence will be presented in Section 4. The results are illustrated with an ARMAX sequence.

For the sake of simplicity, we will omit the variation of n in sequence notation whenever there is no doubt, taking into account that we will keep the designation { Y n } for the stopped clock model and { X n } and { U n } for the sequences that generate it.

2 Inference on { U n }

Assuming that { U n } is not observable, as well as the values of { X n } that are lost, it is of interest to retrieve information about these sequences from the available sequence { Y n } .

Since, for n ≥ 1 and s ≥ 1 , we have

we propose to estimate these probabilities from the respective empirical counterparts of a random sample ( Y ˆ 1 , Y ˆ 2 , … , Y ˆ m ) from { Y n } , i.e.,

which are consistent by the weak law of large numbers. The value of κ can be inferred from

In order to evaluate the finite sample behavior of the aforementioned estimators, we have simulated 1,000 independent replicas with size m = 100 , 1,000, 5,000 of the model in Example 1.1. The absolute bias (abias) and root mean squared error (rmse) are presented in Table 1. The results reveal a good performance of the estimators, even in the case of smaller sample sizes. Parameter κ was always estimated with no error.

3 The extremal index of { Y n }

The sequence { Y n } is stationary because the sequences { X n } and { U n } are stationary and independent from each other. In addition, the common distribution for Y n , n ≥ 1 , is also standard Fréchet, as is the common distribution for X n , since

For any τ > 0 , if we define u n ≡ u n ( τ ) = n / τ , n ≥ 1 , it turns out that E ∑ i = 1 n 1 { Y i > u n } = n P ( Y 1 > u n ) ⟶ n → ∞ τ and n P ( X 1 > u n ) ⟶ n → ∞ τ , so we refer to these levels u n by normalized levels for { Y n } and { X n } .

In this section, in addition to the general assumptions about the model presented in Section 1, we start by assuming that { X n } and { U n } present dependency structures such that variables sufficiently apart can be considered approximately independent. Concretely, we assume that { U n } satisfies the strong-mixing condition (Rosenblatt [9]) and { X n } satisfies condition D ( u n ) (Leadbetter [7]) for normalized levels u n .

The tendency for clustering of values of { Y n } above u n depends on the same tendency within { X n } and the propensity of { U n } for consecutive null values. The clustering tendency can be assessed through the extremal index (Leadbetter, [7]). More precisely, { X n } is said to have extremal index θ X ∈ ( 0 , 1 ] if

If D ( u n ) holds for { X n } , we have

for any integers sequence { k n } , such that,

We can therefore say that

Now we compare the local behavior of sequences { X n } and { Y n } , i.e., of X i and Y i for i ∈ ( j − 1 ) n k n + 1 , … , j n k n , j = 1 , … , k n , with regard to the oscillations of their values in relation to u n . To this end, we will use local dependency conditions D ( s ) ( u n ) . We say that { X n } satisfies D ( s ) ( u n ) , s ≥ 2 , whenever

for some integers sequence { k n } satisfying (4). Condition D ( 1 ) ( u n ) translates into

Observe that if D ( s ) ( u n ) holds for some s ≥ 1 , then D ( m ) ( u n ) also holds for m > s . Condition D ( 1 ) ( u n ) is known as D ′ ( u n ) (Leadbetter et al. [8]) and relates to a unit extremal index, i.e., absence of extreme values clustering. In particular, this is the case of independent variables. Although { X n } satisfies D ′ ( u n ) , this condition is not generally valid for { Y n } . Observe that

For i = 1 and j = κ , we have j ∗ = 1 and the corresponding term becomes n P ( X 1 > u n ) → τ > 0 , as n → ∞ , and this is the reason why, in general, { Y n } does not satisfy D ′ ( u n ) even if { X n } satisfies it.

Under conditions D ( u n ) and D ( s ) ( u n ) with s ≥ 2 , we can also compute the extremal index θ X defined in (3) by (Chernick et al. [1]; Corollary 1.3)

If { X n } and { Y n } have extremal indexes θ X and θ Y , respectively, then θ Y ≤ θ X , since P ( ⋁ i = 1 n X i ≤ n / τ ) ≤ P ( ⋁ i = 1 n Y i ≤ n / τ ) . This corresponds to the intuitively expected, if we remember that the possible repetition of variables X n leads to larger clusters of values above u n . In the following result, we establish a relationship between θ X and θ Y .