PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 88(102) (2010), 87–98 DOI: 10.2298/PIM1002087P

AR(1) TIME SERIES WITH APPROXIMATED BETA MARGINAL

Božidar V. Popović

Communicated by Slobodanka Janković

Abstract. We consider the AR(1) time series model X_{t} − βX_{t−1} = ξ_{t}, β^{−p} ∈ ℕ ∖ {1}, when X_{t} has Beta distribution B(p, q), p ∈ (0, 1], q > 1. Special attention is given to the case p = 1 when the marginal distribution is approximated by the power law distribution closely connected with the Kumaraswamy distribution Kum(p, q), p ∈ (0, 1], q > 1. Using the Laplace transform technique, we prove that for p = 1 the distribution of the innovation process is uniform discrete. For p ∈ (0, 1), the innovation process has a continuous distribution. We also consider estimation issues of the model.

1. Introduction

In the standard time series analysis one assumes that its marginal distribution is Gaussian. However, a Gaussian distribution will not always be appropriate. In earlier works stationary non-Gaussian time series models have been developed for variables with positive and highly skewed distributions. There still remain some situations where Gaussian marginals are inappropriate, i.e., where the marginal time-series variable being modeled, although not skewed or inherently positive valued, has a large kurtosis and long-tailed distributions. There are plenty real situations that can not be modeled by Gaussian distribution like in hydrology, meteorology, information theory, economics, etc. Simple models with exponential marginals or mixed exponential marginals are considered in [9, 29, 1, 2, 13], while another marginals have been discussed like Gamma [9, 19, 7], Laplace [19], uniform [15, 21] and Weibull [19, 7]. Finally, we point out autoregressive processes PBAR and NBAR autoregressive process models constructed by McKenzie’s [10] for positively and respectively, negatively correlated pairs of Beta random variables employing certain properties of the B(p, q) distribution.

2010 Mathematics Subject Classification: Primary 62M10; Secondary 33C15, 66F10, 60E10.

Key words and phrases: Beta distribution, Kumaraswamy distribution, approximated Beta distribution, Kummer function of the first kind, first order autoregressive model.

In this paper, we introduce an autoregressive first order time series model with the marginal Beta distribution B(p, q), p ∈ (0, 1], q > 1 of which the Laplace transform is approximated when the transformation argument is large. The resulting approximation determines a new distribution and for p = 1 results in a discrete uniform distribution for the innovation process. For 0 < p < 1 the distribution of the innovation process is continuous. Special attention is given to the case p = 1, when the approximation becomes a kind of power law distribution, characterized by the related probability density function (PDF) f(x) = q(1 − x)^{q−1} , x ∈ [0, 1]. This distribution coincides with the Beta distribution B(1, q). On the other hand, the B(1, q) generates Kumaraswamy distribution Kum(p, q), p ∈ (0, 1], q > 1 introduced in [24]. This distribution is very important in applications when double bounded random processes arise in practice [27, 28], e.g., it has been applied in modelling the storage volume of a reservoir and system design [26] and turns out to be very important in many hydrological problems. For a complete account of the properties of the Kumaraswamy distribution, consult [16].

A random variable X_{p,q} defined on some standard probability space (Ω, 𝔉, P) having the Kum(p, q) distribution possesses the PDF [8, pp. 169–170] (in its simplest form):

f(x) = pqx^{p−1} (1 − x^{p})^{q−1} · 1_{[0,1]}(x).

The cumulative distribution function’s (CDF) non-constant part is

F (x) = (1 − (1 − x^{p})^{q}) · 1_{[0,1]}(x),

where 1_{[0,1]}(x) stands for the indicator function of the closed unit interval. Furthermore, we have

(1.1)  X_{p,q}^{p} =^{d} Y_{1,q},

where Y_{1,q} has Beta distribution B(1, q), q > 1. Note that both Kum(1, 1) and B(1, 1) represent the uniform U(0, 1) distribution.

The technical part of the research begins with the derivation of the Laplace transform (LT) of the autoregressive model. The resulting LT function is expressed in terms of the Wright hypergeometric function Φ. Therefore we are faced with extremely hard calculations in inverting the Laplace transform. So, we approximate the derived LT of the model, obtaining the inverse LT mutatis mutandis the distribution of the innovation sequence {ξ_{t} : t ∈ ℤ}. The resulting approximate distribution will be referred to as the approximated Beta (AB_{p,q}) and as the approximated power law (APL) for p = 1. Therefore, considering initially a \underline{L}inear time series model with AB_{p,q}, we arrive at a new model called LAB_{p,q} AR(1), which becomes LAPLAR_{q}(1) for p = 1.

Finally, the unknown parameter will be estimated using conditional least squares estimator.

2. Approximated Laplace transform of B(p, q)

Consider a random variable Y_{p,q} defined on a standard probability space (Ω, 𝔉, P); having the B(p, q) distribution. The related PDF is

(2.1)  f(x) = \frac{Γ(p + q)}{Γ(p) Γ(q)} x^{p−1} (1 − x)^{q−1} 1_{[0,1]}(x) .

In what follows Γ(p) stands for the familiar Euler’s gamma function. We write the Laplace transform of some suitable function f as

L_{λ}[f] = ∫_{0}^{∞} e^{−λx} f(x) dx =: φ(λ),

while

L_{x}^{−1} [φ] = \frac{1}{2πi} ∫_{−i∞}^{i∞} e^{xλ} φ(λ) dλ =: f(x)

stands for the inverse Laplace-transform pair of φ(λ).

The Laplace transform of the PDF (2.1) equals

φ_{Y_{p,q}} (λ) = Ee^{−λY_{p,q}} = \frac{Γ(p + q)}{Γ(p) Γ(q)} ∫_{0}^{1} e^{−λx} x^{p−1} (1 − x)^{q−1} dx = F (p, q + p, −λ),

where F denotes the familiar Kummer function of the first kind, that is, the confluent hypergeometric function _{1}F_{1} defined by the series

F (a, b, z) := ∑_{n=0}^{∞} \frac{(a)_{n}}{(b)_{n}} \frac{z^{n}}{n!} .

Here (a)_{n} = a(a + 1) ⋯ (a + n − 1) = Γ(a + n)/Γ(a), (a)_{0} = 1 is the Pochhammer symbol.

Let us prove the formula

(2.2)  φ_{Y_{p,q}} (λ) ∼ \frac{Γ(p + q)}{Γ(q)λ^{p}} (1 − e^{−p(q−1)λ^{p}} ),  λ → ∞,

for all q > 1, where ∼ denotes the asymptotic equality. Indeed, considering the transformation of the integrand in

φ_{Y_{p,q}} (λ) = \frac{Γ(p + q)}{Γ(p) Γ(q)} ∫_{0}^{1} e^{−λx} x^{p−1} (1 − x)^{q−1} dx

= \frac{λ^{−p}}{B(p, q)}∫_{0}^{λ} e^{−x} x^{p−1} ( 1 − \frac{x}{λ})^{q−1} dx,

gives the asymptotics

(2.3)  φ_{Y_{p,q}} (λ) ∼ \frac{Γ(p + q)}{Γ(p) Γ(q)λ^{p}} ∫_{0}^{∞} e^{−x} x^{p−1} ( 1 − \frac{(q − 1)x}{λ} ) dx

= \frac{Γ(p + q)}{Γ(q)λ^{p}} ( 1 − \frac{p(q − 1)}{λ} ) ∼ \frac{Γ(p + q)}{Γ(q)λ^{p}} exp\{ − \frac{p(q − 1)}{λ^{p}} \}

∼ \frac{Γ(p + q)}{Γ(q)λ^{p}} (1 − exp\{ − p(q − 1)λ^{p}\}),

for λ large enough. Hence the asserted result (2.2).

An important special case p = 1 of (2.2) is φ_{Y_{1,q}} (λ) ∼ \frac{q}{λ}(1−e^{−(q−1)λ} ), λ → ∞.

3. The LAPLAR_{q}(1) and LAB_{p,q} AR(1) models

In this section we introduce a new autoregressive model of the first order. Since the LT of Beta distribution one expresses as special function, for the derivation of distribution of innovation sequence {ξ_{t}: t ∈ ℤ}, we will use approximated LT (2.2). In this way, marginal distribution of the model (3.1) is replaced by the AB_{p,q} such that is generated by the model. The LAB_{p,q}AR(1) model is given with the following equations:

(3.1)  X_{t} − βX_{t−1} = ξ_{t}, β^{−p} ∈ ℕ ∖ {1}, p ∈ (0, 1].

Here X_{t} is B(p, q), p ∈ (0, 1], q > 1. Also, we make the following assumptions:

(i) X_{t} is stationary; (ii) {X_{t}} and {ξ_{s}} are independent for t < s, where stationarity is ensured by β ∈ (0, 1).

Let us define a positive integer κ(p) = [β^{−p} ], where [A] stands for the integer part of some A. In what follows, we consider such parameters p ∈ (0, 1] and β ∈ (0, 1) that κ(p) ∈ ℕ.

Now, we derive our principal result-the distribution (or a related approximation) of the innovation sequence ξ_{t}.

Theorem 3.1. Consider the LAPLAR_{q}(1) times series model

X_{t} − βX_{t−1} = ξ_{t}, β = \frac{1}{n}, n ∈ ℕ ∖ \{1\},

such that possesses marginal APL distribution, let q > 1. Then the distribution of the i.i.d. sequence {ξ_{t} : t ∈ ℤ} is approximated by the uniform discrete distribution

P \{ξ_{t} = \frac{(q − 1)j}{n} \} = 1/n, j = \overline{0, n − 1} .

Proof. Let us find the LT of the model (3.1). Using the assumption (ii) upon the independent X_{t} and ξ_{s}, t < s, we have:

φ_{X_{t}} (λ) = Ee^{−λX_{t}} = Ee^{−(βX_{t−1} +ξ_{t})λ} = Ee^{−βX_{t−1} λ} Ee^{−ξ_{t}λ} = φ_{X_{t−1}} (βλ) φ_{ξ_{t}} (λ).

With regard to the assumed stationarity (i) of X_{t} we can conclude φ_{X} (λ) = φ_{X} (βλ)φ_{ξ}(λ). Using asymptotic formula (2.2), we obtain

φ_{ξ}(λ) = \frac{φ_{X} (λ)}{φ_{X} (βλ)} ∼ \frac{1}{n} · \frac{1 − e^{−(q−1)λ}}{1 − e^{−(q−1)βλ}} = \frac{1}{n} · \frac{1 − (e^{−(q−1) \frac{1}{n} λ} )^{n}}{1 − e^{−(q−1) \frac{1}{n} λ}}

= \frac{1}{n} · ∑_{j=0}^{n−1} e^{−(q−1) \frac{j}{n} λ} =: A_{λ}(φ_{ξ}), λ → ∞.

Now, by inverse LT we easily deduce that the distribution of i.i.d. sequence {ξ_{t} : t ∈ ℤ}. The inverse LT of the approximation A_{λ}(φ_{ξ}) we calculate directly:

L_{x}^{−1} [A_{λ}(φ_{ξ})] = \frac{1}{n}L_{x}^{−1} [ ∑_{j=0}^{n−1} e^{−(q−1) \frac{j}{n} λ} ] = \frac{1}{n} ∑_{j=0}^{n−1} δ( x − \frac{(q − 1)j}{n} ) ,

where δ is the well-known Dirac function. Hence, we conclude that the distribution of ξ_{t} is approximated by discrete probability law

(3.2)  ( \underbrace{\begin{matrix}0 & (q − 1)/n & ⋯ & (q − 1)(n − 1)/n \\ n^{−1} & n^{−1} & ⋯ & n^{−1} \end{matrix}}_{n}) ,

which is the assertion of the theorem. □

Remark 3.1. The same model, (3.1), but with the uniform U(0, 1) marginal distribution has been studied in [15]. In this case it has been proved that innovation sequence of (3.1) coincides with discrete distribution (3.2) for q = 2.

Remark 3.2. By equality (1.1) one deduces X_{t}: Kum(p, q) =^{d} X_{t}^{1/p} : B(1, q), that links our results to Kumaraswamy Kum(p, q), q > 1 distribution.

Let us introduce the hypergeometric function Φ(a, α; z) by E. M. Wright defined in the whole complex plane ℂ, for complex a ∈ ℂ and real 0 < α < 1, by the series

(3.3)  Φ(a, −α; z) = ∑_{n=0}^{∞} \frac{z^{n}}{Γ(a − αn) n!} ,

see e.g. [6]. It is not hard to show that

(3.4)  L_{x}^{−1} [λ^{−τ} e^{−T λ^{p}} ] = x^{τ−1} Φ(τ, −p; −T x^{−p} ),  T > 0,

is equivalent to the Humbert–Pollard inverse LT formula [23], [11, Eq. (4)]. Indeed, taking certain p ∈ (0, 1), we deduce (3.4) having

λ^{−τ} e^{−T λ^{p}} = ∑_{n=0}^{∞} \frac{(−T)^{n} λ^{−τ+np}}{n!} = ∑_{n=0}^{∞} \frac{(−T)^{n}}{n!} · \frac{1}{Γ(τ − np)} ∫_{0}^{∞} e^{−λx} x^{τ−np−1} dx

= ∫_{0}^{∞} e^{−λx} x^{τ−1} \{ ∑_{n=0}^{∞} \frac{(−T x^{−p} )^{n}}{Γ(τ − np) n!} \} dx

= ∫_{0}^{∞} e^{−λx} x^{τ−1} Φ(τ, −p; −T x^{−p} ) dx.

Theorem 3.2. Consider the LAB_{p,q} AR(1) times series model

X_{t} − βX_{t−1} = ξ_{t} β^{p} = \frac{1}{n}, n ∈ ℕ ∖ \{1\},

that possesses the marginal AB_{p,q}. Assume that p ∈ (0, 1), q > 1 and κ(p) ∈ ℕ. Then the distribution of the i.i.d. sequence {ξ_{t} : t ∈ ℤ} is approximated by continious distribution having PDF

(3.5)  f_{ξ}(p; x) = \frac{x^{−1}}{∥Φ_{p}∥} ∑_{j=0}^{κ(p)−1} Φ( 0, −p; −\frac{p(q − 1)β^{p}j}{x^{p}} ) ,

where

(3.6)  ∥Φ_{p}∥ := ∑_{j=0}^{κ(p)−1} ∫_{0}^{∞} x^{−1} Φ( 0, −p; −\frac{p(q − 1)β^{p}j}{x^{p}} ) dx.

Proof. Using similar lines to the proof of Theorem 3.1, with the aid of the asymptotic identity (2.3) for large enough λ, we conclude that

(3.7)  φ_{ξ}(λ) = \frac{φ_{X} (λ)}{φ_{X} (βλ)} ∼ β^{p} ∑_{j=0}^{κ(p)−1} exp\{−p(q − 1)β^{p}jλ^{p}\}.

Denote the right-hand side of this expression by B_{λ,p} (φ_{ξ}), say. Now, setting τ = 0 in the Humbert–Pollard formula (3.4) and employing it to the addends in (3.7), one deduces

(3.8)  L_{x}^{−1} [B_{λ,p} (φ_{ξ})] = β^{p} ∑_{j=0}^{κ(p)−1} x^{−1} Φ( 0, −p; −\frac{p(q − 1)β^{p}j}{x^{p}} ) ,

where the Wright function series converges, because p ∈ (0, 1).

The behavior of L_{x}^{−1} [B_{λ,p} (φ_{ξ})] near the origin is controlled by the upper bound

|x^{−1} Φ(0, −p; −x^{−p} )| ⩽ \frac{1}{pπ} \frac{Γ(1/p)}{cos^{1/p}(pπ/2)}, x ⩾ 0,

such that is the special case of the majorant reported in [6, Eq. (32)].

It remains only to show the non-negativity of L_{x}^{−1} [B_{λ,p} (φ_{ξ})]. Indeed, in [11] it was shown that x^{−1} Φ(0, −p; −T x^{−p} ) > 0, T > 0, p ∈ (0, 1), almost everywhere, when x > 0. [6, Theorem 8] strengthened his result to the strict positivity.

Finally, normalizing (3.8), hence the proof of the assertion (3.5). □

Asymptotic behavior of the PDF of the AB_{p,q}, p ∈ (0, 1), when x approaches nil or infinity, follows from the adopted Mikusiński’s results [14, Eqs (4), (5)]

x^{−1} Φ( 0, −p; −\frac{T}{x^{p}} ) ∼ \{ \begin{matrix}\frac{Q_{T}}{x^{p/(2(1−p))}} exp\{ − R_{T}x^{−p/(1−p)} \}, & x → 0+ \\ \frac{p}{Γ(1 − p)} T x^{−p−1} , & x → ∞. \end{matrix}

where

Q_{T} := \frac{(pT)^{1/(2(1−p))}}{\sqrt{2π(1 − p)}} , R_{T} := (1 − p)(p^{p}T)^{1/(1−p)} , T := p(q − 1)β^{p}j.

Let us denote by X_{0} the initial value of the time series {X_{t}: t ∈ ℤ}, and thereafter we can formulate the following result:

Theorem 3.3. Let {ξ_{t}: t ∈ ℤ} be an i.i.d. sequence of random variables having distribution (3.5). If q > 1, p ∈ (0, 1), β^{p} = \frac{1}{n}, n ∈ ℕ ∖ \{1\}, κ(p) ∈ ℕ and X_{0} is from AB_{p,q}, then relation (3.1) defines the time series {X_{t}: t ∈ ℤ} whose marginal distribution is AB_{p,q}.

Remark 3.3. For p = 1 Theorem 3.3 is valid for LAPLAR_{q}(1) model.

Remark 3.4. Since Wright hypergeometric function (3.3) is not defined for α > 1, it is not possible to consider the case p > 1.

3.1. Simulation study. Histograms of beta distribution B(p, q), as well as histogram of AB_{p,q}, for given values of p and q, will be presented in Example 3.1. Having in mind that AB_{p,q} is determined by the model, there was a problem of generating random numbers from AB_{p,q}. The solution is to use the algorithm described in [20]. This algorithm is implemented, by Ridout, in software R [25].

In Example 3.2, we give graphical representation of PDF (3.5) together with PDF of beta distribution for certain parameter values. The normalizing constant ∥Φ_{p}∥ is calculated using adaptive Simpson quadrature. Since integral (3.6) is improper, we apply use the technique from [22, pp. 618–621].

The sample path of the model (3.1) will be simulated in Example 3.3. For this purpose Theorem 3.3 was used. Since the PDF (3.5) is given in the form of special function, it is impossible to build random number generator from PDF. So the initial value X_{0} and random numbers from the distribution whose PDF is (3.5) is generated using the same idea as in Example 3.1.

Example 3.1. Sample of size 5.000 was drawn from B(1/3, 4). Using the algorithm described in [20], a set of 5000 elements from AB_{1/3, 4} is generated. Figure 1 represents histograms of beta distribution and its approximation.

Figure 1. Histogram of Beta distribution and its approximation AB_{p,q}.

Example 3.2. The density function of Beta distribution and PDF of its approximation (3.5), when Laplace transform argument s is large enough, will be presented in Figure 2 for two sets of different parameters.

Example 3.3. In this example we give the sample paths of LAB_{p,q} AR(1) model for different sample sizes Figure 3. Parameters values are: p = 1/3, q = 4 and β = 1/8.

Figure 2. PDF of Beta distribution and of (3.5) for two different parameter arrays near the origin.

Figure 3. Sample paths.

4. Some properties of the LAB_{p,q} AR(1) model

Following the lines of the similar result for linear AR(1) processes by Hamilton [12, pp. 53–56], it is very easy to derive the autocovariance and autocorrelation function of the model (3.1), with lag τ ∈ ℤ. The autocovariance and autocorrelation function are given with the following expressions, respectively:

γ(τ) = β^{|τ|} VarX

ρ(τ) = β^{|τ|} (4.1)

Theorem 4.1. Process X_{t} is not time reversible.

Proof. Using (3.1) and Markov properties of the process X_{t}, it is easy to show that the joint Laplace transform of X_{t} and X_{t−1} , is not symmetric with respect to λ_{1} and λ_{2}. So the assertion of the theorem is proved. □

Theorem 4.2. Difference equation (3.1) has a unique, strictly stationary, F_{t}- measurable and ergodic solution: X_{t} = ∑ _{j=0}^{∞} β^{j} ξ_{t−j} .

Proof. See [17], [18]. □

5. Parameter estimation in LAB_{p,q} AR(1) model

In this section, we estimate the parameters q and β of the LAPLAR_{q}(1) model using estimated autocorrelation function with lag 1, and estimated expectation of random variable X. Also we estimate the β appearing in the LAB_{p,q}AR(1), p ∈ (0, 1] model by a conditional least squares. The properties of this estimator are proved.

5.1. Parameter estimation in LAPLAR_{q}(1). The parameter β can be estimated using (4.1) for p = 1. Thus β = ρ(1), where autocorrelation ρ(1) with lag 1 can be estimated using the following formula:

ρ̂(1) = \frac{∑ _{t=2}^{n} (X_{t} − \overline{X})(X_{t−1} − \overline{X})}{∑ _{t=1}^{n} (X_{t} − \overline{X}) ^{2}} ,

Since X_{t} is B(1, q), and stationarity condition is fulfilled, the expectation of random variable X is

(5.1)  EX = \frac{1}{1 + q}

From (5.1) parameter q can be estimated using q = \frac{\overline{X} −1}{\overline{X}} , where the mean value of the generated sequence \{X_{t}\} is \overline{X}.

5.2. Conditional least squares in LAB_{p,q} AR(1) model. Model (3.1) can be rewritten in the form:

Y_{t} = βY_{t−1} + C_{t}, C_{t} = (β − 1)m + ξ_{t},

where Y_{t} = X_{t} − m and EX_{t} = m. We will assume that parameter m is known.

Let us denote by F_{t−1} , σ-algebra defined by r.v. X_{s}, s ⩽ t − 1. The conditional least squares estimator β̂ of parameter β, based on the sample (Y_{1}, …, Y_{N}), is obtained by minimizing the following function:

(5.2)  ∑_{t=2}^{N} (Y_{t} − E(Y_{t} | F_{t−1} ))^{2} = ∑_{t=2}^{N} (Y_{t} − βY_{t−1} )^{2},

with respect to β. If we equate the first derivative of (5.2) to 0, we get the estimator β̂:

(5.3)  β̂ = \frac{∑ _{t=2}^{N} Y_{t−1} Y_{t}}{∑ _{t=2}^{N} Y_{t−1}^{2}}

Theorem 5.1. The estimator β̂ is strongly consistent estimator of β. Furthermore, \sqrt{N − 1} (β̂ − β) has an asymptotic N (0, (1 − β^{2})) distribution.

Proof. We will consider the difference:

β̂ − β = \frac{(N − 1)^{−1} ∑ _{t=2}^{N} u_{t}Y_{t−1}}{(N − 1)^{−1} ∑ _{t=2}^{N} Y_{t−1}^{2}}

Stationarity and ergodicity of {Y_{t}^{2}} and {Y_{t−1} u_{t}}, followed by the stationarity and ergodicity of {X_{t}}. Expectation V = EY_{t}^{2} is finite.

Since E(R_{t} | F_{t−1} ) = 0 and {Y_{t−1} } is F_{t} measurable, it is valid:

(5.4)  E(u_{t}Y_{t−1} ) = 0

Ergodicity of {Y_{t}^{2}} and {Y_{t−1} u_{t}} implies:

(5.5)  \frac{1}{N − 1} ∑_{t=2}^{N} Y_{t−1}^{2} −→^{a.s.} V, \frac{1}{N − 1} ∑_{t=2}^{N} Y_{t−1} u_{t} −→^{a.s.} 0.

From (5.4) and (5.5) it can be concluded that (β̂ − β) −→^{a.s.} 0. Let us denote by u_{t} = Y_{t} − E(Y_{t} | F_{t−1} ) = Y_{t} − βY_{t−1} and v_{t} = u_{t}^{2}. It can be proved that

E(v_{t} | F_{t−1} ) = (1 − β^{2})(M − m^{2}),

where EX_{t}^{2} = M. Let parameter α be an arbitrary constant from ℝ. Then, we have:

E(αY_{t−1} u_{t})^{2} = E {α^{2}Y_{t−1}^{2} E(v_{t} | F_{t−1} )} = E {α^{2}Y_{t−1}^{2} (1 − β^{2})(M − m^{2})}

Since E(αY_{t−1} u_{t} | F_{t−1} ) = 0, from the Lindeberg–Levy central limit theorem for martingales [3], it follows that (N − 1)^{−1/2} ∑ _{t=2}^{n} αY_{t−1} u_{t} converges to the normal distributed r.v. with mean 0 and variance α^{2}(1 − β^{2})(M − m^{2})EY_{t−1}^{2} . Bearing in mind that α is an arbitrary constant, we can conclude that (N −1)^{−1/2} ∑ _{t=2}^{N} Y_{t−1} u_{t} converges in distribution to the r.v. from normal distribution with mean 0 and variance (1 − β^{2})(M − m^{2})EY_{t−1}^{2} Now, we will consider the following expression:

(5.6)  (N − 1)^{1/2} (β̂ − β) = \frac{(N − 1)^{−1/2} ∑ _{t=2}^{N} Y_{t−1} u_{t}}{(N − 1)^{−1} ∑ _{t=2}^{N} Y_{t−1}^{2}} .

From (5.6) we can conclude that (N − 1)^{1/2} (β̂− β) converges in distribution to r.v. from normal distribution with mean 0 and variance (1 − β^{2}). So the assertion of the theorem is proved. □

Since X_{t} = βX_{t−1} + ξ_{t} > βX_{t−1} , parameter β can be estimated using the following estimator:

(5.7)  β̃ = min_{2⩽t⩽N} \{ \frac{X_{t}}{X_{t−1}} \}

Estimator (5.7) is a consistent estimator and it was analyzed in [4].

5.2.1. Numerical Example. Finally, let us present a numerical simulation of the parameter estimation based on previously given formulae (5.3) and (5.7). It will be assumed that parameters p = 1/3, q = 4 and β = 0.125 are known. Bearing in mind all earlier considerations, 100 samples of sizes 100, 1 000, 5 000, 10 000 and 20 000 were drawn using (3.1), and parameter β was estimated using (5.3) and (5.7). Since we had 100 samples for each size, the mean value of all 100 estimates per each sample size is reported in table [1]. The mean value of estimates which are the result of applying estimators (5.3) and (5.7) are denoted by β_{CLS} and β_{CON} respectively, and their standards deviations are STDEV(β_{CLS} ) and STDEV(β_{CON} ). More details about application of numerical simulations to different stochastic models, can be found in [5].

Table 1. Values of estimated parameter β

Sample size β_{CLS} STDEV(β_{CLS} ) β_{CON} STDEV(β_{CON} )
100  0.1138  0.0923  0.1229  0.0027
1 000  0.1223  0.0309  0.125  0
5 000  0.1245  0.0137  0.125  0
10 000  0.1241  0.0099  0.125  0
20 000  0.1259  0.0066  0.125  0

The simulation shows that the estimator β_{CON} is much better than the estimator β_{CLS} . The estimator β_{CLS} converges very slowly to the true value. It is needed to generate huge samples for better accuracy of β_{CON} .

Acknowledgements

I wish to thank the referees for valuable suggestions and comments which greatly improved the paper. I also thank Professors Tibor K. Pogány and Miroslav M. Ristić for their guidance.

References

1. A. J. Lawrence, The mixed exponential solution to the first order autoregressive model, J. Appl. Probab. 17 (1980), 546–552.

2. A. J. Lawrence, P. A. W. Lewis, A new autoregressive time series model in exponential variables (NEAR(1)), Adv. Appl. Probab. 13 (1980), 826–845.

3. P. Billingsley, The Lindeberg–Levy theorem for martingales, Proc. Am. Math. Soc. 12 (1981), 788–792.

4. B. C. Bell, E. P. Smith, Inference for non-negative autoregressive schemes, Commun. Stat., Theory Methods 15 (1986), 2267–2293.

5. B. V. Popović, Time series models with marginal distribution from pearson system of distributions, MSci Thesis, Faculty of Mathematics, University of Belgrade, 2009. (in Serbian)

6. B. Stanković, On the function of E. M. Wright, Publ. Inst. Math., Nouv. Sér. 10(24) (1970), 113–124.

7. C. H. Sim, Simulation of Weibull and Gamma autoregressive stationary process, Commun. Stat., Simulation Comput. B15 (1986), 1141–1146.

8. D. Ðorić, J. Mališić, V. Jevremović, E. Nikolić-Ðorić, Atlas of Distributions, Faculty of Civil Engineering, University of Belgrade, 2007. (in Serbian)

9. D. P. Gaver, P. A. W. Lewis, First order autoregressive Gamma sequences and point processes, Adv. Appl. Probab. 12 (1980), 727–745.

10. E. McKenzie, An autoregressive process for beta random variables, Manage. Sci. 31 (1985), 988–997.

11. H. Pollard, The representation of exp(−x^{λ}) as a Laplace integral, Bull. Am. Math. Soc.52 (1946), 908–910.

12. J. D. Hamilton, Time Series Analysis, Princeton University Press, 1994.

13. J. Mališić, On exponential autoregressive time series models, in: P. Bauer et al. (eds.), Mathematical Statistics and Probability Theory, Vol. B, Reidel, Dordrecht, 1986, 147–153.

14. J. Mikusiński, On the function whose Laplace transform is exp(−s^{α} ), 0 < α < 1, Stud. Math. 18 (1959), 191–198.

15. M. R. Chernick, A limit theorem for the maximum of autoregressive processes with uniform marginal distribution, Ann. Probab. 9 (1981), 145–149.

16. M. C. Jones, Kumaraswamy’s distribution: A beta-type distribution with some tractability adventages, Stat. Methodol. 6 (2009), 70–81.

17. M. Pourahmadi, On stationarity of the solution of a doubly stochastic model, J. Time Ser. Anal. 7 (1986), 123–131.

18. M. Pourahmadi, Stationarity of the solution of X_{t} = A_{t}X_{t−1} + ξ_{t} and analysis of nongaussian dependent random variables, J. Time Ser. Anal. 9 (1988), 225–239.

19. M. Novković, Autoregressive time series models with gamma and laplace distribution, MSci Thesis, University of Belgrade, Faculty of Mathematics, 1997. (in Serbian)

20. M. S. Ridout, Generating random numbers from a distribution specified by its Laplace transform, Stat. Comput. 19 (2009), 439–450

21. M. M. Ristić, B. Č. Popović, The uniform autoregressive process of the second order (UAR(2)), Stat. Probab. Lett. 57 (2002), 113–119.

22. B. P. Demidovich, I. A. Maron, Computational Mathematics, Mir, Moscow, 1976. (in Russian)

23. P. Humbert, Nouvelles correspondances symboliques, Bull. Sci. Math. 69 (1945), 121–129.

24. P. Kumaraswamy, A generalized probability density function for double-bounded random processes, J. Hydrology 46 (1980), 79Ű-88.

25. R Development Core Team, R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2009.

26. S. G. Fletcher, K. Ponnambalam, Estimation of reservoir yield and storage distribution using moments analysis, J. Hydrology 182 (1996), 259–275.

27. S. Nadarajah, Probability models for unit hydrograph derivation, J. Hydrology 344 (2007), 185–189.

28. S. Nadarajah, On the distribution of Kumaraswamy, J. Hydrology 348 (2008), 568–569.

29. V. Jevremović, Two examples of nonlinear process with mixed exponential marginal distribution, Stat. Probab. Lett. 10 (1990), 221–224.

Statistical Office of the Republic of Serbia (Received 29 12 2009) Belgrade (Revised 10 02 2010 and 19 03 2010) Serbia

 bozidarpopovic@gmail.com