A test for fuzzy exponentiality based on Kullback-Leibler information

Abstract

The exponential distribution has been widely used in engineering, social and biological sciences. In this paper, we propose a new goodness-of-fit test for fuzzy exponentiality using α-pessimistic value. The test statistics is established based on Kullback-Leibler information. By using Monte Carlo method, we obtain the empirical critical points of the test statistic at four different significant levels. To evaluate the performance of the proposed test, we compare it with four commonly used tests through some simulations. Experimental studies show that the proposed test has higher power than other tests in most cases. In particular, for the uniform and linear failure rate alternatives, our method has the best performance. A real data example is investigated to show the application of our test.

Keywords

Triangular fuzzy number power Kullback-Leibler information

1 Introduction

The assumption of exponentiality is usually used in many fields due to its two main properties: the constant failure rate property and the memory less property. Many researchers developed different goodness-of-fit tests for exponential distribution against other life distributions (see, for example, Lilliefors [18], Finkelstein and Schafer [7], Ebrahimi et al. [6], Henze and Meintanis [10], Ghosh and Mitra [8], Majumder and Mitra [19]). However, only a few works focus on the goodness-of-fit tests for fuzzy distribution, even for fuzzy normal distribution. Hesamian and Taheri [13] established fuzzy Kolmogorov-Smirnov one sample test based on fuzzy cumulative distribution function and fuzzy empirical distribution function. Later, Grzegorzewski and Szymanowski [9] extended Hesamian and Taheri’s result to two-sided case. Moreover, they also constructed Cramér-von Mises test and Anderson-Darling test in fuzzy environment. Recently, Zendehdel et al. [22] studied several goodness-of-fit tests for fuzzy exponentiality. By using the α-pessimistic value introduced by Peng and Liu [20], they compared several tests and deduced that Cramér-von Mises test and Anderson-Darling test perform well in terms of power in most cases.

To test the goodness-of-fit for normality or exponentiality, many researchers used Kullback-Leibler information to construct the test statistics (Vasicek [21], Ebrahimi et al. [6] and Balakrishnan et al. [2]). As stated in [6], the goodness-of-fit test based on Kullback-Leibler information has larger power than those on Cramér-von Mises test and Anderson-Darling test. The aim of this paper is to construct the goodness-of-fit test for fuzzy exponential distribution using Kullback-Leibler information. In addition, we compare our test in terms of power with four popular tests: Cramér-von Mises test, Anderson-Darling test, Kolmogorov-Smirnov test and Watson test. The new test statistic is very easy to compute. In addition, experimental studies show that the proposed test has higher power than other tests in most cases. In particular, for the uniform and linear failure rate alternatives, the proposed method has the best performance.

The rest of this paper is organized as follows. In Section 2, we recall some basic theories of fuzzy number and fuzzy random variable. In Section 3, we propose the goodness-of-fit test for fuzzy exponentiality based on Kullback-Leibler information. The critical values of the test statistic are obtained by Monte Carlo simulations in this section. Section 4 is devoted to power comparisons between our test and four frequently used tests. A real data example is illustrated in Section 5. In Section 6, we draw some brief conclusions and the code is given in Appendix.

2 Fuzzy number and fuzzy random variable

In this section, we recall some basic theories of fuzzy number and fuzzy random variable needed in this paper. For more details, see Buckley [4], Klir and Yuan [14], and Lee [17].

Let $𝕌$ be a subset of $ℝ$ . A fuzzy set $\tilde{A}$ on $𝕌$ is defined through its membership function $μ_{\tilde{A}}$ , which is a mapping from $𝕌$ to [0, 1]. For each α ∈ (0, 1], the α-cut (or α-level) of $\tilde{A}$ is defined as $\tilde{A} [α] = {x \in 𝕌 : μ_{\tilde{A}} (x) \geq α}$ , and when α = 0, the α-cut $\tilde{A} [0]$ is defined as the closure of the set ${x \in 𝕌 : μ_{\tilde{A}} (x) > 0}$ ([4 , 17]).

Definition 2.1. ([9, 22]) A fuzzy set $\tilde{A}$ on $ℝ$ is called a fuzzy number if it satisfies that:

$\tilde{A}$ is normal, that is, there exists $x_{\tilde{A}}^{*} \in ℝ$ such that $μ_{\tilde{A}} (x_{\tilde{A}}^{*}) = 1$ ,

$\tilde{A}$ is fuzzy convex, that is, for any $x_{1}, x_{2} \in ℝ$ and any λ ∈ [0, 1], the following relationship holds $μ_{\tilde{A}} (λ x_{1} + (1 - λ) x_{2}) \geq min (μ_{\tilde{A}} (x_{1}), μ_{\tilde{A}} (x_{2})),$

$\tilde{A}$ is upper semi-continuous,

$supp (\tilde{A}) = cl ({x : μ_{\tilde{A} (x)} > 0})$ and cl is the closure operator.

Throughout this paper, we use $F (ℝ)$ to denote the set of all fuzzy numbers on $ℝ$ . In practice, the most commonly used fuzzy number is the triangular fuzzy number whose membership function is given as follows $μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - (a - l)}{l} & a - l \leq x \leq a, \\ \frac{a + r - x}{r} & a \leq x \leq a + r, \\ 0 & a \in ℝ - [a - l, a + r], \end{matrix}$ where $a, l, r \in ℝ$ and l, r > 0. The real number a is called the center value of $\tilde{A}$ . In addition, l and r are called the left and right spreads, respectively. Note that the membership function of the triangular fuzzy number is uniquely determined by these three real numbers. For simplicity, we use $\tilde{A} = (l, a, r)_{T}$ to denote it.

In the following, we introduce the α-pessimistic value of a fuzzy number, which is crucial to state the definitions of fuzzy random variable and fuzzy exponential variable.

Definition 2.2. ([20]) Given a fuzzy number $\tilde{A}$ , for any α ∈ [0, 1], the α-pessimistic value of $\tilde{A}$ is defined as ${\tilde{A}}_{α} = inf {x \in \tilde{A} [0] : C {\tilde{A} \leq x} \geq α},$ (2.1) where $C {\tilde{A} \leq x} = [sup_{y \leq x} μ_{\tilde{A}} (y) + 1 - sup_{y < x} μ_{\tilde{A}} (y)] / 2$ denotes the credibility degree that $\tilde{A}$ is less than or equal to x.

From Peng and Liu [20], the α-pessimistic value ${\tilde{A}}_{α}$ is a non-decreasing function with respect to α. Moreover, the α-cut of $\tilde{A}$ may be rewritten as $\tilde{A} [α] = [{\tilde{A}}_{α / 2}, {\tilde{A}}_{1 - α / 2}]$ . From the above definitions, the α-cut and α-pessimistic value of the triangular fuzzy number $\tilde{A} = (l, a, r)_{T}$ may be rewritten as $\tilde{A} [α] = [a - (1 - α) l, a + (1 - α) r]$ and ${\tilde{A}}_{α} = {\begin{matrix} a - (1 - 2 α) l & 0 \leq α \leq 0.5, \\ a - (1 - 2 α) r & 0.5 < α \leq 1, \end{matrix}$ respectively. For more discussions on α-cut and α-pessimistic value of fuzzy numbers, we refer the reader to [12 , 22].

Example 2.1. Consider a triangular fuzzy number $\tilde{B} = (2, 5, 1)_{T}$ . For any α ∈ [0, 1], the α-cut and α-pessimistic value of $\tilde{B}$ are $\tilde{B} [α] = [3 + 2 α, 6 - α]$ and ${\tilde{B}}_{α} = {\begin{matrix} 3 + 4 α & 0 \leq α \leq 0.5, \\ 4 + 2 α & 0.5 < α \leq 1, \end{matrix}$ respectively. The picture of α-pessimistic value of $\tilde{B}$ is shown in Fig. 1.

Fig. 1

The α-pessimistic value of $\tilde{B} = (2, 5, 1)_{T}$ .

In the following, we introduce the definitions of fuzzy random variable and fuzzy exponential variable based on the α-pessimistic value. For more details, we refer the reader to [11 –13].

Definition 2.3. ([12]) Let $(Ω, A, ¶)$ be a probability space. The mapping $\tilde{X} : Ω \to F (ℝ)$ is called a fuzzy random variable if for any α ∈ [0, 1], the α-pessimistic value ${\tilde{X}}_{α}$ is a crisp random variable on $(Ω, A, ¶)$ .

Remark 2.1. Kwakernaak [15] introduced the definition of fuzzy random variable using the α-cut $\tilde{X} [α] = [{\tilde{X}}_{α}^{L}, {\tilde{X}}_{α}^{U}]$ , where ${\tilde{X}}_{α}^{L}$ and ${\tilde{X}}_{α}^{U}$ denote the lower and upper bounds of $\tilde{X} [α]$ . As stated in [12, 22], Definition 2.3 is equivalent with Kwakernaak’s definition. It’s obvious that the computational procedure with one-dimensional variable ${\tilde{X}}_{α}$ is much easier than that with two-dimensional variable $({\tilde{X}}_{α}^{L}, {\tilde{X}}_{α}^{U})$ .

The fuzzy random variables ${\tilde{X}}_{1}, \dots, {\tilde{X}}_{n}$ on $(Ω, A, ¶)$ are said to be independent and identically distributed (i.i.d.) if for any α ∈ [0, 1], the crisp variables ${\tilde{X}}_{α 1}, \dots, {\tilde{X}}_{α n}$ are i.i.d., where ${\tilde{X}}_{α i}$ denotes the α-pessimistic value of ${\tilde{X}}_{i}$ . If the fuzzy random variables ${\tilde{X}}_{1}, \dots, {\tilde{X}}_{n}$ are i.i.d., we call them a simple fuzzy random sample.

Definition 2.4. ([22]) A fuzzy random variable $\tilde{X}$ on $(Ω, A, ¶)$ is called a fuzzy exponential variable if for any α ∈ [0, 1], ${\tilde{X}}_{α}$ is a crisp exponential variable.

Now, having these basic theories about the fuzzy exponential variable in hand, we are able to propose the goodness-of-fit test for fuzzy exponentiality.

3 The goodness-of-fit test for fuzzy exponentiality

In this section, we propose the goodness-of-fit test for fuzzy exponentiality based on the Kullback-Leibler information. Moreover, we obtain the critical points of the test statistic by using Monte Carlo simulations.

3.1 The test statistic

Let ${\tilde{X}}_{1}, {\tilde{X}}_{2}, \dots, {\tilde{X}}_{n}$ be a simple fuzzy random sample. We will investigate the following issue,

$\begin{matrix} H_{0} : {\tilde{X}}_{i} is fuzzy exponential \leftrightarrow H_{1} : {\tilde{X}}_{i} is not fuzzy \\ exponential . \end{matrix}$ From Definition 2.4, for any α ∈ [0, 1], the sample ${\tilde{X}}_{α 1}, \dots, {\tilde{X}}_{α n}$ is from an exponential population under H₀, where ${\tilde{X}}_{α i}$ denotes the α-pessimistic value of ${\tilde{X}}_{i}$ . Hence, we only need to test the exponentiality for the sample ${\tilde{X}}_{α 1}, \dots, {\tilde{X}}_{α n}$ .

Ebrahimi et al. [6] used Kullback-Leibler information to test the exponentiality. In the following, we extend this method to fuzzy environment and establish the corresponding test statistic. For any α ∈ [0, 1], we use $f_{{\tilde{X}}_{α}} (x)$ to denote the density function of ${\tilde{X}}_{α i}$ . Suppose that f_α (x) denotes the density function of a crisp exponential variable with mean 1/λ_α (λ_α > 0 is a crisp constant), that is, f_α (x) = λ_αe^-λ_αx, x > 0. The Kullback-Leibler discrimination information function between $f_{{\tilde{X}}_{α}} (x)$ and f_α (x) is given by $I (α) = \int_{0}^{\infty} f_{{\tilde{X}}_{α}} (x) log \frac{f_{{\tilde{X}}_{α}} (x)}{f_{α} (x)} dx .$ From [6], I (α) =0 under H₀, while it has large values under H₁. Similar to Ebrahimi et al. [6] and Vasicek [21], I (α) may be estimated by $\begin{matrix} I_{mn} (α) & = - \frac{1}{n} \sum_{i = 1}^{n} log [\frac{n}{2 m} ({\tilde{X}}_{α (i + m)} - {\tilde{X}}_{α (i - m)})] \\ + log {\bar{\tilde{X}}}_{α} + 1, \end{matrix}$ where m is a positive integer satisfying that m < n/2, ${\tilde{X}}_{α (1)} \leq {\tilde{X}}_{α (2)} \leq \dots \leq {\tilde{X}}_{α (n)}$ are the ordered statistics of the sample ${\tilde{X}}_{α 1}, \dots, {\tilde{X}}_{α n}$ , and ${\tilde{X}}_{α (i)} = {\tilde{X}}_{α (1)}$ if i < 1, ${\tilde{X}}_{α (i)} = {\tilde{X}}_{α (n)}$ if i > n. Moreover, ${\bar{\tilde{X}}}_{α}$ denotes the sample mean of ${\tilde{X}}_{α 1}, \dots, {\tilde{X}}_{α n}$ . The integer m is called the window size. Define ${KL}_{mn} (α) = exp {- I_{mn} (α)} .$ (3.1) It’s obvious that 0 < KL_mn (α) <1 for any α ∈ [0, 1] and small values of KL_mn (α) imply that H₁ holds. The test statistic used to discriminate between H₀ and H₁ is defined as ${KL}_{mn} = \int_{0}^{1} {KL}_{mn} (α) d α .$ (3.2) Since 0 < KL_mn (α) <1, the test statistic KL_mn also satisfies that 0 < KL_mn < 1. In addition, small values of KL_mn favour H₁. Hence, we reject H₀ in favour of H₁ at the significance level γ if KL_mn ≤ C_mn (γ), where the critical value C_mn (γ) denotes the lower γ-quantile of the distribution of KL_mn under H₀.

3.2 Critical values

Although it is very easy to compute the test statistic KL_mn based on Equation (3.2), we are unable to obtain its exact distribution under H₀. In the following, we use Monte Carlo simulation to obtain the empirical critical points C_mn (γ) for γ = 0.01, 0.025, 0.05 and 0.1. For each n, we generate 10,000 replications. To determine C_mn (γ), we use the least conservative window size m which yields the maximum critical value and the maximum power. The least conservative window sizes according to n ≤ 100 are listed in Table 1. For more properties of the least conservative window size, see Ebrahimi et al. [6] and Vasicek [21]. The empirical critical points C_mn (γ) are computed by the following five steps:

Table 1
The least conservative window size m corresponding to the sample size n

n 3-4 5-7 8-14 15-24 25-35 36-50 51-70 71-80 81-100

m 1 2 3 4 5 6 7 8 9

In this step, we generate a sample of triangular fuzzy numbers ${\tilde{x}}_{i} = (l_{i}, x_{i}, r_{i})_{T}$ , where l_i = x_iu_i1, r_i = x_iu_i2, and x_i ∼ Exp (1) , u_i1 ∼ U (0.15, 0.2) , u_i2 ∼ U (0.15, 0.2) , i = 1, 2, ⋯ , n. Moreover, x_i, u_i1 and u_i2 are mutually independent.

For i = 1, ⋯ , n and α = 0.01, ⋯ , 1, calculate the α-pessimistic value ${\tilde{x}}_{α i}$ with ${\tilde{x}}_{α i} = {\begin{matrix} x_{i} - (1 - 2 α) l_{i} & 0 \leq α \leq 0.5, \\ x_{i} - (1 - 2 α) r_{i} & 0.5 < α \leq 1 . \end{matrix}$

For each n, choose the least conservative window size m from Table 1. Then, for each α, compute KL_mn (α) based on Equation (3.1).

Compute KL_mn approximately by ${KL}_{mn} \approx \frac{1}{100} \sum_{α = 0.01}^{1} {KL}_{mn} (α) .$ (3.3)

For each m and n, repeat Steps 1-4 10,000 times and determine the lower γ-quantile C_mn (γ) with γ = 0.01, 0.025, 0.05 and 0.01.

Table 2 gives the critical values C_mn (γ) for several sample sizes. It’s easy to see that C_mn (γ) increases as the sample size n increases or the significance level γ increases.

Table 2

Critical values of the KL_mn statistic at significance level γ

n	γ=0.01	γ=0.025	γ=0.05	γ=0.10
3	0.0574	0.0856	0.1202	0.1696
4	0.1064	0.1420	0.1819	0.2352
5	0.1982	0.2504	0.2991	0.3574
6	0.2594	0.3116	0.3586	0.4145
7	0.3059	0.3572	0.4039	0.4563
8	0.3500	0.4007	0.4461	0.4970
9	0.3804	0.4362	0.4802	0.5288
10	0.4198	0.4682	0.5092	0.5557
11	0.4453	0.4946	0.5356	0.5808
12	0.4708	0.5182	0.5597	0.6011
13	0.5011	0.5424	0.5802	0.6206
14	0.5148	0.5614	0.5976	0.6379
15	0.5388	0.5810	0.6130	0.6485
16	0.5499	0.5979	0.6297	0.6648
17	0.5699	0.6083	0.6411	0.6769
18	0.5915	0.6257	0.6533	0.6864
19	0.6005	0.6377	0.6662	0.6985
20	0.6162	0.6503	0.6770	0.7099
25	0.6619	0.6922	0.7175	0.7441
30	0.7037	0.7286	0.7506	0.7736
35	0.7330	0.7556	0.7762	0.7975
40	0.7542	0.7760	0.7932	0.8134
45	0.7709	0.7918	0.8097	0.8271
50	0.7902	0.8068	0.8222	0.8390
60	0.8138	0.8302	0.8438	0.8583
70	0.8307	0.8463	0.8579	0.8723
80	0.8465	0.8608	0.8720	0.8842
90	0.8593	0.8709	0.8809	0.8922
100	0.8698	0.8804	0.8894	0.8998

4 Power comparisons

In this section, to evaluate the performance of our test, we compare it with the following four popular goodness-of-fit tests:

Cramér-von Mises type test statistic

CVM = \frac{1}{100} \sum_{α = 0.01}^{1} [\sum_{i = 1}^{n} (y_{i} - \frac{2 i - 1}{2 n})^{2} + \frac{1}{12 n}],

Anderson-Darling type test statistic

\begin{matrix} - 6 pt & AD = \frac{1}{100} \sum_{α = 0.01}^{1} {- \frac{1}{n} \sum_{i = 1}^{n} [(2 i - 1) log y_{i} \\ - 6 pt & + (2 n + 1 - 2 i) log (1 - y_{i})] - n}, \end{matrix}

Kolmogorov-Smirnov type test statistic

\begin{matrix} - 6 pt & KS = \frac{1}{100} \sum_{α = 0.01}^{1} max {max_{1 \leq i \leq n} {\frac{i}{n} - y_{i}}, \\ - 6 pt & max_{1 \leq i \leq n} {y_{i} - \frac{i - 1}{n}}}, \end{matrix}

Watson type test statistic

\begin{matrix} - 6 pt & U = \frac{1}{100} \sum_{α = 0.01}^{1} [\frac{1}{12 n} \\ - 6 pt & + \sum_{i = 1}^{n} (y_{i} - \bar{y} + 0.5 - \frac{2 i - 1}{2 n})], \end{matrix}

where

y_{i} = F_{0} ({\tilde{x}}_{α (i)}; 1 / {\bar{\tilde{x}}}_{α})

{\bar{\tilde{x}}}_{α} = \frac{1}{n} \sum_{i = 1}^{n} {\tilde{x}}_{α i}

and F₀ (x ; λ) =1 - e^-λx, x > 0, denotes the distribution function of the exponential variable with mean 1/λ. In addition,

\bar{y}

denotes the mean of y_i. The critical values for CVM, AD, KS and U are determined by Zendehdel et al. [22] using Monte Carlo simulation technique. Moreover, large values of CVM, AD, KS and U favour H₁ [22], while small values of KL_mn favour H₁ [6].

We study the the following five alternatives: Weibull distribution W (β, λ), Gamma distribution Ga (β, λ), log-normal distribution LN (υ, σ²), uniform distribution U (0, 1), and linear failure rate distribution LF (β). The above five alternatives have been widely used as alternatives to the exponential distribution by many researchers (e.g. Ebrahimi et al. [6], Henze and Meintanis [10] and Zendehdel et al. [22]). For 1 ≤ i ≤ n, the central value x_i of ${\tilde{x}}_{i} = (l_{i}, x_{i}, r_{i})_{T}$ is generated from the above five alternatives, l_i and r_i are generated in the same way as those of exponential fuzzy numbers, that is, l_i = x_iu_i1 and r_i = x_iu_i2 with u_i1, u_i2 from uniform distribution U (0.15, 0.2). Under each alternative, we generated 10,000 samples of size n equal to 5, 10, 20 and 30. Tables 3–6 show the estimated powers at significance levels γ = 0.01 and 0.05. The powers reported for the KL_mn statistic are based on the critical values given in Table 2. The powers reported for CVM, AD, KS and U statistics are computed with the critical values provided by Zendehdel et al. [22]. From Tables 3–6, we may obtain that none of the tests we studied performs better than other tests. However, in most cases, the statistic KL_mn has better performance than other test statistics. Moreover, for the uniform and linear failure rate alternatives, the power of KL_mn test is uniformly the highest. These results imply that KL_mn test may be adopted in many situations.

Table 3
Monte Carlo power estimates of the KL_mn, CVM, AD, KS and U tests at significance level γ = 0.01, 0.05 with sample size n = 5

Alternatives γ CVM AD KS U KL_mn

W (1.5, Γ (5/3)) 0.01 0.0318 0.0027 0.0275 0.0336 0.0431

0.05 0.1422 0.0744 0.1234 0.1234 0.1760

W (2, Γ (1.5)) 0.01 0.0844 0.0080 0.0697 0.0956 0.1077

0.05 0.3071 0.1756 0.2537 0.2801 0.3581

Ga (2, 2) 0.01 0.0307 0.0026 0.0264 0.0338 0.0395

0.05 0.1443 0.0719 0.1237 0.1248 0.1702

Ga (4, 4) 0.01 0.1107 0.0094 0.0888 0.1194 0.1405

0.05 0.3901 0.2258 0.3308 0.3366 0.4496

LN (-0.2, 0.08) 0.01 0.0548 0.0031 0.0496 0.0655 0.0736

0.05 0.2330 0.1273 0.2054 0.214 0.2783

LN (-0.4, 0.32) 0.01 0.0192 0.0047 0.0174 0.0191 0.0211

0.05 0.0895 0.0472 0.0835 0.083 0.0991

U (0, 1) 0.01 0.0559 0.0070 0.0419 0.0635 0.0778

0.05 0.1939 0.1122 0.1566 0.1796 0.2400

LF (2) 0.01 0.0218 0.0026 0.0168 0.0248 0.0293

0.05 0.0984 0.0577 0.0898 0.0916 0.1247

LF (4) 0.01 0.0263 0.0022 0.0247 0.0335 0.0418

0.05 0.1194 0.0651 0.1147 0.1167 0.1648

Alternatives	γ	CVM	AD	KS	U	KL_mn
W (1.5, Γ (5/3))	0.01	0.0318	0.0027	0.0275	0.0336	0.0431
	0.05	0.1422	0.0744	0.1234	0.1234	0.1760
W (2, Γ (1.5))	0.01	0.0844	0.0080	0.0697	0.0956	0.1077
	0.05	0.3071	0.1756	0.2537	0.2801	0.3581
Ga (2, 2)	0.01	0.0307	0.0026	0.0264	0.0338	0.0395
	0.05	0.1443	0.0719	0.1237	0.1248	0.1702
Ga (4, 4)	0.01	0.1107	0.0094	0.0888	0.1194	0.1405
	0.05	0.3901	0.2258	0.3308	0.3366	0.4496
LN (-0.2, 0.08)	0.01	0.0548	0.0031	0.0496	0.0655	0.0736
	0.05	0.2330	0.1273	0.2054	0.214	0.2783
LN (-0.4, 0.32)	0.01	0.0192	0.0047	0.0174	0.0191	0.0211
	0.05	0.0895	0.0472	0.0835	0.083	0.0991
U (0, 1)	0.01	0.0559	0.0070	0.0419	0.0635	0.0778
	0.05	0.1939	0.1122	0.1566	0.1796	0.2400
LF (2)	0.01	0.0218	0.0026	0.0168	0.0248	0.0293
	0.05	0.0984	0.0577	0.0898	0.0916	0.1247
LF (4)	0.01	0.0263	0.0022	0.0247	0.0335	0.0418
	0.05	0.1194	0.0651	0.1147	0.1167	0.1648

Table 4

Monte Carlo power estimates of the KL_mn, CVM, AD, KS and U tests at significance level γ = 0.01, 0.05 with sample size n = 10

Alternatives	γ	CVM	AD	KS	U	KL_mn
W (1.5, Γ (5/3))	0.01	0.0631	0.0352	0.0610	0.0763	0.0537
	0.05	0.2463	0.1840	0.2150	0.2223	0.3100
W (2, Γ (1.5))	0.01	0.2591	0.1718	0.2185	0.2741	0.3634
	0.05	0.6187	0.5324	0.5116	0.5499	0.6853
Ga (2, 2)	0.01	0.0612	0.0362	0.0601	0.0742	0.1052
	0.05	0.2535	0.1889	0.2156	0.2252	0.3169
Ga (4, 4)	0.01	0.3803	0.2743	0.3342	0.4027	0.4867
	0.05	0.7688	0.6987	0.6788	0.7079	0.8121
LN (-0.2, 0.08)	0.01	0.1448	0.0942	0.1597	0.1763	0.2294
	0.05	0.4651	0.3853	0.4232	0.4352	0.5404
LN (-0.4, 0.32)	0.01	0.0260	0.0178	0.0289	0.0311	0.0378
	0.05	0.1204	0.0933	0.1181	0.1218	0.1558
U (0, 1)	0.01	0.1403	0.0962	0.1248	0.1353	0.2365
	0.05	0.3894	0.3022	0.2935	0.3342	0.5087
LF (2)	0.01	0.0325	0.0177	0.0332	0.0424	0.0604
	0.05	0.1504	0.1091	0.1399	0.1437	0.1963
LF (4)	0.01	0.0491	0.0281	0.0558	0.0611	0.0876
	0.05	0.2043	0.1521	0.1929	0.1927	0.2674

Table 5

Monte Carlo power estimates of the KL_mn, CVM, AD, KS and U tests at significance level γ = 0.01, 0.05 with sample size n = 20

Alternatives	γ	CVM	AD	KS	U	KL_mn
W (1.5, Γ (5/3))	0.01	0.2228	0.1572	0.1591	0.1855	0.2415
	0.05	0.4808	0.4396	0.3800	0.4033	0.5033
W (2, Γ (1.5))	0.01	0.7471	0.6713	0.5823	0.6766	0.7596
	0.05	0.9274	0.9133	0.8418	0.878	0.9296
Ga (2, 2)	0.01	0.2170	0.1552	0.1657	0.1926	0.2476
	0.05	0.4779	0.4492	0.3936	0.4176	0.5099
Ga (4, 4)	0.01	0.9004	0.8736	0.8057	0.8747	0.9029
	0.05	0.9877	0.9879	0.9568	0.9756	0.9836
LN (-0.2, 0.08)	0.01	0.5340	0.4536	0.4591	0.5259	0.5761
	0.05	0.8224	0.8165	0.7639	0.7949	0.8171
LN (-0.4, 0.32)	0.01	0.0565	0.0379	0.0528	0.0706	0.0806
	0.05	0.1885	0.1778	0.1656	0.2192	0.2474
U (0, 1)	0.01	0.3572	0.2720	0.2454	0.3491	0.5524
	0.05	0.6833	0.6404	0.5569	0.5992	0.8068
LF (2)	0.01	0.0865	0.0516	0.0758	0.0826	0.1112
	0.05	0.2823	0.2386	0.2207	0.2322	0.2943
LF (4)	0.01	0.1573	0.1012	0.1294	0.1453	0.1859
	0.05	0.4198	0.3577	0.3220	0.3375	0.4355

Table 6

Monte Carlo power estimates of the KL_mn, CVM, AD, KS and U tests at significance level γ = 0.01, 0.05 with sample size n = 30

Alternatives	γ	CVM	AD	KS	U	KL_mn
W (1.5, Γ (5/3))	0.01	0.3728	0.3423	0.2863	0.3094	0.3820
	0.05	0.6372	0.6419	0.5677	0.5679	0.6456
W (2, Γ (1.5))	0.01	0.9323	0.9337	0.8487	0.8937	0.9348
	0.05	0.9813	0.9804	0.9667	0.9753	0.9880
Ga (2, 2)	0.01	0.3793	0.3530	0.3020	0.3202	0.3865
	0.05	0.6747	0.6698	0.5923	0.5887	0.6494
Ga (4, 4)	0.01	0.9953	0.9938	0.9937	0.9869	0.9955
	0.05	0.9994	0.9996	0.9980	0.999	0.9992
LN (-0.2, 0.08)	0.01	0.8128	0.8150	0.7612	0.7954	0.8227
	0.05	0.9640	0.9721	0.9436	0.9497	0.9697
LN (-0.4, 0.32)	0.01	0.0933	0.0753	0.0832	0.1121	0.1260
	0.05	0.2708	0.2808	0.2598	0.3183	0.3401
U (0, 1)	0.01	0.6178	0.5252	0.4369	0.5425	0.9072
	0.05	0.8597	0.8392	0.7703	0.7853	0.9815
LF (2)	0.01	0.1689	0.1162	0.1339	0.1373	0.1760
	0.05	0.4042	0.3735	0.3357	0.324	0.4210
LF (4)	0.01	0.2906	0.2216	0.2327	0.2405	0.2958
	0.05	0.5913	0.5215	0.4840	0.479	0.5984

5 A practical example

In this section, we apply the proposed test to a practical dataset taken from Lawless [16]. This dataset was also studied by Arefi and Taheri [1] and Zendehdel et al. [22]. It consists of the lifetimes (1000 km) of front disk brake pads of 40 cars. Note that, the lifetimes may not be measured accurately since a disk may work perfectly over a period but be braking for some time. Similar to [1, 22], we use fuzzy numbers ${\tilde{x}}_{i}$ (1 ≤ i ≤ 40) to describe these lifetimes, where ${\tilde{x}}_{i} = (x_{i}, r_{i})_{R}$ and its membership function is $μ_{{\tilde{x}}_{i}} (s) = \frac{x_{i} + r_{i} - s}{r_{i}}, x_{i} \leq s \leq x_{i} + r_{i}, r_{i} > 0,$ with r_i = 0.05x_i. These fuzzy numbers are shown in Table 7. Note that, the α-pessimistic value of ${\tilde{x}}_{i}$ is ${\tilde{x}}_{α i} = x_{i}$ for 0 ≤ α ≤ 0.5, and ${\tilde{x}}_{α i} = x_{i} - (1 - 2 α) r_{i}$ for 0.5 < α ≤ 1.

Table 7
The fuzzy data of the lifetimes (in 1000 km) ${\tilde{x}}_{i} = (x_{i}, r_{i})_{R}$

(86.2, 4.310) _R (45.1, 2.255) _R (52.1, 2.605) _R (54.2, 2.710) _R (59.0, 2.950) _R

(38.4, 1.920) _R (41.0, 2.050) _R (56.4, 2.820) _R (81.3, 4.065) _R (62.4, 3.120) _R

(45.5, 2.275) _R (36.7, 1.835) _R (42.2, 2.110) _R (51.6, 2.580) _R (34.4, 1.720) _R

(22.7, 1.135) _R (22.6, 1.130) _R (40.0, 2.000) _R (38.8, 1.940) _R (50.2, 2.510) _R

(48.8, 2.440) _R (81.7, 4.085) _R (61.5, 3.075) _R (53.6, 2.680) _R (50.7, 2.535) _R

(42.8, 2.140) _R (102.5, 5.125) _R (42.7, 2.135) _R (80.6, 4.030) _R (64.5, 3.225) _R

(73.1, 3.655) _R (28.4, 1.420) _R (46.9, 2.345) _R (45.9, 2.295) _R (33.8, 1.690) _R

(59.8, 2.990) _R (31.7, 1.585) _R (33.9, 1.695) _R (50.6, 2.530) _R (56.7, 2.835) _R

Since n = 40, we choose the least conservative window size m = 6 from Table 1. Then, we obtain that KL_mn = 0.4285 based on Equation (3.3), which is less than 0.7932, the empirical critical value for n = 40 at 5% significance level (see Table 2). Hence, we conclude that the null hypothesis H₀ is rejected, that is, the fuzzy data we considered do not follow a fuzzy exponential distribution, which is consistent with the result in [22].

6 Conclusions

The exponential distribution has been widely used in engineering, social and biological sciences. In this paper, we proposed a new goodness-of-fit test for fuzzy exponentiality using the α-pessimistic value introduced by Peng and Liu [20]. The test statistic is established based on Kullback-Leibler information. Although the test statistic KL_mn is easy to compute, it’s difficult to get its distribution function. We employ Monte Carlo simulations to compute the empirical critical points of KL_mn. For comparison, we select four commonly used goodness-of-fit tests for fuzzy exponentiality: CVM, AD, KS and U test. Simulation results show that KL_mn behaves better than CVM, AD, KS and U in most cases. Moreover, for the uniform and linear failure rate alternatives, KL_mn test has the best performance. Finally, we illustrate the application of the test we proposed by a lifetime data in Lawless [16].

It is noted that the goodness-of-fit test we proposed can be only used for fuzzy numbers. Therefore, constructing the goodness-of-fit tests for exponentiality for intuitionistic fuzzy data and type 2 fuzzy data is an interesting problem and we will investigate it in the future. Recently, Choi et al. [5] constructed a new goodness-of-fit test for exponentiality based on Kullback-Leibler information, which is more powerful than the test of Ebrahimi et al. [6]. Extending the test of Choi et al. [5] to fuzzy environment is another future research direction.

Footnotes

Acknowledgments

The author would like to thank editor-in-chief and two anonymous referees for their valuable comments and suggestions which lead to improve this paper significantly. The author was supported by the National Natural Science Foundation of China (No. 11801317), the Project of Shandong Provincial Higher Educational Science and Technology Program (No. J17KA163).

Appendix

R code for critical values of the KL_mn at significance level γ

k<-10000 #repeated times

n<-20 #sample size

m<-4 #window size

alpha=seq(0.01,1,by=0.01)#α

kl<-rep(0,k)

for(s in 1:k)

{ x<-rexp(n,1) #center points

u1<-runif(n,0.15,0.2)

u2<-runif(n,0.15,0.2)

l<-x*u1 # left spread

r<-x*u2 # right spread

KL<-rep(0,100)# KL_mn (α)

for(j in 1:100)

{ x_alp<-T_alp<-rep(0,n)

if(alpha[j]< =0.5)x_alp<-x-l*(1-2*alpha[j])

if(alpha[j]>0.5)x_alp<-x-r*(1-2*alpha[j])#alpha-pessimistic value

z<-rep(0,n)

x_alp<-sort(x_alp)

for(i in 1:n)

{if(i-m<1)z[i]<-x_alp[i+m]-x_alp[1]

if(i+m>n)z[i]<-x_alp[n]-x_alp[i-m]

if(i+m< = n&i-m> =1)z[i]<-x_alp[i+m]-x_alp[i-m] }

H<-mean(log(n*z/(2*m)))#H_mn

KL[j]<-exp(H)/exp(log(mean(x))+1)#KL_mn

} kl[s]<-mean(KL) }

quantile(kl,γ)

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(86.2, 4.310) _R	(45.1, 2.255) _R	(52.1, 2.605) _R	(54.2, 2.710) _R	(59.0, 2.950) _R
(38.4, 1.920) _R	(41.0, 2.050) _R	(56.4, 2.820) _R	(81.3, 4.065) _R	(62.4, 3.120) _R
(45.5, 2.275) _R	(36.7, 1.835) _R	(42.2, 2.110) _R	(51.6, 2.580) _R	(34.4, 1.720) _R
(22.7, 1.135) _R	(22.6, 1.130) _R	(40.0, 2.000) _R	(38.8, 1.940) _R	(50.2, 2.510) _R
(48.8, 2.440) _R	(81.7, 4.085) _R	(61.5, 3.075) _R	(53.6, 2.680) _R	(50.7, 2.535) _R
(42.8, 2.140) _R	(102.5, 5.125) _R	(42.7, 2.135) _R	(80.6, 4.030) _R	(64.5, 3.225) _R
(73.1, 3.655) _R	(28.4, 1.420) _R	(46.9, 2.345) _R	(45.9, 2.295) _R	(33.8, 1.690) _R
(59.8, 2.990) _R	(31.7, 1.585) _R	(33.9, 1.695) _R	(50.6, 2.530) _R	(56.7, 2.835) _R

A test for fuzzy exponentiality based on Kullback-Leibler information

Abstract

Keywords

1 Introduction

2 Fuzzy number and fuzzy random variable

3.1 The test statistic

Table 1 The least conservative window size m corresponding to the sample size n n 3-4 5-7 8-14 15-24 25-35 36-50 51-70 71-80 81-100 m 1 2 3 4 5 6 7 8 9

Footnotes

Acknowledgments

Appendix

References

Table 1
The least conservative window size m corresponding to the sample size n

n 3-4 5-7 8-14 15-24 25-35 36-50 51-70 71-80 81-100

m 1 2 3 4 5 6 7 8 9