In modeling reliability data, the exponential distribution is commonly used due to its simplicity. For estimating the parameter of the exponential distribution, classical estimators including maximum likelihood estimator represent the most commonly used method and are well known to be efficient. However, the maximum likelihood estimator is highly sensitive in the presence of contamination or outliers. In this study, a robust and efficient estimator of the exponential distribution parameter was proposed based on the probability integral transform statistic. To examine the robustness of this new estimator, asymptotic variance, breakdown point, and gross error sensitivity were derived. This new estimator offers reasonable protection against outliers besides being simple to compute. Furthermore, a simulation study was conducted to compare the performance of this new estimator with the maximum likelihood estimator, weighted likelihood estimator, and M-scale estimator in the presence of outliers. Finally, a statistical analysis of three reliability data sets was conducted to demonstrate the performance of the proposed estimator.

Exponential distribution is the most widely used parametric distribution for modeling reliability and failure time data due to its mathematical simplicity and ability to create a realistic failure time model [

Assume

and

where

and

The reliability data can be sometimes contaminated with outliers. Outliers are observations much deviated from the bulk of the data [

Several robust methods have been proposed in the literature for estimating the parameter of exponential distribution. Thall [

This study aims to develop a new robust estimator for the parameter of exponential distribution based on the probability integral transform statistic, which offers reasonable protection against outliers. Generally, probability integral transform statistic is used to transform the random variable of any continuous distribution to a standard uniform distribution [

The rest of this paper is structured as follows. Section 2 provides a brief explanation of M-estimators and then presents the new robust estimation method for the rate parameter of exponential distribution. Section 3 compares the performance of the proposed estimator with several estimators in the presence of outliers through a simulation study. Section 4 applies the proposed method for estimating the rate parameter of exponential distribution to real data sets. Finally, Section 5 concludes the paper.

In this section, a brief explanation of the M-estimators is given followed by the introduction of the new estimator developed based on the probability integral transform statistical approach. Furthermore, the robustness of this new estimator was compared with the ordinary MLE and discussed in this section.

M-estimators are generalized ML estimators that provide tools for measuring the robustness of the maximum likelihood-type estimates. As described in Huber [

or

is known as an M-estimator.

Assume that

where

Note that

The PITSE for the rate parameter of exponential distribution is a class of M-estimator with

To investigate the properties of PITSE, the function introduced by Huber [

Thus, based on

The MLE is well known to be efficient in the sense that it has minimum asymptotic variance. For this reason, MLE is useful in providing a quantitative benchmark on the measure of efficiency. Note that the MLE for the rate parameter given in

The ARE of PITSE is defined as the ratio of the asymptotic variance of the MLE to the asymptotic variance of the PITSE. In other words, the ARE measures the relative efficiency of the estimator

As the value of

The breakdown point (BP) is a useful measure of the robustness of a statistical approach in which the degree of sensitivity of an estimator to data contamination is measured. According to Huber [

Note that the MLE, namely

The breakdown point (BP) is a useful measure of the robustness of a statistical approach in which the degree of sensitivity of Gross error sensitivity (GES) is also an important measure of the robustness of an estimator. According to Hampel et al. [

where

An estimator with small GES should be more robust than that with larger GES. The MLE, which is

As the value of

As the value of

As the value of

0.165 | 0.288 | 0.463 | 0.632 | 0.809 | 1.000 | 1.211 | 1.449 | 1.721 | |
---|---|---|---|---|---|---|---|---|---|

ARE (%) | 98 | 95 | 90 | 85 | 80 | 75 | 70 | 65 | 60 |

UBP | 0.14 | 0.22 | 0.32 | 0.39 | 0.45 | 0.50 | 0.55 | 0.59 | 0.63 |

LBP | 0.86 | 0.78 | 0.68 | 061 | 0.55 | 0.50 | 0.45 | 0.41 | 0.37 |

GES | 7.07 |
4.47 |
3.16 |
2.58 |
2.24 |
2 |
2.21 |
2.45 |
2.72 |

In this section, a simulation study to compare the performance of the MLE, PITSE, WLE, and M-scale estimator in the presence of outliers is conducted. The design and results of the simulation study are given in the next two following subsections. Then, the guidelines for selecting the suitable ARE of the PITSE for practical application purposes are provided.

In the simulation study, the methods considered for comparison were MLE, PITSE (90% and 70% ARE), WLE (90% and 70% ARE), and M-scale estimator (70% ARE). The data sets were simulated from an exponential distribution, Exp(

The performance of the estimators was assessed in terms of percentage relative root mean square error (

where

Results based on the simulation study are presented in

MLE | PITSE 90% ARE | WLE 90% ARE | PITSE 70% ARE | WLE 70% ARE | M-scale 70% ARE | ||
---|---|---|---|---|---|---|---|

0.5 | 1 | 38.50 | 19.93 | 20.97 | 21.80 | 21.45 | 23.50 |

3 | 62.05 | 27.79 | 26.16 | 24.20 | 24.18 | 24.00 | |

5 | 73.45 | 42.70 | 44.35 | 29.84 | 40.18 | 28.71 | |

1 | 1 | 38.17 | 20.07 | 21.22 | 21.97 | 21.78 | 23.63 |

3 | 61.96 | 27.67 | 26.03 | 24.19 | 24.18 | 24.10 | |

5 | 73.46 | 42.39 | 44.21 | 29.46 | 39.77 | 28.35 | |

1.5 | 1 | 38.47 | 19.80 | 20.93 | 21.90 | 21.54 | 23.41 |

3 | 61.92 | 27.54 | 25.93 | 24.10 | 24.12 | 24.05 | |

5 | 73.42 | 42.38 | 43.96 | 29.40 | 39.61 | 28.19 | |

2 | 1 | 38.37 | 19.66 | 20.76 | 21.76 | 21.41 | 23.73 |

3 | 62.13 | 27.53 | 25.92 | 24.00 | 23.98 | 24.07 | |

5 | 73.44 | 42.34 | 43.97 | 29.34 | 39.71 | 28.17 |

The simulation results for the case of large sample sizes are shown in

PITSE 90% ARE | WLE 90% ARE | PITSE 70% ARE | WLE 70% ARE | M-scale 70% ARE | |||
---|---|---|---|---|---|---|---|

0.5 | 1 | 29.73 | 14.98 | 15.26 | 15.93 | 15.69 | 17.69 |

3 | 51.20 | 18.50 | 17.30 | 16.95 | 16.34 | 17.86 | |

5 | 63.23 | 27.52 | 27.05 | 20.80 | 23.69 | 20.09 | |

1 | 1 | 29.72 | 15.21 | 15.46 | 16.25 | 15.97 | 18.03 |

3 | 51.16 | 18.64 | 17.29 | 16.98 | 16.38 | 18.23 | |

5 | 63.32 | 27.61 | 27.23 | 21.02 | 23.97 | 20.21 | |

1.5 | 1 | 29.51 | 15.11 | 15.36 | 16.21 | 15.81 | 17.82 |

3 | 51.23 | 18.71 | 17.43 | 17.05 | 16.46 | 18.12 | |

5 | 63.30 | 27.72 | 27.26 | 21.05 | 23.95 | 20.28 | |

2 | 1 | 29.92 | 15.11 | 15.26 | 15.90 | 15.67 | 17.84 |

3 | 50.97 | 18.63 | 17.20 | 16.90 | 16.30 | 18.05 | |

5 | 63.28 | 27.01 | 26.95 | 20.90 | 23.60 | 20.14 |

MLE | PITSE 90% ARE | WLE 90% ARE | PITSE 70% ARE | WLE 70% ARE | M-scale 70% ARE | ||
---|---|---|---|---|---|---|---|

0.5 | 1 | 24.49 | 12.60 | 12.62 | 13.26 | 12.92 | 14.92 |

3 | 43.73 | 14.67 | 13.75 | 13.93 | 13.24 | 15.14 | |

5 | 55.72 | 20.86 | 19.27 | 16.83 | 17.07 | 16.53 | |

1 | 1 | 24.20 | 12.74 | 12.79 | 13.42 | 13.12 | 15.02 |

3 | 43.69 | 14.49 | 13.69 | 13.90 | 13.23 | 15.11 | |

5 | 55.88 | 20.82 | 19.31 | 16.64 | 16.97 | 16.30 | |

1.5 | 1 | 24.39 | 12.83 | 12.84 | 13.48 | 13.21 | 15.21 |

3 | 43.72 | 14.63 | 13.90 | 14.02 | 13.41 | 15.33 | |

5 | 55.72 | 20.85 | 19.21 | 16.80 | 17.07 | 16.50 | |

2 | 1 | 24.47 | 12.85 | 12.86 | 13.51 | 13.18 | 15.20 |

3 | 43.71 | 14.67 | 13.83 | 14.00 | 13.35 | 15.28 | |

5 | 55.88 | 20.96 | 19.45 | 16.76 | 17.23 | 16.49 |

MLE | PITSE 90% ARE | WLE 90% ARE | PITSE 70% ARE | WLE 70% ARE | M-scale 70% ARE | ||
---|---|---|---|---|---|---|---|

0.5 | 1 | 24.49 | 12.60 | 12.62 | 13.26 | 12.92 | 14.92 |

5 | 43.73 | 14.67 | 13.75 | 13.93 | 13.24 | 15.14 | |

10 | 55.72 | 20.86 | 19.27 | 16.83 | 17.07 | 16.53 | |

1 | 1 | 24.20 | 12.74 | 12.79 | 13.42 | 13.12 | 15.02 |

5 | 43.69 | 14.49 | 13.69 | 13.90 | 13.23 | 15.11 | |

10 | 55.88 | 20.82 | 19.31 | 16.64 | 16.97 | 16.30 | |

1.5 | 1 | 24.39 | 12.83 | 12.84 | 13.48 | 13.21 | 15.21 |

5 | 43.72 | 14.63 | 13.90 | 14.02 | 13.41 | 15.33 | |

10 | 55.72 | 20.85 | 19.21 | 16.80 | 17.07 | 16.50 | |

2 | 1 | 24.47 | 12.85 | 12.86 | 13.51 | 13.18 | 15.20 |

5 | 43.71 | 14.67 | 13.83 | 14.00 | 13.35 | 15.28 | |

10 | 55.88 | 20.96 | 19.45 | 16.76 | 17.23 | 16.49 |

MLE | PITSE 90% ARE | WLE 90% ARE | PITSE 70% ARE | WLE 70% ARE | M-scale 70% ARE | ||
---|---|---|---|---|---|---|---|

0.5 | 1 | 17.34 | 6.51 | 6.13 | 6.99 | 6.40 | 7.22 |

5 | 48.46 | 14.22 | 14.33 | 10.37 | 12.16 | 9.64 | |

10 | 65.16 | 26.44 | 36.60 | 16.48 | 32.33 | 15.46 | |

1 | 1 | 17.41 | 6.48 | 6.05 | 7.00 | 6.32 | 7.25 |

5 | 48.25 | 14.17 | 14.26 | 10.34 | 12.07 | 9.63 | |

10 | 65.16 | 26.45 | 36.65 | 16.49 | 32.30 | 15.48 | |

1.5 | 1 | 17.28 | 6.49 | 6.13 | 6.99 | 6.39 | 7.21 |

5 | 48.40 | 14.18 | 14.33 | 10.34 | 12.09 | 9.60 | |

10 | 65.14 | 26.42 | 36.63 | 16.47 | 32.30 | 15.46 | |

2 | 1 | 17.61 | 6.53 | 6.16 | 7.00 | 6.43 | 7.22 |

5 | 48.38 | 14.23 | 14.41 | 10.36 | 12.19 | 9.61 | |

10 | 65.11 | 26.50 | 36.65 | 16.53 | 32.39 | 15.51 |

MLE | PITSE 90% ARE | WLE 90% ARE | PITSE 70% ARE | WLE 70% ARE | M-scale 70% ARE | ||
---|---|---|---|---|---|---|---|

0.5 | 1 | 16.74 | 5.22 | 4.80 | 5.41 | 5.10 | 5.57 |

5 | 48.39 | 13.81 | 15.26 | 9.52 | 12.91 | 8.62 | |

10 | 65.34 | 26.29 | 38.86 | 16.07 | 34.77 | 14.95 | |

1 | 1 | 16.76 | 5.23 | 4.90 | 5.43 | 5.10 | 5.57 |

5 | 48.46 | 13.83 | 15.28 | 9.57 | 12.91 | 8.66 | |

10 | 65.23 | 26.30 | 38.78 | 16.13 | 34.66 | 15.00 | |

1.5 | 1 | 16.87 | 5.26 | 4.93 | 5.45 | 5.13 | 5.61 |

5 | 48.38 | 13.81 | 15.26 | 9.54 | 12.89 | 8.65 | |

10 | 65.24 | 26.30 | 38.80 | 16.12 | 34.71 | 15.01 | |

2 | 1 | 16.87 | 5.24 | 4.89 | 5.43 | 5.11 | 5.56 |

5 | 48.46 | 13.87 | 15.30 | 9.60 | 12.91 | 8.70 | |

10 | 65.29 | 26.35 | 38.80 | 16.19 | 34.67 | 15.09 |

In practice, the suitable ARE of PITSE can be selected based on sample size and number or proportion of outliers. Based on a comprehensive simulation study, the guidelines for selecting the suitable ARE of PITSE in practical application are provided in

Sample size | Number or proportion of outliers | Suitable ARE |
---|---|---|

For both small and large sample sizes | When there are no outliers in the data | 98% ARE |

Small sample size, |
When the number of outliers |
70%–90% AREs |

When the number of outliers |
||

Small sample size, |
When the number of outliers |
80%–90% AREs |

When the number of outliers |
70%–80% AREs | |

When the number of outliers |
||

Large sample size, |
When the number of outliers |
80%–90% AREs |

When the number of outliers |
70%–80% AREs | |

When the number of outliers |
||

Large sample size, |
When proportion of outliers |
80%–90% AREs |

When proportion of outliers |
70%–80% AREs | |

When proportion of outliers |

It should be noted that when there are a large number or proportion of outliers in a data set, PITSE with ARE not less than 60% should be used to make sure that the PITSE has a reasonable efficiency in estimating the rate parameter of exponential distribution.

In this section, three applications of exponential distribution are proposed using three real data sets to compare the performance of MLE, PITSE, WLE, and M-scale estimator. This comparative study considered several ARE levels for PITSE and WLE, while a fixed ARE level of 70% was used for the M-scale estimator. The first data set (Set 1) was obtained from Linhart et al. [

Data | Sample size | Mean | Median | Standard deviation | Skewness | Kurtosis | No. of outliers (proportion) |
---|---|---|---|---|---|---|---|

Set 1 | 30 | 59.60 | 22.00 | 71.88 | 1.61 | 1.64 | 3 (10.00%) |

Set 2 | 126 | 1.73 | 0.70 | 2.65 | 2.65 | 7.22 | 3 (2.38%) |

Set 3 | 213 | 93.14 | 57 | 106.76 | 2.10 | 4.85 | 4 (1.88%) |

Data | Method | Estimated parameter ( |
KS statistic | |
---|---|---|---|---|

Set 1 | MLE | 0.01678 | 0.2132 | 0.1309 |

PITSE (75% ARE) | 0.02066 | 0.1551 | 0.4665 | |

PITSE (70% ARE) | 0.02127 | |||

WLE (75% ARE) | 0.01899 | 0.1794 | 0.2889 | |

WLE (70% ARE) | 0.01899 | 0.1794 | 0.2889 | |

M-scale (70% ARE) | 0.02362 | |||

Set 2 | MLE | 0.57944 | 0.1949 | 0.0001 |

PITSE (85% ARE) | 0.81268 | |||

PITSE (80% ARE) | 0.85963 | 0.1091 | 0.0999 | |

WLE (85% ARE) | 0.77143 | 0.1115 | 0.0874 | |

WLE (80% ARE) | 0.77143 | 0.1115 | 0.0874 | |

M-scale (70% ARE) | 0.97727 | 0.1322 | 0.0244 | |

Set 3 | MLE | 0.01074 | 0.0726 | 0.2113 |

PITSE (90% ARE) | 0.01143 | |||

PITSE (85% ARE) | 0.01163 | 0.0566 | 0.5035 | |

WLE (90% ARE) | 0.01074 | 0.0726 | 0.2113 | |

WLE (85% ARE) | 0.01074 | 0.0726 | 0.2113 | |

M-scale (70% ARE) | 0.01252 | 0.0746 | 0.1865 |

Note: The best method is written in bold.

To identify the presence of outliers in all data sets, the generalized boxplot method [

To compare the performance of all considered methods in estimating the parameter of exponential distribution, the Kolmogorov–Smirnov (KS) tests were employed as a goodness-of-fit assessment. The best method was determined by choosing the smallest values of KS statistics as well as the highest

Since the measures of reliability based on the exponential distribution depend on the parameter

In this study, a robust and efficient estimator for the parameter of exponential distribution called PITSE has been introduced based on probability integral transform statistic. The asymptotic variance, BP, and GES were derived to study the PITSE properties. The advantage of PITSE is that it is conceptually simple and easy to compute. According to the simulation study, PITSE performed better than MLE and was comparable with WLE and M-scale estimator when outliers are present in the data set. However, in the case of a high degree of contamination, the performance of WLE was worse than PITSE and M-scale estimator. On the other hand, the M-scale estimator only has the maximum ARE of about 71%, which makes it unsuitable for estimating the parameter of exponential distribution in the case of a small degree of contamination. Therefore, the PITSE introduced in this study was considered a viable alternative for estimating the parameter of exponential distribution in the presence of outliers. Finally, the application on three real data sets showed that the PITSE provided desirable protection against outliers. The R commands for PITSE are available in Appendix B.

Although the PITSE proposed in this study was considered viable for estimating the exponential parameter, there existed a limitation regarding the scope of the current work. It was the reliability modeling under exponential model assumptions that was limited to certain cases (constant failure rate) since the exponential distribution is a simple model consisting of one parameter. There are two-parameter distributions such as Weibull that could provide a better fit to the reliability data, hence providing a better reliability estimation. Therefore, for future work, the robust and efficient estimator for the Weibull parameters can be developed based on the probability integral transform statistical approach. From there, it is believed that a better reliability estimation can be obtained particularly for the case when the outliers are present in the data set.

The authors would like to thank the editor for the time spent reviewing this manuscript.

Set 1:

23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95

Set 2:

0.01, 0.01, 0.01, 0.01, 0.01, 0.02, 0.02, 0.02, 0.02, 0.03, 0.04, 0.06, 0.08, 0.1, 0.1, 0.12, 0.12, 0.12, 0.13, 0.14, 0.15, 0.15, 0.15, 0.16, 0.16, 0.17, 0.18, 0.18, 0.19, 0.2, 0.21, 0.22, 0.23, 0.25, 0.26, 0.28, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.39, 0.41, 0.41, 0.42, 0.43, 0.44, 0.44, 0.45, 0.45, 0.5, 0.53, 0.56, 0.58, 0.58, 0.61, 0.62, 0.62, 0.62, 0.64, 0.66, 0.7, 0.7, 0.7, 0.72, 0.77, 0.78, 0.78, 0.8, 0.82, 0.83, 0.85, 0.86, 0.96, 0.97, 0.98, 0.99, 1.05, 1.06, 1.07, 1.18, 1.35, 1.36, 1.42, 1.55, 1.59, 1.65, 1.73, 1.77, 1.79, 1.8, 1.91, 2.09, 2.14, 2.15, 2.15, 2.31, 2.33, 2.36, 2.43, 2.45, 2.5, 2.51, 2.58, 2.64, 2.68, 3.08, 3.94, 4.12, 4.33, 4.42, 4.53, 4.88, 4.97, 5.11, 5.32, 5.55, 6.63, 6.89, 7.62, 11.41, 11.76, 11.85, 12.36, 13.22

Set 3:

194, 15, 41, 29, 33, 181, 413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118, 34, 31, 18, 18, 67, 57, 62, 7, 22, 34, 90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29, 118, 25, 156, 310, 76, 26, 44, 23, 62, 130, 208, 70, 101, 208, 74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27, 153, 26, 326, 55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33, 15, 104, 35, 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95, 97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82, 54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24, 50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36, 22, 139, 210, 97, 30, 23, 13, 14, 359, 9, 12, 270, 603, 3, 104, 2, 438, 50, 254, 5, 283, 35, 12, 130, 493, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130, 230, 66, 61, 34, 487, 18, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 102, 209

### PITSE for the rate parameter ###

f < -function(data,l,k){

n < -length(data)

fx < -(sum(exp( − l * k * data))/n) − (1/(k + 1))

return(fx)

}

#solve using secant method

# k – tuning parameter

# l1 – 1

pitse < -function(data,k, l1, l2, num = 1000, eps = 1e-05, eps1 = 1e-05)

{

i = 0

while ((abs(l1 - l2) > eps) && (i < num)) {

c = l2 − f(data,l2,k) * (l2 − l1)/(f(data,l2,k) − f(data,l1,k))

l1 = l2

l2 = c

i = i + 1

}

rate < -l2

return(rate)

}