Exact Linear Likelihood Inference for Laplace

Exact Linear Likelihood Inference for Laplace Prof. N. Balakrishnan McMaster University, Hamilton, Canada bala@mcmaster.ca p. 1/52

Pierre-Simon Laplace 1749 1827 p. 2/52

Laplace s Biography Born: On March 23, 1749, in Normandy, France p. 3/52

Laplace s Biography Born: On March 23, 1749, in Normandy, France Died: On March 5, 1827, in Paris, France p. 3/52

Laplace s Biography Born: On March 23, 1749, in Normandy, France Died: On March 5, 1827, in Paris, France Nationality: French p. 3/52

Laplace s Biography Born: On March 23, 1749, in Normandy, France Died: On March 5, 1827, in Paris, France Nationality: French Alma mater: University of Caen, France p. 3/52

Laplace s Biography Born: On March 23, 1749, in Normandy, France Died: On March 5, 1827, in Paris, France Nationality: French Alma mater: University of Caen, France Advisors: Jean d Alembert, Christophe Gadbled p. 3/52

In collaboration with Colleen Cutler (Univ. of Waterloo) G. Iliopoulos (Univ. of Piraeus) Aaron Childs (McMaster Univ.) X. Zhu (Xian-Jiaotong Liverpool Univ) R. Ambagaspitiya (Univ. of Calgary) Kai Liu (McMaster Univ.) p. 4/52

Goals 1. To provide historical details on Laplace distribution; p. 5/52

Goals 1. To provide historical details on Laplace distribution; 2. To describe results on order statistics; p. 5/52

Goals 1. To provide historical details on Laplace distribution; 2. To describe results on order statistics; 3. To present results for Type-II censoring; p. 5/52

Goals 1. To provide historical details on Laplace distribution; 2. To describe results on order statistics; 3. To present results for Type-II censoring; 4. To present results on outlier-models and robustness; 5. To present results for Type-I censoring; p. 5/52

Historical Details! p. 6/52

Distribution The Laplace distribution has its pdf as f(x) = 1 2σ e x µ /σ, < x <, where µ and σ are the location and scale parameters. p. 7/52

Distribution The Laplace distribution has its pdf as f(x) = 1 2σ e x µ /σ, < x <, where µ and σ are the location and scale parameters. It is also known as Double Exponential Distribution. The distribution is symmetric about µ. p. 7/52

Distribution The quantile function is µ+σln(2u), 0 < u 1 2 Q(u) = 1 µ σln{2(1 u)}, < u < 1. 2 p. 8/52

Distribution The quantile function is µ+σln(2u), 0 < u 1 2 Q(u) = 1 µ σln{2(1 u)}, < u < 1. 2 Chapter 24 of Johnson, Kotz and Balakrishnan (1995) and the book of Kotz, Kozubowski and Podgórski (2001) both provide detailed overviews of various developments on theory, methods and applications of the Laplace distribution and its generalizations. p. 8/52

ML Estimation Let x 1,,x n be a random sample from the Laplace distribution. p. 9/52

ML Estimation Let x 1,,x n be a random sample from the Laplace distribution. Then, the likelihood function is { n L = f(x i ) = 1 2 n σ exp 1 n σ i=1 n i=1 x i µ }. p. 9/52

ML Estimation Let x 1,,x n be a random sample from the Laplace distribution. Then, the likelihood function is { n L = f(x i ) = 1 2 n σ exp 1 n σ i=1 n i=1 x i µ }. Evidently, L is maximized with respect to µ when n i=1 x i µ is minimized. This happens when µ is the central most value (if n is odd), or any value in the interval of the two middle most values (if n is even). p. 9/52

ML Estimation Thus, the MLE of µ is taken to be the sample median x m+1:n, n = 2m+1 ˆµ = 1 (x 2 m:n +x m+1:n ), n = 2m. p. 10/52

ML Estimation Thus, the MLE of µ is taken to be the sample median x m+1:n, n = 2m+1 ˆµ = 1 (x 2 m:n +x m+1:n ), n = 2m. Next, upon maximizing L with respect to σ, we obtain the MLE of σ to be ˆσ = 1 n n i=1 x i Median. Note that the MLEs in this case naturally turn out to be linear estimators, meaning they are linear functions of order statistics. p. 10/52

Historical Note In fact, the Laplace distribution was discovered by Laplace (1774) as the distribution form for which the likelihood function is maximized by setting the location parameter µ equal to the median of the observed values of an odd number of i.i.d. observations. Furthermore, Laplace went on by replacing the median by the arithmetic mean as the value maximizing the likelihood function and derived the corresponding distribution function to be the normal distribution; see Stigler (1975). p. 11/52

Here comes censoring! p. 12/52

Symmetric Censoring Suppose we have available only a Type-II symmetrically censored sample of the form x r+1:n < < x n r:n, with the smallest r and the largest r order statistics having been censored. Then, the corresponding likelihood function is L = n! (r!) 2 [F(x r+1:n){1 F(x n r:n )}] r n r i=r+1 f(x i:n ). p. 13/52

Symmetric Censoring 1. Case µ < x r+1:n : In this case, the likelihood is L = [ C 1 σ n 2r 2 e(µ x n r:n)/σ 1 4 e(2µ x r+1:n x n r:n)/σ [e { ] n r i=r+1 x i:n+(n 2r)µ}/σ. ] r p. 14/52

Symmetric Censoring 1. Case µ < x r+1:n : In this case, the likelihood is L = [ C 1 σ n 2r 2 e(µ x n r:n)/σ 1 4 e(2µ x r+1:n x n r:n)/σ [e { ] n r i=r+1 x i:n+(n 2r)µ}/σ. ] r It is easy to show that the second and third terms are both increasing functions of µ for < µ < x r+1:n. p. 14/52

Symmetric Censoring 2. Case µ > x n r:n : In this case, the likelihood is L = C σ n 2r [ 1 2 e(x r+1:n µ)/σ 1 4 e(x r+1:n+x n r:n 2µ)/σ [e { n r i=r+1 x i:n (n 2r)µ}/σ ]. ] r p. 15/52

Symmetric Censoring 2. Case µ > x n r:n : In this case, the likelihood is L = C σ n 2r [ 1 2 e(x r+1:n µ)/σ 1 4 e(x r+1:n+x n r:n 2µ)/σ [e { n r i=r+1 x i:n (n 2r)µ}/σ ]. ] r In this case, it is easy to show that second and the third terms are both decreasing functions of µ for x n r:n < µ <. p. 15/52

Symmetric Censoring 3. Case x r+1: µ x n r:n : In this case, L = C n r:n x r+1:n)/σ e n r σ n 2re r(x i=r+1 xi:n µ /σ. p. 16/52

Symmetric Censoring 3. Case x r+1: µ x n r:n : In this case, L = C n r:n x r+1:n)/σ e n r σ n 2re r(x i=r+1 xi:n µ /σ. From this expression, it is clear that the MLE of µ is once again the sample median given by x m+1:n, n = 2m+1 ˆµ = any value in [x m:n,x m+1:n ], n = 2m. p. 16/52

Symmetric Censoring Next, upon maximizing L with respect to σ, we obtain p. 17/52

Symmetric Censoring Next, upon maximizing L with respect to σ, we obtain [ 2m+1 r 1 ˆσ = x i:2m+1 +rx 2m+1 r:2m+1 2m+1 2r i=m+2 ] m x i:2m+1 rx r+1:2m+1, n = 2m+1; ˆσ = i=r+1 1 2m 2r m i=r+1 [ 2m r i=m+1 x i:2m +rx 2m r:2m x i:2m rx r+1:2m ], n = 2m. p. 17/52

Right Censoring Let us now consider a Type-II right censored sample of the form x 1:n < < x n r:n, with largest r order statistics having been censored. p. 18/52

Right Censoring Let us now consider a Type-II right censored sample of the form x 1:n < < x n r:n, with largest r order statistics having been censored. Then, the corresponding likelihood function is L = n! r! {1 F(x n r:n)} r n r i=1 f(x i:n ). p. 18/52

Right Censoring 1. Case r n : In this case, note that the median of the 2 sample is available. So, the MLE of µ is once again the sample median. p. 19/52

Right Censoring 1. Case r n : In this case, note that the median of the 2 sample is available. So, the MLE of µ is once again the sample median. 2. Case r > n : In this case, the median of the sample 2 is not available. It is easy to show in this case that the likelihood function is increasing over µ in the interval (,x 1:n ) and also increasing in the interval (x 1:n,x n r:n ]. p. 19/52

Right Censoring The likelihood function in this case is { 1 1 2 e(x n r:n µ)/σ L = C σ n r } r e n r i=1 (µ x i:n)/σ. p. 20/52

Right Censoring The likelihood function in this case is { 1 1 2 e(x n r:n µ)/σ L = C σ n r } r e n r i=1 (µ x i:n)/σ. When this function is maximized for µ, we determine the MLE of µ (when r > n 2 ) as ˆµ = x n r:n + ˆσ ( log n/2 ). n r Substituting these expressions of MLEs for µ in L and then maximizing it with respect to σ, we get expressions for the MLE of σ. p. 20/52

Right Censoring For example, in the case when r n, we obtain the 2 MLE of σ as follows: p. 21/52

Right Censoring For example, in the case when r n, we obtain the 2 MLE of σ as follows: ˆσ = ˆσ = 1 n r 1 n r [ n r i=m+2 [ n r i=m+1 x i:n x i:n ] m x i:n +rx n r:n, n = 2m+1; i=1 ] m x i:n +rx n r:n, n = 2m. i=1 Similar expressions for ˆσ can be presented for the case when r > n by using the MLE of µ presented earlier 2 [Balakrishnan and Cutler (1996)]. p. 21/52

How about order statistics and moments? p. 22/52

Order Statistics Let X 1:n < < X n:n be the order statistics from a standard Laplace distribution. p. 23/52

Order Statistics Let X 1:n < < X n:n be the order statistics from a standard Laplace distribution. Let Y 1:n < < Y n:n be the order statistics from the folded distribution with pdf and cdf as p(x) = 2f(x) and P(x) = 2F(x) 1, x > 0. In other words, these order statistics are from the standard exponential distribution. Let (µ (k) i:n,µ i,j:n) and (ν (k) i:n,ν i,j:n) denote the single and product moments of order statistics from Laplace and exponential distributions, respectively. p. 23/52

Order Statistics Then, Govindarajulu (1963) established the following relationships between these two sets of moments: p. 24/52

Order Statistics Then, Govindarajulu (1963) established the following relationships between these two sets of moments: { µ (k) i:n = 1 i 1 2 n r=0 ( ) n ν (k) i r:n r r +( 1)k n r=i ( ) } n ν (k) r i+1:r r ; p. 24/52

Order Statistics Then, Govindarajulu (1963) established the following relationships between these two sets of moments: { µ (k) i:n = 1 i 1 2 n r=0 { µ i,j:n = 1 i 1 2 n ( ) n ν (k) i r:n r r +( 1)k r=0 j 1 r=i n r=i ( ) n ν i r,j r:n r + r r=j } )ν r i+1:r ν j r:n r ( n r n ( ) } n ν (k) r i+1:r r ( ) n ν r j+1,r i+1:r r. ; p. 24/52

Order Statistics A simple and elegant probabilistic proof was given by Balakrishnan, Govindarajulu and Balasubramanian (1993). p. 25/52

Order Statistics A simple and elegant probabilistic proof was given by Balakrishnan, Govindarajulu and Balasubramanian (1993). Let D denote the number of X i s that are 0. p. 25/52

Order Statistics A simple and elegant probabilistic proof was given by Balakrishnan, Govindarajulu and Balasubramanian (1993). Let D denote the number of X i s that are 0. Then, the following properties hold: 1. D Bin ( n, 2) 1 ; 2. Given D = i 1, X i:n < < X n:n are distributed exactly as order statistics from a sample of size n i+1 from exponential distribution; p. 25/52

Order Statistics 3. Also, given D = i 1, X i 1:n < < X 1:n are distributed exactly as order statistics from a sample of size i 1 from exponential distribution; p. 26/52

Order Statistics 3. Also, given D = i 1, X i 1:n < < X 1:n are distributed exactly as order statistics from a sample of size i 1 from exponential distribution; 4. Furthermore, given D = i 1, these two sets of order statistics are mutually independent. Then, the above four properties readily yield the relations presented earlier; in addition, this approach leads to some generalizations of Govindarajulu s results as well. p. 26/52

What about BLUEs? p. 27/52

Bias, MSE & JRE BLUEs of µ and σ were tabulated by Govindarajulu (1966) for Type-II symmetrically censored samples. p. 28/52

Bias, MSE & JRE BLUEs of µ and σ were tabulated by Govindarajulu (1966) for Type-II symmetrically censored samples. The MLEs ˆµ and ˆσ are symmetric and skew-symmetric estimators, respectively. The same is true for the BLUEs µ and σ. So, both pairs of estimators are uncorrelated. While ˆµ, µ and σ are all unbiased, ˆσ is biased. p. 28/52

Bias, MSE & JRE n r V(ˆµ)/σ 2 B(ˆσ)/σ MSE(ˆσ)/σ 2 JRE 5 0 0.3512 0.1354 0.1895 100.96 1 0.2361 0.3117 114.06 10 0 0.1452 0.0551 0.0790 109.77 1 0.0689 0.1127 106.59 2 0.0921 0.1618 106.06 3 0.1488 0.2434 114.44 15 0 0.0963 0.0392 0.0445 111.72 1 0.0452 0.0614 106.79 2 0.0534 0.0799 104.37 3 0.0654 0.1031 103.66 4 0.0853 0.1363 105.12 5 0.1258 0.1935 110.63 6 0.2441 0.3259 131.62 p. 29/52

Bias, MSE & JRE Bias in ˆσ decreases as n increases, for fixed r. p. 30/52

Bias, MSE & JRE Bias in ˆσ decreases as n increases, for fixed r. (ˆµ, ˆσ) is jointly more efficient than (µ, σ ) for small censored sample sizes considered here. p. 30/52

Bias, MSE & JRE Bias in ˆσ decreases as n increases, for fixed r. (ˆµ, ˆσ) is jointly more efficient than (µ, σ ) for small censored sample sizes considered here. Furthermore, unlike the BLUEs µ and σ, the MLEs ˆµ and ˆσ are explicit linear estimators. p. 30/52

Outliers? No problem!! p. 31/52

Outlier-Model Let X 1:n < < X n:n denote the order statistics from a single-outlier Laplace model, with the outlier having a different scale parameter α. p. 32/52

Outlier-Model Let X 1:n < < X n:n denote the order statistics from a single-outlier Laplace model, with the outlier having a different scale parameter α. Let Y 1:n < < Y n:n denote the order statistics from a single-outlier exponential model, with the outlier having the same scale parameter α. Let Z 1:n < < Z n:n denote the order statistics from the standard exponential distribution. p. 32/52

Outlier-Model Then, Balakrishnan (1989) established the following generalized relations between these three sets of moments of order statistics: p. 33/52

Outlier-Model Then, Balakrishnan (1989) established the following generalized relations between these three sets of moments of order statistics: µ (k) i:n = 1 2 n + { i 1 r=1 i 1 r=0 ( n 1 ) r 1 ( n 1 ) r ν (k) i r:n r +( 1)k ν (k) i r:n r +( 1)k n 1 r=i ( n 1 ) r n ( n 1 ) r=i r 1 ν (k) r i+1:r ν (k) r i+1:r } ; p. 33/52

Outlier-Model Then, Balakrishnan (1989) established the following generalized relations between these three sets of moments of order statistics: µ (k) i:n = µ (k) i,j:n = 1 2 n + 1 2 n { i 1 r=1 ( n 1 ) r 1 ν (k) i r:n r +( 1)k n 1 r=i ( n 1 ) i 1 ( n 1 ) n ( + ν (k) n 1 ) i r:n r r +( 1)k r 1 r=0 r=i i 1 ( n 1 n 1 )νi r,j r:n r + ( n 1 ) j 1 r=i i 1 r=0 r=1 r 1 r=j ( n 1 ) νj r:n r r 1 ν r i+1:r ( n 1 ) ν i r,j r:n r + r j 1 r=i r n ( n 1 ) r=j r 1 r ν (k) r i+1:r ν (k) r i+1:r } ν r j+1,r i+1:r ( n 1 ) νr i+1:r r ν j r:n r ν r j+1,r i+1:r ;. p. 33/52

Outlier-Model These results were used by Balakrishnan and Ambagaspitiya (1988) to study the robustness features of various linear estimators of both location and scale parameters of the Laplace distribution with respect to the presence of a scale-outlier in the sample. Of course, these developments can also be extended to the case of multiple outliers. They compared various estimators of location and scale parameters in the presence of a single-outlier with its scale parameter as ασ, and these results are presented next for the case when n = 10. p. 34/52

Outlier-Model 1 σ 2 (Variance of ˆµ), when n = 10 ˆµ 1.0 2.0 4.0 6.0 8.0 10.0 Mean 0.2000 0.2600 0.5000 0.9000 1.4600 2.1800 Median 0.1452 0.1630 0.1751 0.1797 0.1822 0.1837 BLUE(0) 0.1399 0.1581 0.1717 0.1773 0.1805 0.1825 BLUE(1) 0.1399 0.1580 0.1716 0.1771 0.1801 0.1819 BLUE(2) 0.1399 0.1580 0.1714 0.1768 0.1797 0.1815 TrimMean(1) 0.1617 0.1889 0.2191 0.2344 0.2435 0.2495 TrimMean(2) 0.1463 0.1670 0.1845 0.1921 0.1962 0.1989 WinsMean(1) 0.1430 0.1632 0.1806 0.1882 0.1926 0.1953 WinsMean(2) 0.1611 0.1857 0.2086 0.2189 0.2247 0.2284 LinWMean(1) 0.1430 0.1632 0.1806 0.1882 0.1926 0.1953 LinWMean(2) 0.1401 0.1585 0.1724 0.1780 0.1811 0.1830 α GastMean 0.1420 0.1605 0.1743 0.1798 0.1828 0.1847 p. 35/52

Outlier-Model 1 σ 2 (MSE of ˆσ), when n = 10 α ˆσ 1.0 2.0 4.0 6.0 8.0 10.0 BLUE(0) 0.1062 0.1500 0.3735 0.7768 1.3613 2.1266 BLUE(1) 0.1350 0.1342 0.2091 0.2342 0.2498 0.2604 BLUE(2) 0.1857 0.1804 0.2520 0.2682 0.2776 0.2836 RSE(0) 0.1064 0.1511 0.3823 0.8025 1.4118 2.2102 RSE(1) 0.1378 0.1675 0.2097 0.2332 0.2478 0.2576 RSE(2) 0.1908 0.2250 0.2625 0.2807 0.2912 0.2980 RSE(3) 0.3053 0.3501 0.3887 0.4054 0.4146 0.4204 p. 36/52

Exact Inference for Type-II Right Censoring p. 37/52

Exact Inference 1. Case r n : In this case, the joint MGF of (ˆµ, ˆσ) can 2 be derived as follows [Balakrishnan and Zhu (2016)]: p. 38/52

Exact Inference 1. Case r n : In this case, the joint MGF of (ˆµ, ˆσ) can 2 be derived as follows [Balakrishnan and Zhu (2016)]: E ( e t 1ˆµ+t 2ˆσ ) = r 1 j=0 + r 1 j n r l=0 l=0 p 1 e t 1µ (1 s 2 σ) j (1+s 2 σ) (r 1 j) { 1 (s 1 s 2 l)σ r(n r +1+l) } 1 p 2 e t 1µ (1 s 2 σ) (r 1) ( 1+ t 1σ l+r ) 1, p. 38/52

Exact Inference p 1 p 2 = ( 1) l c jr 2 n r 1 j (n r +1+l) 1, l = ( 1) l c jr 2 r l n r (l+r) 1, l c jr = c jr = s 1 = n! j!(r 1 j)!(n r)!, n! (r 1)!(n r)!, ( r 1 ln n ) r 2r +1 t 1 + r 1 r t 2, s 2 = 1 r ln n 2r t 1 + 1 r t 2. p. 39/52

Exact Inference p 1 p 2 = ( 1) l c jr 2 n r 1 j (n r +1+l) 1, l = ( 1) l c jr 2 r l n r (l+r) 1, l c jr = c jr = s 1 = n! j!(r 1 j)!(n r)!, n! (r 1)!(n r)!, ( r 1 ln n ) r 2r +1 t 1 + r 1 r t 2, s 2 = 1 r ln n 2r t 1 + 1 r t 2. Analogous results for other cases have also been derived by Balakrishnan and Zhu (2016). p. 39/52

Exact Inference From the joint MGF of (ˆµ, ˆσ), the Bias and MSE of the estimators ˆµ and ˆσ can be easily obtained. p. 40/52

Exact Inference From the joint MGF of (ˆµ, ˆσ), the Bias and MSE of the estimators ˆµ and ˆσ can be easily obtained. Also, the correlation between the estimators ˆµ and ˆσ can be obtained. From the joint MGF of (ˆµ, ˆσ), upon setting t 2 = 0 and t 1 = 0, respectively, the marginal MGFs of ˆµ and ˆσ can be deduced. p. 40/52

Exact Inference ˆµ ˆσ r 1 d = µ+ d = + r 1 j=0 + j=0 n r l=0 r 1 j l=0 n r l=0 r 1 j l=0 p 2 [Γ p 1 [Γ p 2 Γ p 1 [Γ ( j, bσ r ( r 1, bσ r ) )+Γ ( r 1 j, bσ r ( )] σ +E, l+r ) ( )] r +b(r l 1)σ +E r(n r+l+1) ( j, σ ) +Γ ( r 1 j, σ ) ( )] (r l 1)σ +E r r r(n r +l+1) ( r 1, σ ) ; r p. 41/52

Exact Inference ˆµ ˆσ r 1 d = µ+ d = + r 1 j=0 j=0 n r l=0 r 1 j l=0 r 1 j l=0 p 2 [Γ p 1 [Γ p 1 [Γ ( j, bσ r ( r 1, bσ r ) )+Γ ( r 1 j, bσ r ( )] σ +E, l+r ) ( )] r +b(r l 1)σ +E r(n r+l+1) ( j, σ ) +Γ ( r 1 j, σ ) ( )] (r l 1)σ +E r r r(n r +l+1) + n r l=0 p 2 Γ ( r 1, σ ) ; r in the above, Γ and E denote Gamma and Exponential random variables, Γ and E denote the negative of these variables, and Γ(0, σ r ), Γ (0, σ r ), E(0), E (0) are all degenerate at the point 0. p. 41/52

Example The data below, given by Mann and Fertig (1973), are the lifetimes of 13 aeroplane components with the last 3 components having been censored: p. 42/52

Example The data below, given by Mann and Fertig (1973), are the lifetimes of 13 aeroplane components with the last 3 components having been censored: 0.22 0.50 0.88 1.00 1.32 1.33 1.54 1.76 2.50 3.00 Here, we analyze these data by assuming a Laplace distribution. We computed the MLEs based on this Type-II censored sample, their mean square errors (MSE) and correlation, and the 95% confidence intervals based on the exact formulae presented earlier. p. 42/52

Example MLEs of µ and σ and their MSEs and correlation coefficient r ˆµ ˆσ MSE(ˆµ) MSE(ˆσ) Corr(ˆµ,ˆσ) 10 1.5400 1.1010 0.1379 0.1182 0.0039 p. 43/52

Example MLEs of µ and σ and their MSEs and correlation coefficient r ˆµ ˆσ MSE(ˆµ) MSE(ˆσ) Corr(ˆµ,ˆσ) 10 1.5400 1.1010 0.1379 0.1182 0.0039 Exact and simulated 95% CIs for µ and σ Exact results Simulated results r µ σ µ σ 10 (0.7828, 2.2972) (0.4827, 1.7936) (0.7843, 2.2920) (0.4837, 1.7893) p. 43/52

How about Type-I Censoring? p. 44/52

Likelihood Function Let x 1:n < x 2:n < < x n:n denote the ordered lifetimes of n units under a life-test. p. 45/52

Likelihood Function Let x 1:n < x 2:n < < x n:n denote the ordered lifetimes of n units under a life-test. Suppose the life-test is terminated at a fixed time T. p. 45/52

Likelihood Function Let x 1:n < x 2:n < < x n:n denote the ordered lifetimes of n units under a life-test. Suppose the life-test is terminated at a fixed time T. Then, the data observed will be (x 1:n < < x D:n ), where D is the random number of failures until T. The corresponding likelihood function is d L = C d f(x i:n ){1 F(T)} n d, x 1:n < < x d:n < T. i=1 p. 45/52

MLEs By maximizing L, the MLEs of (µ,σ) have been derived by Zhu and Balakrishnan (2016) as follows: p. 46/52

MLEs By maximizing L, the MLEs of (µ,σ) have been derived by Zhu and Balakrishnan (2016) as follows: ˆµ = ˆσ = [X m:n,x m+1:n ], n = 2m,d m+1; X m+1:n, n = 2m+1,d m+1; [X m:n,t], n = 2m,d = m; T + ˆσlog( n 2d ), d < n 2 ; [ 1 (n d)t + d d i=m+1 X i:n ] m i=1 X i:n, n = 2m,d m; [ 1 (n d)t + d d i=m+2 X i:n ] m i=1 X i:n, n = 2m+1,d m+1; 1 di=1 (T X d i:n ), d < n 2. p. 46/52

Example Let us again consider the data of Mann and Fertig (1973), but with the life-test terminating at time T = 2.75: p. 47/52

Example Let us again consider the data of Mann and Fertig (1973), but with the life-test terminating at time T = 2.75: 0.22 0.50 0.88 1.00 1.32 1.33 1.54 1.76 2.50 p. 47/52

Example Let us again consider the data of Mann and Fertig (1973), but with the life-test terminating at time T = 2.75: 0.22 0.50 0.88 1.00 1.32 1.33 1.54 1.76 2.50 Let us analyze these data by assuming a Laplace distribution. We computed the MLEs based on this Type-I censored sample, their mean square errors (MSE) and covariance, and the 95% confidence intervals based on the exact results. p. 47/52

Example MLEs of µ and σ and their MSEs and covariance T ˆµ ˆσ MSE(ˆµ) MSE(ˆσ) Cov(ˆµ,ˆσ) 2.75 1.5400 1.1122 0.1407 0.1164 0.0021 p. 48/52

Example MLEs of µ and σ and their MSEs and covariance T ˆµ ˆσ MSE(ˆµ) MSE(ˆσ) Cov(ˆµ,ˆσ) 2.75 1.5400 1.1122 0.1407 0.1164 0.0021 Exact and simulated 90% CIs for µ and σ Exact results Simulated results T µ σ µ σ 2.75 (0.9228, 2.1572) (0.6053, 1.7073) (0.9188, 2.1583) (0.6066, 1.6872) p. 48/52

What else? p. 49/52

Other Work Results for Type-II symmetric censoring developed by Iliopoulos and Balakrishnan (2011) are simpler; p. 50/52

Other Work Results for Type-II symmetric censoring developed by Iliopoulos and Balakrishnan (2011) are simpler; The results on Type-II and Type-I censoring have been generalized to hybrid Type-II and Type-I censoring; From the joint MGF of the MLEs, exact inference has been developed for reliability, quantile and cumulative hazard functions, and likelihood predictors; Exact inference under progressive Type-II censoring based on BLUEs has been developed by Liu, Zhu and Balakrishnan (2018). p. 50/52

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References Kotz, S., Kozubowski, T.J. and Podgórski, K. (2001). The Laplace Distribution and Generalizations, Birkhäuser. Laplace, P.S. (1774). Mémories de mathématique et de physique presentés à l Academie royale des scienes, 6. Liu, K., Zhu, X. and Balakrishnan, N. (2018). Metrika, 81. Mann, N.R. and Fertig, K.W. (1973). Technometrics, 15. Stigler, S.M. (1975). Biometrika, 62. Zhu, X. and Balakrishnan, N. (2016). IEEE Trans. Reliab., 65. p. 52/52