Statistical inferences and applications of the half exponential power distribution

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1 Statistical inferences and alications of the half exonential ower distribution Wenhao Gui Deartment of Mathematics and Statistics, University of Minnesota Duluth, Duluth MN 558, USA Abstract In this aer, we investigate the statistical inferences and alications of the half exonential ower distribution for the first time. The roosed model defined on the non-negative reals extends the half normal distribution and is more flexible. The characterization and roerties involving moments and some measures based on moments of this distribution are derived. The inference asects using methods of moment and maximum likelihood are resented. We also study the erformance of the estimators using Monte Carlo simulation. Finally, we illustrate it with two real alications. Keywords: half exonential ower distribution, moment estimation, maximum likelihood, monte carlo Introduction The well known exonential ower(ep) distribution or the generalized normal distribution has the following density function f(x) = x Γ(, < x < () )e Where > 0 is the shae arameter. This family consists of a wide range of symmetric distributions and allows continuous variation from normality to nonnormality. It includes the normal distribution Z N(0, ) as the secial case when = and the Lalace distribution when =. Nadarajah Nadarajah (005) rovided a comrehensive treatment of its mathematical roerties. Its tails can be more latykurtic ( > ) or more letokurtic ( < ) than the normal distribution ( = ). The distribution has been widely used in Bayes analysis and robustness studies. See Box and Tiao (96), Genc (007), Goodman and Kotz (973) and Tiao and Lund (970). On the other hand, since the most oular models used to describe the life time rocess are defined on non-negative measurements, which motivates us to take a ositive truncation wgui@d.umn.edu

2 in the model () and develo a half exonential ower distribution (HEP). As far as we know, this model has not been reviously studied although, we believe, it lays an imortant role in data analysis. The resulting non-negative half exonential ower distribution generalizes the half normal (HN) distribution and it is more flexible. In our work, we aim to investigate the statistical features of the non-negative model and aly it to fit the lifetime data. The rest of this aer is organized as follows: in Section, we resent the new distribution and study its roerties. Section 3 discusses the inference, moments and maximum likelihood estimation for the arameters. In Section 4, we discuss a useful technique, a half normal lot with a simulated enveloe, to assess the model adequacy. Simulation studies are erformed in Section 5. Section 6 gives two illustrative examles and reorts the results. Section 7 concludes our work. The half exonential ower distribution. The density and hazard function Definition.. A random variable X has a half exonential ower slash distribution if its density function with scale arameter σ > 0 takes f(x) = x σγ( σ, x 0 () )e where σ > 0 and > 0. We denote it as X HEP (σ, ). Density: f(x) = 0.5 = = = 8 h(x) = 0.5 = = = x x (a) Density function (b) Hazard function Figure : The density and hazard rate functions of HEP (σ, ) for σ = Figure a dislays some lots of the density function of the half exonential ower distribution with various arameters.

3 The cumulative distribution function of the half exonential ower distribution X HEP (σ, ) is given as follows. For x 0, F (x) = x 0 f X (u)du = x 0 u σγ( σ du = γ(, x )e Γ( ) (3) where γ(, ) is the lower incomlete gamma function, defined as γ(s, x) = x 0 ts e t dt. The hazard rate function (also known failure rate function) of the half exonential ower distribution is given by, for x 0 σ ) h(x) = e x σ f(x) F (x) = [ σ Γ( ) γ(, x σ ) ] (4) Since Γ(s) γ(s, x) x s e x, as x, we obtain h(x) x. Therefore, the hazard σ rate function is increasing for and decreasing for 0 < <. Figure b dislays some lots of the hazard rate function of the half exonential ower distribution with various arameters.. Moments and Measures based on moments Proosition.. Let X HEP (σ, ), for k =,, 3,..., the k th non-central moments are given by µ k = EX k = k σ k Γ( ) Γ(k + ) (5) The following results are immediate consequences of (5). Corollary.3. Let X HEP (σ, ). The mean and variance of X are given by EX = σ Γ( )Γ( ), and (6) ] σ [Γ( )Γ( 3 ) [Γ( )] Var(X) = [Γ( (7) )] Corollary.4. Let X HEP (σ, ). The skewness and kurtosis coefficients of X are given by β = [Γ( )]3 3Γ( )Γ( )Γ( 3 ) + [Γ( )] Γ( 4 ) (Γ( )Γ( 3 ) [Γ( )] ) 3/ (8) β = 3[Γ( )]4 + 6Γ( )[Γ( )] Γ( 3 ) 4[Γ( )] Γ( )Γ( 4 ) + [Γ( )]3 Γ( 5 ) (Γ( )Γ( 3 ) [Γ( )] ) (9) Figure shows the skewness and kurtosis coefficients with various arameters for the HEP model. 3

4 Skewness β log( β ) (a) Skewness coefficient (b) Skewness coefficient in log scale Kurtosis β log(β ) (c) Kurtosis coefficient (d) Kurtosis coefficient in log scale Figure : The lot for the skewness and kurtosis coefficients with various arameters 3 Inference 3. Moment estimation Let X, X,..., X n is a random samle from the distribution HEP (σ, ). From (5), we have σ EX = σ Γ( )Γ( ) and EX = Γ( 3). Relacing EX and Γ( ) EX by the corresonding samle 4

5 estimators, we obtain the moment equations. X = n X = n n i= n i= The estimate ˆ is the solution to the equation which can be solved numerically. and the estimate ˆσ is given by X i = σ Γ( )Γ( ) (0) X i = σ Γ( ) Γ(3 ) () Γ( )Γ( 3 ) [Γ( )] = X X () ˆσ = XΓ( ˆ ) ˆ ˆ Γ( ˆ ) (3) It is clear that, for the secial case when is known, estimator ˆσ is unbiased and its mean squared error (MSE) is given by MSE(ˆσ) = σ [Γ( )Γ( 3 ) [Γ( )] ] n[γ( )] (4) In the following roosition, we resent the asymtotic roerty of the moment estimators. Proosition 3.. Let X, X,..., X n is a random samle of size n from the distribution HEP (σ, ), and let θ = (σ, ), then, if µ 6 = EX 6 < and ˆθ is the moment estimator of θ, we have n(ˆθ θ) d N (0, H Σ[H ] T ) (5) as n. where Σ = ({µ i+j µ i µ j } ij ) and H is given by ) whose entries are given by H = H(θ) = ( µ σ µ σ µ σ = Γ( Γ( ) ) µ µ (6) + µ σγ( = )[ + log ψ( ) + ψ( )] Γ( ) µ σ = / σγ( 3 ) Γ( ) + µ s = Γ( 3)[ + log ψ( ) + 3ψ( 3)] Γ( ) 5

6 where ψ() is the digamma function defined as the logarithmic derivative of the gamma function, ψ(x) = d log Γ(x) = Γ (x). dx Γ(x) Remark 3.. A consistent estimator for the asymtotic covariance matrix H Σ[H ] T can be obtained by relacing arameters by their corresonding moment estimators. 3. Maximum likelihood estimation In this section, we consider the maximum likelihood estimation about the arameter θ = (σ, ) of the HEP model defined in (). The log-likelihood for a random samle x, x,..., x n is l(θ) = log n i= f(x i ) = n( ) log n log σ n log Γ( ) σ n x i (7) By taking the artial derivatives of the log-likelihood function with resect to σ, resectively and equalizing the obtained exressions to zero, the following maximum likelihood estimating equations are obtained l σ = n σ + σ + l = n x i = 0 (8) i= n(log + ) + nψ( ) + + log σ σ n i= x i σ i= n x i log x i = 0 (9) In general, there are no exlicit solutions for the maximum likelihood estimating equations above. The estimates can be obtained by means of numerical rocedures such as Newton- Rahson method. The rogram R rovides the nonlinear otimization routine otim for solving such roblems. For asymtotic inference of θ = (σ, ), we need the Fisher information matrix I(θ). It is known that its inverse is the asymtotic variance matrix of the maximum likelihood estimators. For the case of a single observation (n = ), we take the second order derivatives of the log-likelihood function in (7). i= l σσ = σ + x σ+ l σ = σ + x (log x log σ) l = 4 σ [ 3σ + σ + x + σ log + x log σ + 3 x [log σ] x log x 3 x log σ log x + 3 x [log x] + σ ψ( ) + σ ψ ( ) ] 6

7 Using the facts Ex = σ E(x log x) = σ [ log σ + log + ψ( + )] E(x [log x] ) = σ [( log σ + log + ψ( + )) + ψ ( + )] we can obtain the elements of the Fisher information matrix I = El σσ = σ (0) I = El σ = log + ψ( + ) σ I = El σ = log + ψ( + ) σ () () I = El = + [log + ψ( + )] + ψ ( + ) + ψ ( ) 4 (3) Proosition 3.3. Let X, X,..., X n is a random samle of size n from the distribution HEP (σ, ), and let θ = (σ, ) and ˆθ is the maximum likelihood estimator of θ, we have n(ˆθ θ) d N (0, I(θ) ) (4) 4 Assessment of model adequacy In this section, we introduce a useful tool, a half normal lot with a simulated enveloe which will be used to evaluate the HEP model in Section 6. The advantage of this technique is its ease of interretation without knowing the distribution of the residuals. Atkinson (985) roosed this diagnostic lot to detect otential outliers and influential observations in linear regression models. A simulated enveloe is added to the lot to aid overall assessment, whereby the observed residuals are exected to lie within the boundary of the enveloe if the resumed model has been correctly secified. The method of simulated enveloe and its corresonding transformations have been widely alied in many alications, see Flack and Flores (989), Ferrari and Cribari-Neto (004) and da Silva Ferreira et al. (0) etc. The simulated enveloe technique comares the observed statistics with those of the data generated from the roosed model. Any sizeble dearture of the observed residuals from the simulated quantities may be thought as evidence against the adequacy of the roosed model. Here is the rocedure to roduce the half normal lot with simulated enveloes.. Fit the model to the observed data(samle size=n).. Generate a samle of n observations based on the fitted model. 7

8 3. Fit the model to the generated samle above and comute the ordered absolute values of the standard residuals. 4. Reeat the above stes k times. 5. Consider the n sets of the k ordered statistics; calculate the average, minimum and maximum values across each set. 6. Plot these values together with the ordered residuals from the original data against the half normal scores Φ ((i + n /8)/(n + /)). The minimum and maximum values of the k ordered statistics constitute a simulated enveloe to guide assessment of the model adequacy. Atkinson (985) suggested using k = 9 since there is a 5% chance to detect the largest residual being outside the boundary of the simulated enveloe. Moreover, other tyes of residuals such as deviance or score residual may be used in the rocedure. For examle, da Silva Ferreira et al. (0) used the Mahalanobis distance to assess their models. The horizontal axis can also show other variables such as index etc. 5 Simulation study In this section, we conduct some simulations and study the roerties of the estimators numerically. We erform a simulation to illustrate the behaviors of the moment and MLE estimators for arameters θ = (σ, ), resectively. The simulation is conducted by the software R. We generate 000 samles of size n = 00, n = 50 and n = 00 from the HEP (σ, ) distribution for fixed arameters σ and. The random numbers can be generated as follows: We first generate random numbers Y from an exonential ower distribution with µ = 0, σ and, the rocedures can be found in Chiodi (986); then we take the absolute value of the random numbers, X = Y. It follows that X HEP (σ, ). The estimators are comuted using the results in Section 3. The emirical means and standard deviations of the estimators are resented in Table and Table resectively. The simulation studies show that the arameters are well estimated and the estimates are asymtotically unbiased. The emirical MSEs decrease as samle size increases as exected. Further, MLE s are more efficient than moment estimators. 6 Real Data Illustration In this section, we analyze two real dataset to fit with the roosed model. The alications demonstrate that the HEP model fits the data better than the HN model. 8

9 Table : Emirical means and SD for the moment estimators of σ and n = 00 n = 50 n = 00 σ ˆσ(SD) ˆ(SD) ˆσ(SD) ˆ(SD) ˆσ(SD) ˆ(SD).06 (0.74).0643 (0.949).0099 (0.077).0450 (0.675).0084 (0.0935).0380 (0.46).0046 (0.04).0544 (0.3443) (0.086).0369 (0.367).0034 (0.0745).0484 (0.869) (0.0844) (0.433) (0.074) (0.4089).0044 (0.0640) (0.3970).0365 (0.499).0660 (0.959).0390 (0.099).0559 (0.635).033 (0.87).0443 (0.505).0090 (0.983).076 (0.3453).0 (0.70).054 (0.37).004 (0.44).037 (0.84) (0.660) (0.4338).003 (0.39) (0.4054).06 (0.75) (0.3974) Table : Emirical means and SD for the MLE estimators of σ and n = 00 n = 50 n = 00 σ ˆσ(SD) ˆ(SD) ˆσ(SD) ˆ(SD) ˆσ(SD) ˆ(SD).09 (0.7).055 (0.055).034 (0.079).0397 (0.695).006 (0.0890).070 (0.40).053 (0.06).08 (0.668).0048 (0.0883).0995 (0.440).0063 (0.0770).0876 (0.3644) (0.0) (.364).0099 (0.086) (0.774).0068 (0.0736) 3.54 (0.6405).00 (0.63).0566 (0.07).0309 (0.78).0409 (0.697).053 (0.766).04 (0.37).050 (0.66).944 (0.64).036 (0.798).94 (0.4469).003 (0.53).0695 (0.3449) (0.35) (.456).04 (0.68) (0.86).08 (0.43) 3.9 (0.7) 6. Alication The data are the lasma ferritin concentration measurements of 0 athletes collected at the Australian Institute of Sort. This data set has been studied by several authors, see Azzalini and Dalla Valle (996), Cook and Weisberc (994) and Elal-Olivero et al. (009). The descritive statistics for the data set are shown in Table 3. where b and b are the samle skewness and kurtosis coefficients. Notice that the data set resent non-negative measurements. Table 3: Summary for the lasma ferritin concentration measurements samle size mean standard deviation b b We fit the dataset with the half normal and the half exonential ower distribution, resectively, using maximum likelihood method. The MLE estimators are comuted using R and the results are reorted in Table 4. The usual Akaike information criterion (AIC) and Bayesian information criterion (BIC) to measure of the goodness of fit are also comuted. AIC = k log L and BIC = k log n log L. where k is the number of arameters in the distribution and L is the maximized value of the likelihood function. The results indicate that HEP model has the lower values for the AIC and BIC statistics and thus it is a better model. Figure 3a and 3b dislay the fitted models using the MLE estimates. The diagnostic rocedure introduced in Section 4 is imlemented for both models. The simulated enveloe lots are shown in Figure 4a and Figure 4b. Most of the observed residuals are either near or outside the boundary of the enveloe, indicating inadequacy of the fitted HN model. On the other hand, the observed residuals corresonding to the HEP model in Figure 4b are well within the simulated enveloe, indicating that the HEP model rovides a better fit to the data. 9

10 Table 4: Maximum likelihood arameter estimates(with (SD)) of the HN and HEP models for the lasma ferritin concentration data Model ˆσ ˆ loglik AIC BIC HN (3.0588) HEP (6.496) (0.338) Density HEP HN Distribution function ECDF HEP HN lasma ferritin concentration lasma ferritin concentration (a) Histogram and fitted curves (b) Emirical and fitted CDF Figure 3: Models fitted for the lasma ferr concentration data set standard residuals 0 3 standard residuals scores (a) Half normal scores (b) Half exonential ower 6. Alication Figure 4: Simulated enveloes for on HN and HEP models We consider the stress-ruture data set, the life of fatigue fracture of Kevlar 49/eoxy that are subject to the ressure at the 90% level. The data set has been reviously studied by 0

11 Andrews and Herzberg (985), Barlow et al. (988) and Olmos et al. (0). Table 5 summarizes the data set. This data set also shows non-negative asymmetry. Same as before, we fit the data set with the half normal and the half exonential ower distribution, resectively, using maximum likelihood method. The results are reorted in Table 6. The AIC and BIC are resented as well and the results show that HEP model fits better. Figure 5a and 5b dislay the fitted models using the MLE estimates. Table 5: Summary for the life of fatigue fracture samle size mean standard deviation b b The diagnostic rocedure introduced in Section 4 is imlemented for both models. The simulated enveloe lots are shown in Figure 6a and Figure 6b. The observed residuals corresonding to the HEP model in Figure 6b are well within the simulated enveloe, indicating that the HEP model rovides a better fit to the data. Table 6: Maximum likelihood arameter estimates(with (SD)) of the HN and HEP models for the life of fatigue fracture data Model ˆσ ˆ loglik AIC BIC HN (0.064) HEP (0.98) (0.677) Density HEP HN Distribution function ECDF HEP HN life of fatigue fracture life of fatigue fracture (a) Histogram and fitted curves (b) Emirical and fitted CDF Figure 5: Models fitted for the life of fatigue fracture data set

12 standard residuals standard residuals scores (a) Half normal scores (b) Half exonential ower Figure 6: Simulated enveloes for on HN and HEP models 7 Concluding Remarks In this article, we have studied the half exonential ower distribution HEP (σ, ) in detail. This non-negative distribution contains the half normal distribution as its secial case. Probabilistic and inferential roerties are studied. A simulation is conducted and demonstrates the good erformance of the moment and maximum likelihood estimators. We aly the model to two real datasets, illustrating that the roosed model is aroriate and flexible in real alications. There are a number of ossible extensions of the current work. Mixture modeling using the roosed distributions is the most natural extension. Other extensions of the current work include a generalization of the distribution to multivariate settings. A Aendix A. Proofs of Proositions () Proof of Proosition.. Proof. EX k = 0 x k σγ( )e x σ dx = σγ( ) = σγ( k+ Γ( k + )σk+ ) = k σ k Γ( ) Γ(k + ) 0 x k e x σ dx

13 () Proof of Proosition 3.. Proof. This result follows directly by using standard large samle theory for moment estimators, as discussed in Sen and Singer Sen and Singer (993). (3) Proof of Proosition 3.3. Proof. It follows directly by the large samle theory for maximum likelihood estimators and the Fisher information matrix given above. References Andrews, D. and Herzberg, A. (985). Data: a collection of roblems from many fields for the student and research worker, volume 8. Sringer-Verlag New York. Atkinson, A. (985). Plots, transformations, and regression: an introduction to grahical methods of diagnostic regression analysis. Clarendon Press Oxford. Azzalini, A. (985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, ages Azzalini, A. and Dalla Valle, A. (996). Biometrika, 83(4): The multivariate skew-normal distribution. Barlow, R., Toland, R., and Freeman, T. (988). A bayesian analysis of the stress-ruture life of kevlar/eoxy sherical ressure vessels. Accelerated Life Testing and Exerts Oinions in Reliability. Box, G. and Tiao, G. (96). A further look at robustness via bayes s theorem. Biometrika, 49(3/4): Castillo, N., Sanhueza, A., et al. (0). On the fern ndez-steel distribution: Inference and alication. Comutational Statistics & Data Analysis. Chiodi, M. (986). Procedures for generating seudo-random numbers from a normal distribution of order ( > ). Statistica Alicata, :7 6. Cook, R. and Weisberc, S. (994). An introduction to regression grahic? Methods, 7:640. da Silva Ferreira, C., Bolfarine, H., and Lachos, V. (0). Skew scale mixtures of normal distributions: Proerties and estimation. Statistical Methodology, 8():54 7. Elal-Olivero, D. (00). Alha-skew-normal distribution. Proyecciones (Antofagasta), 9(3):

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